/* * Copyright (c) 2008-2012 Stefan Krah. All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * * 1. Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS "AS IS" AND * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF * SUCH DAMAGE. */ #include "mpdecimal.h" #include #include #include #include #include #include "basearith.h" #include "bits.h" #include "convolute.h" #include "crt.h" #include "memory.h" #include "typearith.h" #include "umodarith.h" #ifdef PPRO #if defined(_MSC_VER) #include #pragma fenv_access(on) #elif !defined(__OpenBSD__) && !defined(__NetBSD__) /* C99 */ #include #pragma STDC FENV_ACCESS ON #endif #endif #if defined(_MSC_VER) #define ALWAYS_INLINE __forceinline #elif defined(LEGACY_COMPILER) #define ALWAYS_INLINE #undef inline #define inline #else #ifdef TEST_COVERAGE #define ALWAYS_INLINE #else #define ALWAYS_INLINE inline __attribute__ ((always_inline)) #endif #endif #define MPD_NEWTONDIV_CUTOFF 1024L #define MPD_NEW_STATIC(name, flags, exp, digits, len) \ mpd_uint_t name##_data[MPD_MINALLOC_MAX]; \ mpd_t name = {flags|MPD_STATIC|MPD_STATIC_DATA, exp, digits, \ len, MPD_MINALLOC_MAX, name##_data} #define MPD_NEW_CONST(name, flags, exp, digits, len, alloc, initval) \ mpd_uint_t name##_data[alloc] = {initval}; \ mpd_t name = {flags|MPD_STATIC|MPD_CONST_DATA, exp, digits, \ len, alloc, name##_data} #define MPD_NEW_SHARED(name, a) \ mpd_t name = {(a->flags&~MPD_DATAFLAGS)|MPD_STATIC|MPD_SHARED_DATA, \ a->exp, a->digits, a->len, a->alloc, a->data} static mpd_uint_t data_one[1] = {1}; static mpd_uint_t data_zero[1] = {0}; static const mpd_t one = {MPD_STATIC|MPD_CONST_DATA, 0, 1, 1, 1, data_one}; static const mpd_t minus_one = {MPD_NEG|MPD_STATIC|MPD_CONST_DATA, 0, 1, 1, 1, data_one}; static const mpd_t zero = {MPD_STATIC|MPD_CONST_DATA, 0, 1, 1, 1, data_zero}; static inline void _mpd_check_exp(mpd_t *dec, const mpd_context_t *ctx, uint32_t *status); static void _settriple(mpd_t *result, uint8_t sign, mpd_uint_t a, mpd_ssize_t exp); static inline mpd_ssize_t _mpd_real_size(mpd_uint_t *data, mpd_ssize_t size); static void _mpd_qadd(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status); static inline void _mpd_qmul(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status); static void _mpd_base_ndivmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b, uint32_t *status); static inline void _mpd_qpow_uint(mpd_t *result, const mpd_t *base, mpd_uint_t exp, uint8_t resultsign, const mpd_context_t *ctx, uint32_t *status); static mpd_uint_t mpd_qsshiftr(mpd_t *result, const mpd_t *a, mpd_ssize_t n); /******************************************************************************/ /* Performance critical inline functions */ /******************************************************************************/ #ifdef CONFIG_64 /* Digits in a word, primarily useful for the most significant word. */ ALWAYS_INLINE int mpd_word_digits(mpd_uint_t word) { if (word < mpd_pow10[9]) { if (word < mpd_pow10[4]) { if (word < mpd_pow10[2]) { return (word < mpd_pow10[1]) ? 1 : 2; } return (word < mpd_pow10[3]) ? 3 : 4; } if (word < mpd_pow10[6]) { return (word < mpd_pow10[5]) ? 5 : 6; } if (word < mpd_pow10[8]) { return (word < mpd_pow10[7]) ? 7 : 8; } return 9; } if (word < mpd_pow10[14]) { if (word < mpd_pow10[11]) { return (word < mpd_pow10[10]) ? 10 : 11; } if (word < mpd_pow10[13]) { return (word < mpd_pow10[12]) ? 12 : 13; } return 14; } if (word < mpd_pow10[18]) { if (word < mpd_pow10[16]) { return (word < mpd_pow10[15]) ? 15 : 16; } return (word < mpd_pow10[17]) ? 17 : 18; } return (word < mpd_pow10[19]) ? 19 : 20; } #else ALWAYS_INLINE int mpd_word_digits(mpd_uint_t word) { if (word < mpd_pow10[4]) { if (word < mpd_pow10[2]) { return (word < mpd_pow10[1]) ? 1 : 2; } return (word < mpd_pow10[3]) ? 3 : 4; } if (word < mpd_pow10[6]) { return (word < mpd_pow10[5]) ? 5 : 6; } if (word < mpd_pow10[8]) { return (word < mpd_pow10[7]) ? 7 : 8; } return (word < mpd_pow10[9]) ? 9 : 10; } #endif /* Adjusted exponent */ ALWAYS_INLINE mpd_ssize_t mpd_adjexp(const mpd_t *dec) { return (dec->exp + dec->digits) - 1; } /* Etiny */ ALWAYS_INLINE mpd_ssize_t mpd_etiny(const mpd_context_t *ctx) { return ctx->emin - (ctx->prec - 1); } /* Etop: used for folding down in IEEE clamping */ ALWAYS_INLINE mpd_ssize_t mpd_etop(const mpd_context_t *ctx) { return ctx->emax - (ctx->prec - 1); } /* Most significant word */ ALWAYS_INLINE mpd_uint_t mpd_msword(const mpd_t *dec) { assert(dec->len > 0); return dec->data[dec->len-1]; } /* Most significant digit of a word */ inline mpd_uint_t mpd_msd(mpd_uint_t word) { int n; n = mpd_word_digits(word); return word / mpd_pow10[n-1]; } /* Least significant digit of a word */ ALWAYS_INLINE mpd_uint_t mpd_lsd(mpd_uint_t word) { return word % 10; } /* Coefficient size needed to store 'digits' */ ALWAYS_INLINE mpd_ssize_t mpd_digits_to_size(mpd_ssize_t digits) { mpd_ssize_t q, r; _mpd_idiv_word(&q, &r, digits, MPD_RDIGITS); return (r == 0) ? q : q+1; } /* Number of digits in the exponent. Not defined for MPD_SSIZE_MIN. */ inline int mpd_exp_digits(mpd_ssize_t exp) { exp = (exp < 0) ? -exp : exp; return mpd_word_digits(exp); } /* Canonical */ ALWAYS_INLINE int mpd_iscanonical(const mpd_t *dec UNUSED) { return 1; } /* Finite */ ALWAYS_INLINE int mpd_isfinite(const mpd_t *dec) { return !(dec->flags & MPD_SPECIAL); } /* Infinite */ ALWAYS_INLINE int mpd_isinfinite(const mpd_t *dec) { return dec->flags & MPD_INF; } /* NaN */ ALWAYS_INLINE int mpd_isnan(const mpd_t *dec) { return dec->flags & (MPD_NAN|MPD_SNAN); } /* Negative */ ALWAYS_INLINE int mpd_isnegative(const mpd_t *dec) { return dec->flags & MPD_NEG; } /* Positive */ ALWAYS_INLINE int mpd_ispositive(const mpd_t *dec) { return !(dec->flags & MPD_NEG); } /* qNaN */ ALWAYS_INLINE int mpd_isqnan(const mpd_t *dec) { return dec->flags & MPD_NAN; } /* Signed */ ALWAYS_INLINE int mpd_issigned(const mpd_t *dec) { return dec->flags & MPD_NEG; } /* sNaN */ ALWAYS_INLINE int mpd_issnan(const mpd_t *dec) { return dec->flags & MPD_SNAN; } /* Special */ ALWAYS_INLINE int mpd_isspecial(const mpd_t *dec) { return dec->flags & MPD_SPECIAL; } /* Zero */ ALWAYS_INLINE int mpd_iszero(const mpd_t *dec) { return !mpd_isspecial(dec) && mpd_msword(dec) == 0; } /* Test for zero when specials have been ruled out already */ ALWAYS_INLINE int mpd_iszerocoeff(const mpd_t *dec) { return mpd_msword(dec) == 0; } /* Normal */ inline int mpd_isnormal(const mpd_t *dec, const mpd_context_t *ctx) { if (mpd_isspecial(dec)) return 0; if (mpd_iszerocoeff(dec)) return 0; return mpd_adjexp(dec) >= ctx->emin; } /* Subnormal */ inline int mpd_issubnormal(const mpd_t *dec, const mpd_context_t *ctx) { if (mpd_isspecial(dec)) return 0; if (mpd_iszerocoeff(dec)) return 0; return mpd_adjexp(dec) < ctx->emin; } /* Odd word */ ALWAYS_INLINE int mpd_isoddword(mpd_uint_t word) { return word & 1; } /* Odd coefficient */ ALWAYS_INLINE int mpd_isoddcoeff(const mpd_t *dec) { return mpd_isoddword(dec->data[0]); } /* 0 if dec is positive, 1 if dec is negative */ ALWAYS_INLINE uint8_t mpd_sign(const mpd_t *dec) { return dec->flags & MPD_NEG; } /* 1 if dec is positive, -1 if dec is negative */ ALWAYS_INLINE int mpd_arith_sign(const mpd_t *dec) { return 1 - 2 * mpd_isnegative(dec); } /* Radix */ ALWAYS_INLINE long mpd_radix(void) { return 10; } /* Dynamic decimal */ ALWAYS_INLINE int mpd_isdynamic(mpd_t *dec) { return !(dec->flags & MPD_STATIC); } /* Static decimal */ ALWAYS_INLINE int mpd_isstatic(mpd_t *dec) { return dec->flags & MPD_STATIC; } /* Data of decimal is dynamic */ ALWAYS_INLINE int mpd_isdynamic_data(mpd_t *dec) { return !(dec->flags & MPD_DATAFLAGS); } /* Data of decimal is static */ ALWAYS_INLINE int mpd_isstatic_data(mpd_t *dec) { return dec->flags & MPD_STATIC_DATA; } /* Data of decimal is shared */ ALWAYS_INLINE int mpd_isshared_data(mpd_t *dec) { return dec->flags & MPD_SHARED_DATA; } /* Data of decimal is const */ ALWAYS_INLINE int mpd_isconst_data(mpd_t *dec) { return dec->flags & MPD_CONST_DATA; } /******************************************************************************/ /* Inline memory handling */ /******************************************************************************/ /* Fill destination with zeros */ ALWAYS_INLINE void mpd_uint_zero(mpd_uint_t *dest, mpd_size_t len) { mpd_size_t i; for (i = 0; i < len; i++) { dest[i] = 0; } } /* Free a decimal */ ALWAYS_INLINE void mpd_del(mpd_t *dec) { if (mpd_isdynamic_data(dec)) { mpd_free(dec->data); } if (mpd_isdynamic(dec)) { mpd_free(dec); } } /* * Resize the coefficient. Existing data up to 'nwords' is left untouched. * Return 1 on success, 0 otherwise. * * Input invariant: MPD_MINALLOC <= result->alloc. * * Case nwords == result->alloc: * 'result' is unchanged. Return 1. * * Case nwords > result->alloc: * Case realloc success: * The value of 'result' does not change. Return 1. * Case realloc failure: * 'result' is NaN, status is updated with MPD_Malloc_error. Return 0. * * Case nwords < result->alloc: * Case is_static_data or realloc failure [1]: * 'result' is unchanged. Return 1. * Case realloc success: * The value of result is undefined (expected). Return 1. * * * [1] In that case the old (now oversized) area is still valid. */ ALWAYS_INLINE int mpd_qresize(mpd_t *result, mpd_ssize_t nwords, uint32_t *status) { assert(!mpd_isconst_data(result)); /* illegal operation for a const */ assert(!mpd_isshared_data(result)); /* illegal operation for a shared */ assert(MPD_MINALLOC <= result->alloc); nwords = (nwords <= MPD_MINALLOC) ? MPD_MINALLOC : nwords; if (nwords == result->alloc) { return 1; } if (mpd_isstatic_data(result)) { if (nwords > result->alloc) { return mpd_switch_to_dyn(result, nwords, status); } return 1; } return mpd_realloc_dyn(result, nwords, status); } /* Same as mpd_qresize, but the complete coefficient (including the old * memory area!) is initialized to zero. */ ALWAYS_INLINE int mpd_qresize_zero(mpd_t *result, mpd_ssize_t nwords, uint32_t *status) { assert(!mpd_isconst_data(result)); /* illegal operation for a const */ assert(!mpd_isshared_data(result)); /* illegal operation for a shared */ assert(MPD_MINALLOC <= result->alloc); nwords = (nwords <= MPD_MINALLOC) ? MPD_MINALLOC : nwords; if (nwords != result->alloc) { if (mpd_isstatic_data(result)) { if (nwords > result->alloc) { return mpd_switch_to_dyn_zero(result, nwords, status); } } else if (!mpd_realloc_dyn(result, nwords, status)) { return 0; } } mpd_uint_zero(result->data, nwords); return 1; } /* * Reduce memory size for the coefficient to MPD_MINALLOC. In theory, * realloc may fail even when reducing the memory size. But in that case * the old memory area is always big enough, so checking for MPD_Malloc_error * is not imperative. */ ALWAYS_INLINE void mpd_minalloc(mpd_t *result) { assert(!mpd_isconst_data(result)); /* illegal operation for a const */ assert(!mpd_isshared_data(result)); /* illegal operation for a shared */ if (!mpd_isstatic_data(result) && result->alloc > MPD_MINALLOC) { uint8_t err = 0; result->data = mpd_realloc(result->data, MPD_MINALLOC, sizeof *result->data, &err); if (!err) { result->alloc = MPD_MINALLOC; } } } int mpd_resize(mpd_t *result, mpd_ssize_t nwords, mpd_context_t *ctx) { uint32_t status = 0; if (!mpd_qresize(result, nwords, &status)) { mpd_addstatus_raise(ctx, status); return 0; } return 1; } int mpd_resize_zero(mpd_t *result, mpd_ssize_t nwords, mpd_context_t *ctx) { uint32_t status = 0; if (!mpd_qresize_zero(result, nwords, &status)) { mpd_addstatus_raise(ctx, status); return 0; } return 1; } /******************************************************************************/ /* Set attributes of a decimal */ /******************************************************************************/ /* Set digits. Assumption: result->len is initialized and > 0. */ inline void mpd_setdigits(mpd_t *result) { mpd_ssize_t wdigits = mpd_word_digits(mpd_msword(result)); result->digits = wdigits + (result->len-1) * MPD_RDIGITS; } /* Set sign */ ALWAYS_INLINE void mpd_set_sign(mpd_t *result, uint8_t sign) { result->flags &= ~MPD_NEG; result->flags |= sign; } /* Copy sign from another decimal */ ALWAYS_INLINE void mpd_signcpy(mpd_t *result, mpd_t *a) { uint8_t sign = a->flags&MPD_NEG; result->flags &= ~MPD_NEG; result->flags |= sign; } /* Set infinity */ ALWAYS_INLINE void mpd_set_infinity(mpd_t *result) { result->flags &= ~MPD_SPECIAL; result->flags |= MPD_INF; } /* Set qNaN */ ALWAYS_INLINE void mpd_set_qnan(mpd_t *result) { result->flags &= ~MPD_SPECIAL; result->flags |= MPD_NAN; } /* Set sNaN */ ALWAYS_INLINE void mpd_set_snan(mpd_t *result) { result->flags &= ~MPD_SPECIAL; result->flags |= MPD_SNAN; } /* Set to negative */ ALWAYS_INLINE void mpd_set_negative(mpd_t *result) { result->flags |= MPD_NEG; } /* Set to positive */ ALWAYS_INLINE void mpd_set_positive(mpd_t *result) { result->flags &= ~MPD_NEG; } /* Set to dynamic */ ALWAYS_INLINE void mpd_set_dynamic(mpd_t *result) { result->flags &= ~MPD_STATIC; } /* Set to static */ ALWAYS_INLINE void mpd_set_static(mpd_t *result) { result->flags |= MPD_STATIC; } /* Set data to dynamic */ ALWAYS_INLINE void mpd_set_dynamic_data(mpd_t *result) { result->flags &= ~MPD_DATAFLAGS; } /* Set data to static */ ALWAYS_INLINE void mpd_set_static_data(mpd_t *result) { result->flags &= ~MPD_DATAFLAGS; result->flags |= MPD_STATIC_DATA; } /* Set data to shared */ ALWAYS_INLINE void mpd_set_shared_data(mpd_t *result) { result->flags &= ~MPD_DATAFLAGS; result->flags |= MPD_SHARED_DATA; } /* Set data to const */ ALWAYS_INLINE void mpd_set_const_data(mpd_t *result) { result->flags &= ~MPD_DATAFLAGS; result->flags |= MPD_CONST_DATA; } /* Clear flags, preserving memory attributes. */ ALWAYS_INLINE void mpd_clear_flags(mpd_t *result) { result->flags &= (MPD_STATIC|MPD_DATAFLAGS); } /* Set flags, preserving memory attributes. */ ALWAYS_INLINE void mpd_set_flags(mpd_t *result, uint8_t flags) { result->flags &= (MPD_STATIC|MPD_DATAFLAGS); result->flags |= flags; } /* Copy flags, preserving memory attributes of result. */ ALWAYS_INLINE void mpd_copy_flags(mpd_t *result, const mpd_t *a) { uint8_t aflags = a->flags; result->flags &= (MPD_STATIC|MPD_DATAFLAGS); result->flags |= (aflags & ~(MPD_STATIC|MPD_DATAFLAGS)); } /* Initialize a workcontext from ctx. Set traps, flags and newtrap to 0. */ static inline void mpd_workcontext(mpd_context_t *workctx, const mpd_context_t *ctx) { workctx->prec = ctx->prec; workctx->emax = ctx->emax; workctx->emin = ctx->emin; workctx->round = ctx->round; workctx->traps = 0; workctx->status = 0; workctx->newtrap = 0; workctx->clamp = ctx->clamp; workctx->allcr = ctx->allcr; } /******************************************************************************/ /* Getting and setting parts of decimals */ /******************************************************************************/ /* Flip the sign of a decimal */ static inline void _mpd_negate(mpd_t *dec) { dec->flags ^= MPD_NEG; } /* Set coefficient to zero */ void mpd_zerocoeff(mpd_t *result) { mpd_minalloc(result); result->digits = 1; result->len = 1; result->data[0] = 0; } /* Set the coefficient to all nines. */ void mpd_qmaxcoeff(mpd_t *result, const mpd_context_t *ctx, uint32_t *status) { mpd_ssize_t len, r; _mpd_idiv_word(&len, &r, ctx->prec, MPD_RDIGITS); len = (r == 0) ? len : len+1; if (!mpd_qresize(result, len, status)) { return; } result->len = len; result->digits = ctx->prec; --len; if (r > 0) { result->data[len--] = mpd_pow10[r]-1; } for (; len >= 0; --len) { result->data[len] = MPD_RADIX-1; } } /* * Cut off the most significant digits so that the rest fits in ctx->prec. * Cannot fail. */ static void _mpd_cap(mpd_t *result, const mpd_context_t *ctx) { uint32_t dummy; mpd_ssize_t len, r; if (result->len > 0 && result->digits > ctx->prec) { _mpd_idiv_word(&len, &r, ctx->prec, MPD_RDIGITS); len = (r == 0) ? len : len+1; if (r != 0) { result->data[len-1] %= mpd_pow10[r]; } len = _mpd_real_size(result->data, len); /* resize to fewer words cannot fail */ mpd_qresize(result, len, &dummy); result->len = len; mpd_setdigits(result); } if (mpd_iszero(result)) { _settriple(result, mpd_sign(result), 0, result->exp); } } /* * Cut off the most significant digits of a NaN payload so that the rest * fits in ctx->prec - ctx->clamp. Cannot fail. */ static void _mpd_fix_nan(mpd_t *result, const mpd_context_t *ctx) { uint32_t dummy; mpd_ssize_t prec; mpd_ssize_t len, r; prec = ctx->prec - ctx->clamp; if (result->len > 0 && result->digits > prec) { if (prec == 0) { mpd_minalloc(result); result->len = result->digits = 0; } else { _mpd_idiv_word(&len, &r, prec, MPD_RDIGITS); len = (r == 0) ? len : len+1; if (r != 0) { result->data[len-1] %= mpd_pow10[r]; } len = _mpd_real_size(result->data, len); /* resize to fewer words cannot fail */ mpd_qresize(result, len, &dummy); result->len = len; mpd_setdigits(result); if (mpd_iszerocoeff(result)) { /* NaN0 is not a valid representation */ result->len = result->digits = 0; } } } } /* * Get n most significant digits from a decimal, where 0 < n <= MPD_UINT_DIGITS. * Assumes MPD_UINT_DIGITS == MPD_RDIGITS+1, which is true for 32 and 64 bit * machines. * * The result of the operation will be in lo. If the operation is impossible, * hi will be nonzero. This is used to indicate an error. */ static inline void _mpd_get_msdigits(mpd_uint_t *hi, mpd_uint_t *lo, const mpd_t *dec, unsigned int n) { mpd_uint_t r, tmp; assert(0 < n && n <= MPD_RDIGITS+1); _mpd_div_word(&tmp, &r, dec->digits, MPD_RDIGITS); r = (r == 0) ? MPD_RDIGITS : r; /* digits in the most significant word */ *hi = 0; *lo = dec->data[dec->len-1]; if (n <= r) { *lo /= mpd_pow10[r-n]; } else if (dec->len > 1) { /* at this point 1 <= r < n <= MPD_RDIGITS+1 */ _mpd_mul_words(hi, lo, *lo, mpd_pow10[n-r]); tmp = dec->data[dec->len-2] / mpd_pow10[MPD_RDIGITS-(n-r)]; *lo = *lo + tmp; if (*lo < tmp) (*hi)++; } } /******************************************************************************/ /* Gathering information about a decimal */ /******************************************************************************/ /* The real size of the coefficient without leading zero words. */ static inline mpd_ssize_t _mpd_real_size(mpd_uint_t *data, mpd_ssize_t size) { while (size > 1 && data[size-1] == 0) { size--; } return size; } /* Return number of trailing zeros. No errors are possible. */ mpd_ssize_t mpd_trail_zeros(const mpd_t *dec) { mpd_uint_t word; mpd_ssize_t i, tz = 0; for (i=0; i < dec->len; ++i) { if (dec->data[i] != 0) { word = dec->data[i]; tz = i * MPD_RDIGITS; while (word % 10 == 0) { word /= 10; tz++; } break; } } return tz; } /* Integer: Undefined for specials */ static int _mpd_isint(const mpd_t *dec) { mpd_ssize_t tz; if (mpd_iszerocoeff(dec)) { return 1; } tz = mpd_trail_zeros(dec); return (dec->exp + tz >= 0); } /* Integer */ int mpd_isinteger(const mpd_t *dec) { if (mpd_isspecial(dec)) { return 0; } return _mpd_isint(dec); } /* Word is a power of 10 */ static int mpd_word_ispow10(mpd_uint_t word) { int n; n = mpd_word_digits(word); if (word == mpd_pow10[n-1]) { return 1; } return 0; } /* Coefficient is a power of 10 */ static int mpd_coeff_ispow10(const mpd_t *dec) { if (mpd_word_ispow10(mpd_msword(dec))) { if (_mpd_isallzero(dec->data, dec->len-1)) { return 1; } } return 0; } /* All digits of a word are nines */ static int mpd_word_isallnine(mpd_uint_t word) { int n; n = mpd_word_digits(word); if (word == mpd_pow10[n]-1) { return 1; } return 0; } /* All digits of the coefficient are nines */ static int mpd_coeff_isallnine(const mpd_t *dec) { if (mpd_word_isallnine(mpd_msword(dec))) { if (_mpd_isallnine(dec->data, dec->len-1)) { return 1; } } return 0; } /* Odd decimal: Undefined for non-integers! */ int mpd_isodd(const mpd_t *dec) { mpd_uint_t q, r; assert(mpd_isinteger(dec)); if (mpd_iszerocoeff(dec)) return 0; if (dec->exp < 0) { _mpd_div_word(&q, &r, -dec->exp, MPD_RDIGITS); q = dec->data[q] / mpd_pow10[r]; return mpd_isoddword(q); } return dec->exp == 0 && mpd_isoddword(dec->data[0]); } /* Even: Undefined for non-integers! */ int mpd_iseven(const mpd_t *dec) { return !mpd_isodd(dec); } /******************************************************************************/ /* Getting and setting decimals */ /******************************************************************************/ /* Internal function: Set a static decimal from a triple, no error checking. */ static void _ssettriple(mpd_t *result, uint8_t sign, mpd_uint_t a, mpd_ssize_t exp) { mpd_set_flags(result, sign); result->exp = exp; _mpd_div_word(&result->data[1], &result->data[0], a, MPD_RADIX); result->len = (result->data[1] == 0) ? 1 : 2; mpd_setdigits(result); } /* Internal function: Set a decimal from a triple, no error checking. */ static void _settriple(mpd_t *result, uint8_t sign, mpd_uint_t a, mpd_ssize_t exp) { mpd_minalloc(result); mpd_set_flags(result, sign); result->exp = exp; _mpd_div_word(&result->data[1], &result->data[0], a, MPD_RADIX); result->len = (result->data[1] == 0) ? 1 : 2; mpd_setdigits(result); } /* Set a special number from a triple */ void mpd_setspecial(mpd_t *result, uint8_t sign, uint8_t type) { mpd_minalloc(result); result->flags &= ~(MPD_NEG|MPD_SPECIAL); result->flags |= (sign|type); result->exp = result->digits = result->len = 0; } /* Set result of NaN with an error status */ void mpd_seterror(mpd_t *result, uint32_t flags, uint32_t *status) { mpd_minalloc(result); mpd_set_qnan(result); mpd_set_positive(result); result->exp = result->digits = result->len = 0; *status |= flags; } /* quietly set a static decimal from an mpd_ssize_t */ void mpd_qsset_ssize(mpd_t *result, mpd_ssize_t a, const mpd_context_t *ctx, uint32_t *status) { mpd_uint_t u; uint8_t sign = MPD_POS; if (a < 0) { if (a == MPD_SSIZE_MIN) { u = (mpd_uint_t)MPD_SSIZE_MAX + (-(MPD_SSIZE_MIN+MPD_SSIZE_MAX)); } else { u = -a; } sign = MPD_NEG; } else { u = a; } _ssettriple(result, sign, u, 0); mpd_qfinalize(result, ctx, status); } /* quietly set a static decimal from an mpd_uint_t */ void mpd_qsset_uint(mpd_t *result, mpd_uint_t a, const mpd_context_t *ctx, uint32_t *status) { _ssettriple(result, MPD_POS, a, 0); mpd_qfinalize(result, ctx, status); } /* quietly set a static decimal from an int32_t */ void mpd_qsset_i32(mpd_t *result, int32_t a, const mpd_context_t *ctx, uint32_t *status) { mpd_qsset_ssize(result, a, ctx, status); } /* quietly set a static decimal from a uint32_t */ void mpd_qsset_u32(mpd_t *result, uint32_t a, const mpd_context_t *ctx, uint32_t *status) { mpd_qsset_uint(result, a, ctx, status); } #ifdef CONFIG_64 /* quietly set a static decimal from an int64_t */ void mpd_qsset_i64(mpd_t *result, int64_t a, const mpd_context_t *ctx, uint32_t *status) { mpd_qsset_ssize(result, a, ctx, status); } /* quietly set a static decimal from a uint64_t */ void mpd_qsset_u64(mpd_t *result, uint64_t a, const mpd_context_t *ctx, uint32_t *status) { mpd_qsset_uint(result, a, ctx, status); } #endif /* quietly set a decimal from an mpd_ssize_t */ void mpd_qset_ssize(mpd_t *result, mpd_ssize_t a, const mpd_context_t *ctx, uint32_t *status) { mpd_minalloc(result); mpd_qsset_ssize(result, a, ctx, status); } /* quietly set a decimal from an mpd_uint_t */ void mpd_qset_uint(mpd_t *result, mpd_uint_t a, const mpd_context_t *ctx, uint32_t *status) { _settriple(result, MPD_POS, a, 0); mpd_qfinalize(result, ctx, status); } /* quietly set a decimal from an int32_t */ void mpd_qset_i32(mpd_t *result, int32_t a, const mpd_context_t *ctx, uint32_t *status) { mpd_qset_ssize(result, a, ctx, status); } /* quietly set a decimal from a uint32_t */ void mpd_qset_u32(mpd_t *result, uint32_t a, const mpd_context_t *ctx, uint32_t *status) { mpd_qset_uint(result, a, ctx, status); } #if defined(CONFIG_32) && !defined(LEGACY_COMPILER) /* set a decimal from a uint64_t */ static void _c32setu64(mpd_t *result, uint64_t u, uint8_t sign, uint32_t *status) { mpd_uint_t w[3]; uint64_t q; int i, len; len = 0; do { q = u / MPD_RADIX; w[len] = (mpd_uint_t)(u - q * MPD_RADIX); u = q; len++; } while (u != 0); if (!mpd_qresize(result, len, status)) { return; } for (i = 0; i < len; i++) { result->data[i] = w[i]; } mpd_set_sign(result, sign); result->exp = 0; result->len = len; mpd_setdigits(result); } static void _c32_qset_u64(mpd_t *result, uint64_t a, const mpd_context_t *ctx, uint32_t *status) { _c32setu64(result, a, MPD_POS, status); mpd_qfinalize(result, ctx, status); } /* set a decimal from an int64_t */ static void _c32_qset_i64(mpd_t *result, int64_t a, const mpd_context_t *ctx, uint32_t *status) { uint64_t u; uint8_t sign = MPD_POS; if (a < 0) { if (a == INT64_MIN) { u = (uint64_t)INT64_MAX + (-(INT64_MIN+INT64_MAX)); } else { u = -a; } sign = MPD_NEG; } else { u = a; } _c32setu64(result, u, sign, status); mpd_qfinalize(result, ctx, status); } #endif /* CONFIG_32 && !LEGACY_COMPILER */ #ifndef LEGACY_COMPILER /* quietly set a decimal from an int64_t */ void mpd_qset_i64(mpd_t *result, int64_t a, const mpd_context_t *ctx, uint32_t *status) { #ifdef CONFIG_64 mpd_qset_ssize(result, a, ctx, status); #else _c32_qset_i64(result, a, ctx, status); #endif } /* quietly set a decimal from a uint64_t */ void mpd_qset_u64(mpd_t *result, uint64_t a, const mpd_context_t *ctx, uint32_t *status) { #ifdef CONFIG_64 mpd_qset_uint(result, a, ctx, status); #else _c32_qset_u64(result, a, ctx, status); #endif } #endif /* !LEGACY_COMPILER */ /* * Quietly get an mpd_uint_t from a decimal. Assumes * MPD_UINT_DIGITS == MPD_RDIGITS+1, which is true for * 32 and 64 bit machines. * * If the operation is impossible, MPD_Invalid_operation is set. */ static mpd_uint_t _mpd_qget_uint(int use_sign, const mpd_t *a, uint32_t *status) { mpd_t tmp; mpd_uint_t tmp_data[2]; mpd_uint_t lo, hi; if (mpd_isspecial(a)) { *status |= MPD_Invalid_operation; return MPD_UINT_MAX; } if (mpd_iszero(a)) { return 0; } if (use_sign && mpd_isnegative(a)) { *status |= MPD_Invalid_operation; return MPD_UINT_MAX; } if (a->digits+a->exp > MPD_RDIGITS+1) { *status |= MPD_Invalid_operation; return MPD_UINT_MAX; } if (a->exp < 0) { if (!_mpd_isint(a)) { *status |= MPD_Invalid_operation; return MPD_UINT_MAX; } /* At this point a->digits+a->exp <= MPD_RDIGITS+1, * so the shift fits. */ tmp.data = tmp_data; tmp.flags = MPD_STATIC|MPD_STATIC_DATA; tmp.alloc = 2; mpd_qsshiftr(&tmp, a, -a->exp); tmp.exp = 0; a = &tmp; } _mpd_get_msdigits(&hi, &lo, a, MPD_RDIGITS+1); if (hi) { *status |= MPD_Invalid_operation; return MPD_UINT_MAX; } if (a->exp > 0) { _mpd_mul_words(&hi, &lo, lo, mpd_pow10[a->exp]); if (hi) { *status |= MPD_Invalid_operation; return MPD_UINT_MAX; } } return lo; } /* * Sets Invalid_operation for: * - specials * - negative numbers (except negative zero) * - non-integers * - overflow */ mpd_uint_t mpd_qget_uint(const mpd_t *a, uint32_t *status) { return _mpd_qget_uint(1, a, status); } /* Same as above, but gets the absolute value, i.e. the sign is ignored. */ mpd_uint_t mpd_qabs_uint(const mpd_t *a, uint32_t *status) { return _mpd_qget_uint(0, a, status); } /* quietly get an mpd_ssize_t from a decimal */ mpd_ssize_t mpd_qget_ssize(const mpd_t *a, uint32_t *status) { mpd_uint_t u; int isneg; u = mpd_qabs_uint(a, status); if (*status&MPD_Invalid_operation) { return MPD_SSIZE_MAX; } isneg = mpd_isnegative(a); if (u <= MPD_SSIZE_MAX) { return isneg ? -((mpd_ssize_t)u) : (mpd_ssize_t)u; } else if (isneg && u+(MPD_SSIZE_MIN+MPD_SSIZE_MAX) == MPD_SSIZE_MAX) { return MPD_SSIZE_MIN; } *status |= MPD_Invalid_operation; return MPD_SSIZE_MAX; } #ifdef CONFIG_64 /* quietly get a uint64_t from a decimal */ uint64_t mpd_qget_u64(const mpd_t *a, uint32_t *status) { return mpd_qget_uint(a, status); } /* quietly get an int64_t from a decimal */ int64_t mpd_qget_i64(const mpd_t *a, uint32_t *status) { return mpd_qget_ssize(a, status); } #else /* quietly get a uint32_t from a decimal */ uint32_t mpd_qget_u32(const mpd_t *a, uint32_t *status) { return mpd_qget_uint(a, status); } /* quietly get an int32_t from a decimal */ int32_t mpd_qget_i32(const mpd_t *a, uint32_t *status) { return mpd_qget_ssize(a, status); } #endif /******************************************************************************/ /* Filtering input of functions, finalizing output of functions */ /******************************************************************************/ /* * Check if the operand is NaN, copy to result and return 1 if this is * the case. Copying can fail since NaNs are allowed to have a payload that * does not fit in MPD_MINALLOC. */ int mpd_qcheck_nan(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { if (mpd_isnan(a)) { *status |= mpd_issnan(a) ? MPD_Invalid_operation : 0; mpd_qcopy(result, a, status); mpd_set_qnan(result); _mpd_fix_nan(result, ctx); return 1; } return 0; } /* * Check if either operand is NaN, copy to result and return 1 if this * is the case. Copying can fail since NaNs are allowed to have a payload * that does not fit in MPD_MINALLOC. */ int mpd_qcheck_nans(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { if ((a->flags|b->flags)&(MPD_NAN|MPD_SNAN)) { const mpd_t *choice = b; if (mpd_issnan(a)) { choice = a; *status |= MPD_Invalid_operation; } else if (mpd_issnan(b)) { *status |= MPD_Invalid_operation; } else if (mpd_isqnan(a)) { choice = a; } mpd_qcopy(result, choice, status); mpd_set_qnan(result); _mpd_fix_nan(result, ctx); return 1; } return 0; } /* * Check if one of the operands is NaN, copy to result and return 1 if this * is the case. Copying can fail since NaNs are allowed to have a payload * that does not fit in MPD_MINALLOC. */ static int mpd_qcheck_3nans(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_t *c, const mpd_context_t *ctx, uint32_t *status) { if ((a->flags|b->flags|c->flags)&(MPD_NAN|MPD_SNAN)) { const mpd_t *choice = c; if (mpd_issnan(a)) { choice = a; *status |= MPD_Invalid_operation; } else if (mpd_issnan(b)) { choice = b; *status |= MPD_Invalid_operation; } else if (mpd_issnan(c)) { *status |= MPD_Invalid_operation; } else if (mpd_isqnan(a)) { choice = a; } else if (mpd_isqnan(b)) { choice = b; } mpd_qcopy(result, choice, status); mpd_set_qnan(result); _mpd_fix_nan(result, ctx); return 1; } return 0; } /* Check if rounding digit 'rnd' leads to an increment. */ static inline int _mpd_rnd_incr(const mpd_t *dec, mpd_uint_t rnd, const mpd_context_t *ctx) { int ld; switch (ctx->round) { case MPD_ROUND_DOWN: case MPD_ROUND_TRUNC: return 0; case MPD_ROUND_HALF_UP: return (rnd >= 5); case MPD_ROUND_HALF_EVEN: return (rnd > 5) || ((rnd == 5) && mpd_isoddcoeff(dec)); case MPD_ROUND_CEILING: return !(rnd == 0 || mpd_isnegative(dec)); case MPD_ROUND_FLOOR: return !(rnd == 0 || mpd_ispositive(dec)); case MPD_ROUND_HALF_DOWN: return (rnd > 5); case MPD_ROUND_UP: return !(rnd == 0); case MPD_ROUND_05UP: ld = (int)mpd_lsd(dec->data[0]); return (!(rnd == 0) && (ld == 0 || ld == 5)); default: /* Without a valid context, further results will be undefined. */ return 0; /* GCOV_NOT_REACHED */ } } /* * Apply rounding to a decimal that has been right-shifted into a full * precision decimal. If an increment leads to an overflow of the precision, * adjust the coefficient and the exponent and check the new exponent for * overflow. */ static inline void _mpd_apply_round(mpd_t *dec, mpd_uint_t rnd, const mpd_context_t *ctx, uint32_t *status) { if (_mpd_rnd_incr(dec, rnd, ctx)) { /* We have a number with exactly ctx->prec digits. The increment * can only lead to an overflow if the decimal is all nines. In * that case, the result is a power of ten with prec+1 digits. * * If the precision is a multiple of MPD_RDIGITS, this situation is * detected by _mpd_baseincr returning a carry. * If the precision is not a multiple of MPD_RDIGITS, we have to * check if the result has one digit too many. */ mpd_uint_t carry = _mpd_baseincr(dec->data, dec->len); if (carry) { dec->data[dec->len-1] = mpd_pow10[MPD_RDIGITS-1]; dec->exp += 1; _mpd_check_exp(dec, ctx, status); return; } mpd_setdigits(dec); if (dec->digits > ctx->prec) { mpd_qshiftr_inplace(dec, 1); dec->exp += 1; dec->digits = ctx->prec; _mpd_check_exp(dec, ctx, status); } } } /* * Apply rounding to a decimal. Allow overflow of the precision. */ static inline void _mpd_apply_round_excess(mpd_t *dec, mpd_uint_t rnd, const mpd_context_t *ctx, uint32_t *status) { if (_mpd_rnd_incr(dec, rnd, ctx)) { mpd_uint_t carry = _mpd_baseincr(dec->data, dec->len); if (carry) { if (!mpd_qresize(dec, dec->len+1, status)) { return; } dec->data[dec->len] = 1; dec->len += 1; } mpd_setdigits(dec); } } /* * Apply rounding to a decimal that has been right-shifted into a decimal * with full precision or less. Return failure if an increment would * overflow the precision. */ static inline int _mpd_apply_round_fit(mpd_t *dec, mpd_uint_t rnd, const mpd_context_t *ctx, uint32_t *status) { if (_mpd_rnd_incr(dec, rnd, ctx)) { mpd_uint_t carry = _mpd_baseincr(dec->data, dec->len); if (carry) { if (!mpd_qresize(dec, dec->len+1, status)) { return 0; } dec->data[dec->len] = 1; dec->len += 1; } mpd_setdigits(dec); if (dec->digits > ctx->prec) { mpd_seterror(dec, MPD_Invalid_operation, status); return 0; } } return 1; } /* Check a normal number for overflow, underflow, clamping. If the operand is modified, it will be zero, special or (sub)normal with a coefficient that fits into the current context precision. */ static inline void _mpd_check_exp(mpd_t *dec, const mpd_context_t *ctx, uint32_t *status) { mpd_ssize_t adjexp, etiny, shift; int rnd; adjexp = mpd_adjexp(dec); if (adjexp > ctx->emax) { if (mpd_iszerocoeff(dec)) { dec->exp = ctx->emax; if (ctx->clamp) { dec->exp -= (ctx->prec-1); } mpd_zerocoeff(dec); *status |= MPD_Clamped; return; } switch (ctx->round) { case MPD_ROUND_HALF_UP: case MPD_ROUND_HALF_EVEN: case MPD_ROUND_HALF_DOWN: case MPD_ROUND_UP: case MPD_ROUND_TRUNC: mpd_setspecial(dec, mpd_sign(dec), MPD_INF); break; case MPD_ROUND_DOWN: case MPD_ROUND_05UP: mpd_qmaxcoeff(dec, ctx, status); dec->exp = ctx->emax - ctx->prec + 1; break; case MPD_ROUND_CEILING: if (mpd_isnegative(dec)) { mpd_qmaxcoeff(dec, ctx, status); dec->exp = ctx->emax - ctx->prec + 1; } else { mpd_setspecial(dec, MPD_POS, MPD_INF); } break; case MPD_ROUND_FLOOR: if (mpd_ispositive(dec)) { mpd_qmaxcoeff(dec, ctx, status); dec->exp = ctx->emax - ctx->prec + 1; } else { mpd_setspecial(dec, MPD_NEG, MPD_INF); } break; default: /* debug */ abort(); /* GCOV_NOT_REACHED */ } *status |= MPD_Overflow|MPD_Inexact|MPD_Rounded; } /* fold down */ else if (ctx->clamp && dec->exp > mpd_etop(ctx)) { /* At this point adjexp=exp+digits-1 <= emax and exp > etop=emax-prec+1: * (1) shift = exp -emax+prec-1 > 0 * (2) digits+shift = exp+digits-1 - emax + prec <= prec */ shift = dec->exp - mpd_etop(ctx); if (!mpd_qshiftl(dec, dec, shift, status)) { return; } dec->exp -= shift; *status |= MPD_Clamped; if (!mpd_iszerocoeff(dec) && adjexp < ctx->emin) { /* Underflow is impossible, since exp < etiny=emin-prec+1 * and exp > etop=emax-prec+1 would imply emax < emin. */ *status |= MPD_Subnormal; } } else if (adjexp < ctx->emin) { etiny = mpd_etiny(ctx); if (mpd_iszerocoeff(dec)) { if (dec->exp < etiny) { dec->exp = etiny; mpd_zerocoeff(dec); *status |= MPD_Clamped; } return; } *status |= MPD_Subnormal; if (dec->exp < etiny) { /* At this point adjexp=exp+digits-1 < emin and exp < etiny=emin-prec+1: * (1) shift = emin-prec+1 - exp > 0 * (2) digits-shift = exp+digits-1 - emin + prec < prec */ shift = etiny - dec->exp; rnd = (int)mpd_qshiftr_inplace(dec, shift); dec->exp = etiny; /* We always have a spare digit in case of an increment. */ _mpd_apply_round_excess(dec, rnd, ctx, status); *status |= MPD_Rounded; if (rnd) { *status |= (MPD_Inexact|MPD_Underflow); if (mpd_iszerocoeff(dec)) { mpd_zerocoeff(dec); *status |= MPD_Clamped; } } } /* Case exp >= etiny=emin-prec+1: * (1) adjexp=exp+digits-1 < emin * (2) digits < emin-exp+1 <= prec */ } } /* Transcendental functions do not always set Underflow reliably, * since they only use as much precision as is necessary for correct * rounding. If a result like 1.0000000000e-101 is finalized, there * is no rounding digit that would trigger Underflow. But we can * assume Inexact, so a short check suffices. */ static inline void mpd_check_underflow(mpd_t *dec, const mpd_context_t *ctx, uint32_t *status) { if (mpd_adjexp(dec) < ctx->emin && !mpd_iszero(dec) && dec->exp < mpd_etiny(ctx)) { *status |= MPD_Underflow; } } /* Check if a normal number must be rounded after the exponent has been checked. */ static inline void _mpd_check_round(mpd_t *dec, const mpd_context_t *ctx, uint32_t *status) { mpd_uint_t rnd; mpd_ssize_t shift; /* must handle specials: _mpd_check_exp() can produce infinities or NaNs */ if (mpd_isspecial(dec)) { return; } if (dec->digits > ctx->prec) { shift = dec->digits - ctx->prec; rnd = mpd_qshiftr_inplace(dec, shift); dec->exp += shift; _mpd_apply_round(dec, rnd, ctx, status); *status |= MPD_Rounded; if (rnd) { *status |= MPD_Inexact; } } } /* Finalize all operations. */ void mpd_qfinalize(mpd_t *result, const mpd_context_t *ctx, uint32_t *status) { if (mpd_isspecial(result)) { if (mpd_isnan(result)) { _mpd_fix_nan(result, ctx); } return; } _mpd_check_exp(result, ctx, status); _mpd_check_round(result, ctx, status); } /******************************************************************************/ /* Copying */ /******************************************************************************/ /* Internal function: Copy a decimal, share data with src: USE WITH CARE! */ static inline void _mpd_copy_shared(mpd_t *dest, const mpd_t *src) { dest->flags = src->flags; dest->exp = src->exp; dest->digits = src->digits; dest->len = src->len; dest->alloc = src->alloc; dest->data = src->data; mpd_set_shared_data(dest); } /* * Copy a decimal. In case of an error, status is set to MPD_Malloc_error. */ int mpd_qcopy(mpd_t *result, const mpd_t *a, uint32_t *status) { if (result == a) return 1; if (!mpd_qresize(result, a->len, status)) { return 0; } mpd_copy_flags(result, a); result->exp = a->exp; result->digits = a->digits; result->len = a->len; memcpy(result->data, a->data, a->len * (sizeof *result->data)); return 1; } /* * Copy to a decimal with a static buffer. The caller has to make sure that * the buffer is big enough. Cannot fail. */ static void mpd_qcopy_static(mpd_t *result, const mpd_t *a) { if (result == a) return; memcpy(result->data, a->data, a->len * (sizeof *result->data)); mpd_copy_flags(result, a); result->exp = a->exp; result->digits = a->digits; result->len = a->len; } /* * Return a newly allocated copy of the operand. In case of an error, * status is set to MPD_Malloc_error and the return value is NULL. */ mpd_t * mpd_qncopy(const mpd_t *a) { mpd_t *result; if ((result = mpd_qnew_size(a->len)) == NULL) { return NULL; } memcpy(result->data, a->data, a->len * (sizeof *result->data)); mpd_copy_flags(result, a); result->exp = a->exp; result->digits = a->digits; result->len = a->len; return result; } /* * Copy a decimal and set the sign to positive. In case of an error, the * status is set to MPD_Malloc_error. */ int mpd_qcopy_abs(mpd_t *result, const mpd_t *a, uint32_t *status) { if (!mpd_qcopy(result, a, status)) { return 0; } mpd_set_positive(result); return 1; } /* * Copy a decimal and negate the sign. In case of an error, the * status is set to MPD_Malloc_error. */ int mpd_qcopy_negate(mpd_t *result, const mpd_t *a, uint32_t *status) { if (!mpd_qcopy(result, a, status)) { return 0; } _mpd_negate(result); return 1; } /* * Copy a decimal, setting the sign of the first operand to the sign of the * second operand. In case of an error, the status is set to MPD_Malloc_error. */ int mpd_qcopy_sign(mpd_t *result, const mpd_t *a, const mpd_t *b, uint32_t *status) { uint8_t sign_b = mpd_sign(b); /* result may equal b! */ if (!mpd_qcopy(result, a, status)) { return 0; } mpd_set_sign(result, sign_b); return 1; } /******************************************************************************/ /* Comparisons */ /******************************************************************************/ /* * For all functions that compare two operands and return an int the usual * convention applies to the return value: * * -1 if op1 < op2 * 0 if op1 == op2 * 1 if op1 > op2 * * INT_MAX for error */ /* Convenience macro. If a and b are not equal, return from the calling * function with the correct comparison value. */ #define CMP_EQUAL_OR_RETURN(a, b) \ if (a != b) { \ if (a < b) { \ return -1; \ } \ return 1; \ } /* * Compare the data of big and small. This function does the equivalent * of first shifting small to the left and then comparing the data of * big and small, except that no allocation for the left shift is needed. */ static int _mpd_basecmp(mpd_uint_t *big, mpd_uint_t *small, mpd_size_t n, mpd_size_t m, mpd_size_t shift) { #if defined(__GNUC__) && !defined(__INTEL_COMPILER) && !defined(__clang__) /* spurious uninitialized warnings */ mpd_uint_t l=l, lprev=lprev, h=h; #else mpd_uint_t l, lprev, h; #endif mpd_uint_t q, r; mpd_uint_t ph, x; assert(m > 0 && n >= m && shift > 0); _mpd_div_word(&q, &r, (mpd_uint_t)shift, MPD_RDIGITS); if (r != 0) { ph = mpd_pow10[r]; --m; --n; _mpd_divmod_pow10(&h, &lprev, small[m--], MPD_RDIGITS-r); if (h != 0) { CMP_EQUAL_OR_RETURN(big[n], h) --n; } for (; m != MPD_SIZE_MAX; m--,n--) { _mpd_divmod_pow10(&h, &l, small[m], MPD_RDIGITS-r); x = ph * lprev + h; CMP_EQUAL_OR_RETURN(big[n], x) lprev = l; } x = ph * lprev; CMP_EQUAL_OR_RETURN(big[q], x) } else { while (--m != MPD_SIZE_MAX) { CMP_EQUAL_OR_RETURN(big[m+q], small[m]) } } return !_mpd_isallzero(big, q); } /* Compare two decimals with the same adjusted exponent. */ static int _mpd_cmp_same_adjexp(const mpd_t *a, const mpd_t *b) { mpd_ssize_t shift, i; if (a->exp != b->exp) { /* Cannot wrap: a->exp + a->digits = b->exp + b->digits, so * a->exp - b->exp = b->digits - a->digits. */ shift = a->exp - b->exp; if (shift > 0) { return -1 * _mpd_basecmp(b->data, a->data, b->len, a->len, shift); } else { return _mpd_basecmp(a->data, b->data, a->len, b->len, -shift); } } /* * At this point adjexp(a) == adjexp(b) and a->exp == b->exp, * so a->digits == b->digits, therefore a->len == b->len. */ for (i = a->len-1; i >= 0; --i) { CMP_EQUAL_OR_RETURN(a->data[i], b->data[i]) } return 0; } /* Compare two numerical values. */ static int _mpd_cmp(const mpd_t *a, const mpd_t *b) { mpd_ssize_t adjexp_a, adjexp_b; /* equal pointers */ if (a == b) { return 0; } /* infinities */ if (mpd_isinfinite(a)) { if (mpd_isinfinite(b)) { return mpd_isnegative(b) - mpd_isnegative(a); } return mpd_arith_sign(a); } if (mpd_isinfinite(b)) { return -mpd_arith_sign(b); } /* zeros */ if (mpd_iszerocoeff(a)) { if (mpd_iszerocoeff(b)) { return 0; } return -mpd_arith_sign(b); } if (mpd_iszerocoeff(b)) { return mpd_arith_sign(a); } /* different signs */ if (mpd_sign(a) != mpd_sign(b)) { return mpd_sign(b) - mpd_sign(a); } /* different adjusted exponents */ adjexp_a = mpd_adjexp(a); adjexp_b = mpd_adjexp(b); if (adjexp_a != adjexp_b) { if (adjexp_a < adjexp_b) { return -1 * mpd_arith_sign(a); } return mpd_arith_sign(a); } /* same adjusted exponents */ return _mpd_cmp_same_adjexp(a, b) * mpd_arith_sign(a); } /* Compare the absolutes of two numerical values. */ static int _mpd_cmp_abs(const mpd_t *a, const mpd_t *b) { mpd_ssize_t adjexp_a, adjexp_b; /* equal pointers */ if (a == b) { return 0; } /* infinities */ if (mpd_isinfinite(a)) { if (mpd_isinfinite(b)) { return 0; } return 1; } if (mpd_isinfinite(b)) { return -1; } /* zeros */ if (mpd_iszerocoeff(a)) { if (mpd_iszerocoeff(b)) { return 0; } return -1; } if (mpd_iszerocoeff(b)) { return 1; } /* different adjusted exponents */ adjexp_a = mpd_adjexp(a); adjexp_b = mpd_adjexp(b); if (adjexp_a != adjexp_b) { if (adjexp_a < adjexp_b) { return -1; } return 1; } /* same adjusted exponents */ return _mpd_cmp_same_adjexp(a, b); } /* Compare two values and return an integer result. */ int mpd_qcmp(const mpd_t *a, const mpd_t *b, uint32_t *status) { if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_isnan(a) || mpd_isnan(b)) { *status |= MPD_Invalid_operation; return INT_MAX; } } return _mpd_cmp(a, b); } /* * Compare a and b, convert the usual integer result to a decimal and * store it in 'result'. For convenience, the integer result of the comparison * is returned. Comparisons involving NaNs return NaN/INT_MAX. */ int mpd_qcompare(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { int c; if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(result, a, b, ctx, status)) { return INT_MAX; } } c = _mpd_cmp(a, b); _settriple(result, (c < 0), (c != 0), 0); return c; } /* Same as mpd_compare(), but signal for all NaNs, i.e. also for quiet NaNs. */ int mpd_qcompare_signal(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { int c; if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(result, a, b, ctx, status)) { *status |= MPD_Invalid_operation; return INT_MAX; } } c = _mpd_cmp(a, b); _settriple(result, (c < 0), (c != 0), 0); return c; } /* Compare the operands using a total order. */ int mpd_cmp_total(const mpd_t *a, const mpd_t *b) { mpd_t aa, bb; int nan_a, nan_b; int c; if (mpd_sign(a) != mpd_sign(b)) { return mpd_sign(b) - mpd_sign(a); } if (mpd_isnan(a)) { c = 1; if (mpd_isnan(b)) { nan_a = (mpd_isqnan(a)) ? 1 : 0; nan_b = (mpd_isqnan(b)) ? 1 : 0; if (nan_b == nan_a) { if (a->len > 0 && b->len > 0) { _mpd_copy_shared(&aa, a); _mpd_copy_shared(&bb, b); aa.exp = bb.exp = 0; /* compare payload */ c = _mpd_cmp_abs(&aa, &bb); } else { c = (a->len > 0) - (b->len > 0); } } else { c = nan_a - nan_b; } } } else if (mpd_isnan(b)) { c = -1; } else { c = _mpd_cmp_abs(a, b); if (c == 0 && a->exp != b->exp) { c = (a->exp < b->exp) ? -1 : 1; } } return c * mpd_arith_sign(a); } /* * Compare a and b according to a total order, convert the usual integer result * to a decimal and store it in 'result'. For convenience, the integer result * of the comparison is returned. */ int mpd_compare_total(mpd_t *result, const mpd_t *a, const mpd_t *b) { int c; c = mpd_cmp_total(a, b); _settriple(result, (c < 0), (c != 0), 0); return c; } /* Compare the magnitude of the operands using a total order. */ int mpd_cmp_total_mag(const mpd_t *a, const mpd_t *b) { mpd_t aa, bb; _mpd_copy_shared(&aa, a); _mpd_copy_shared(&bb, b); mpd_set_positive(&aa); mpd_set_positive(&bb); return mpd_cmp_total(&aa, &bb); } /* * Compare the magnitude of a and b according to a total order, convert the * the usual integer result to a decimal and store it in 'result'. * For convenience, the integer result of the comparison is returned. */ int mpd_compare_total_mag(mpd_t *result, const mpd_t *a, const mpd_t *b) { int c; c = mpd_cmp_total_mag(a, b); _settriple(result, (c < 0), (c != 0), 0); return c; } /* Determine an ordering for operands that are numerically equal. */ static inline int _mpd_cmp_numequal(const mpd_t *a, const mpd_t *b) { int sign_a, sign_b; int c; sign_a = mpd_sign(a); sign_b = mpd_sign(b); if (sign_a != sign_b) { c = sign_b - sign_a; } else { c = (a->exp < b->exp) ? -1 : 1; c *= mpd_arith_sign(a); } return c; } /******************************************************************************/ /* Shifting the coefficient */ /******************************************************************************/ /* * Shift the coefficient of the operand to the left, no check for specials. * Both operands may be the same pointer. If the result length has to be * increased, mpd_qresize() might fail with MPD_Malloc_error. */ int mpd_qshiftl(mpd_t *result, const mpd_t *a, mpd_ssize_t n, uint32_t *status) { mpd_ssize_t size; assert(!mpd_isspecial(a)); assert(n >= 0); if (mpd_iszerocoeff(a) || n == 0) { return mpd_qcopy(result, a, status); } size = mpd_digits_to_size(a->digits+n); if (!mpd_qresize(result, size, status)) { return 0; /* result is NaN */ } _mpd_baseshiftl(result->data, a->data, size, a->len, n); mpd_copy_flags(result, a); result->exp = a->exp; result->digits = a->digits+n; result->len = size; return 1; } /* Determine the rounding indicator if all digits of the coefficient are shifted * out of the picture. */ static mpd_uint_t _mpd_get_rnd(const mpd_uint_t *data, mpd_ssize_t len, int use_msd) { mpd_uint_t rnd = 0, rest = 0, word; word = data[len-1]; /* special treatment for the most significant digit if shift == digits */ if (use_msd) { _mpd_divmod_pow10(&rnd, &rest, word, mpd_word_digits(word)-1); if (len > 1 && rest == 0) { rest = !_mpd_isallzero(data, len-1); } } else { rest = !_mpd_isallzero(data, len); } return (rnd == 0 || rnd == 5) ? rnd + !!rest : rnd; } /* * Same as mpd_qshiftr(), but 'result' is an mpd_t with a static coefficient. * It is the caller's responsibility to ensure that the coefficient is big * enough. The function cannot fail. */ static mpd_uint_t mpd_qsshiftr(mpd_t *result, const mpd_t *a, mpd_ssize_t n) { mpd_uint_t rnd; mpd_ssize_t size; assert(!mpd_isspecial(a)); assert(n >= 0); if (mpd_iszerocoeff(a) || n == 0) { mpd_qcopy_static(result, a); return 0; } if (n >= a->digits) { rnd = _mpd_get_rnd(a->data, a->len, (n==a->digits)); mpd_zerocoeff(result); } else { result->digits = a->digits-n; size = mpd_digits_to_size(result->digits); rnd = _mpd_baseshiftr(result->data, a->data, a->len, n); result->len = size; } mpd_copy_flags(result, a); result->exp = a->exp; return rnd; } /* * Inplace shift of the coefficient to the right, no check for specials. * Returns the rounding indicator for mpd_rnd_incr(). * The function cannot fail. */ mpd_uint_t mpd_qshiftr_inplace(mpd_t *result, mpd_ssize_t n) { uint32_t dummy; mpd_uint_t rnd; mpd_ssize_t size; assert(!mpd_isspecial(result)); assert(n >= 0); if (mpd_iszerocoeff(result) || n == 0) { return 0; } if (n >= result->digits) { rnd = _mpd_get_rnd(result->data, result->len, (n==result->digits)); mpd_zerocoeff(result); } else { rnd = _mpd_baseshiftr(result->data, result->data, result->len, n); result->digits -= n; size = mpd_digits_to_size(result->digits); /* reducing the size cannot fail */ mpd_qresize(result, size, &dummy); result->len = size; } return rnd; } /* * Shift the coefficient of the operand to the right, no check for specials. * Both operands may be the same pointer. Returns the rounding indicator to * be used by mpd_rnd_incr(). If the result length has to be increased, * mpd_qcopy() or mpd_qresize() might fail with MPD_Malloc_error. In those * cases, MPD_UINT_MAX is returned. */ mpd_uint_t mpd_qshiftr(mpd_t *result, const mpd_t *a, mpd_ssize_t n, uint32_t *status) { mpd_uint_t rnd; mpd_ssize_t size; assert(!mpd_isspecial(a)); assert(n >= 0); if (mpd_iszerocoeff(a) || n == 0) { if (!mpd_qcopy(result, a, status)) { return MPD_UINT_MAX; } return 0; } if (n >= a->digits) { rnd = _mpd_get_rnd(a->data, a->len, (n==a->digits)); mpd_zerocoeff(result); } else { result->digits = a->digits-n; size = mpd_digits_to_size(result->digits); if (result == a) { rnd = _mpd_baseshiftr(result->data, a->data, a->len, n); /* reducing the size cannot fail */ mpd_qresize(result, size, status); } else { if (!mpd_qresize(result, size, status)) { return MPD_UINT_MAX; } rnd = _mpd_baseshiftr(result->data, a->data, a->len, n); } result->len = size; } mpd_copy_flags(result, a); result->exp = a->exp; return rnd; } /******************************************************************************/ /* Miscellaneous operations */ /******************************************************************************/ /* Logical And */ void mpd_qand(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { const mpd_t *big = a, *small = b; mpd_uint_t x, y, z, xbit, ybit; int k, mswdigits; mpd_ssize_t i; if (mpd_isspecial(a) || mpd_isspecial(b) || mpd_isnegative(a) || mpd_isnegative(b) || a->exp != 0 || b->exp != 0) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (b->digits > a->digits) { big = b; small = a; } if (!mpd_qresize(result, big->len, status)) { return; } /* full words */ for (i = 0; i < small->len-1; i++) { x = small->data[i]; y = big->data[i]; z = 0; for (k = 0; k < MPD_RDIGITS; k++) { xbit = x % 10; x /= 10; ybit = y % 10; y /= 10; if (xbit > 1 || ybit > 1) { goto invalid_operation; } z += (xbit&ybit) ? mpd_pow10[k] : 0; } result->data[i] = z; } /* most significant word of small */ x = small->data[i]; y = big->data[i]; z = 0; mswdigits = mpd_word_digits(x); for (k = 0; k < mswdigits; k++) { xbit = x % 10; x /= 10; ybit = y % 10; y /= 10; if (xbit > 1 || ybit > 1) { goto invalid_operation; } z += (xbit&ybit) ? mpd_pow10[k] : 0; } result->data[i++] = z; /* scan the rest of y for digits > 1 */ for (; k < MPD_RDIGITS; k++) { ybit = y % 10; y /= 10; if (ybit > 1) { goto invalid_operation; } } /* scan the rest of big for digits > 1 */ for (; i < big->len; i++) { y = big->data[i]; for (k = 0; k < MPD_RDIGITS; k++) { ybit = y % 10; y /= 10; if (ybit > 1) { goto invalid_operation; } } } mpd_clear_flags(result); result->exp = 0; result->len = _mpd_real_size(result->data, small->len); mpd_qresize(result, result->len, status); mpd_setdigits(result); _mpd_cap(result, ctx); return; invalid_operation: mpd_seterror(result, MPD_Invalid_operation, status); } /* Class of an operand. Returns a pointer to the constant name. */ const char * mpd_class(const mpd_t *a, const mpd_context_t *ctx) { if (mpd_isnan(a)) { if (mpd_isqnan(a)) return "NaN"; else return "sNaN"; } else if (mpd_ispositive(a)) { if (mpd_isinfinite(a)) return "+Infinity"; else if (mpd_iszero(a)) return "+Zero"; else if (mpd_isnormal(a, ctx)) return "+Normal"; else return "+Subnormal"; } else { if (mpd_isinfinite(a)) return "-Infinity"; else if (mpd_iszero(a)) return "-Zero"; else if (mpd_isnormal(a, ctx)) return "-Normal"; else return "-Subnormal"; } } /* Logical Xor */ void mpd_qinvert(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_uint_t x, z, xbit; mpd_ssize_t i, digits, len; mpd_ssize_t q, r; int k; if (mpd_isspecial(a) || mpd_isnegative(a) || a->exp != 0) { mpd_seterror(result, MPD_Invalid_operation, status); return; } digits = (a->digits < ctx->prec) ? ctx->prec : a->digits; _mpd_idiv_word(&q, &r, digits, MPD_RDIGITS); len = (r == 0) ? q : q+1; if (!mpd_qresize(result, len, status)) { return; } for (i = 0; i < len; i++) { x = (i < a->len) ? a->data[i] : 0; z = 0; for (k = 0; k < MPD_RDIGITS; k++) { xbit = x % 10; x /= 10; if (xbit > 1) { goto invalid_operation; } z += !xbit ? mpd_pow10[k] : 0; } result->data[i] = z; } mpd_clear_flags(result); result->exp = 0; result->len = _mpd_real_size(result->data, len); mpd_qresize(result, result->len, status); mpd_setdigits(result); _mpd_cap(result, ctx); return; invalid_operation: mpd_seterror(result, MPD_Invalid_operation, status); } /* Exponent of the magnitude of the most significant digit of the operand. */ void mpd_qlogb(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } mpd_setspecial(result, MPD_POS, MPD_INF); } else if (mpd_iszerocoeff(a)) { mpd_setspecial(result, MPD_NEG, MPD_INF); *status |= MPD_Division_by_zero; } else { mpd_qset_ssize(result, mpd_adjexp(a), ctx, status); } } /* Logical Or */ void mpd_qor(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { const mpd_t *big = a, *small = b; mpd_uint_t x, y, z, xbit, ybit; int k, mswdigits; mpd_ssize_t i; if (mpd_isspecial(a) || mpd_isspecial(b) || mpd_isnegative(a) || mpd_isnegative(b) || a->exp != 0 || b->exp != 0) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (b->digits > a->digits) { big = b; small = a; } if (!mpd_qresize(result, big->len, status)) { return; } /* full words */ for (i = 0; i < small->len-1; i++) { x = small->data[i]; y = big->data[i]; z = 0; for (k = 0; k < MPD_RDIGITS; k++) { xbit = x % 10; x /= 10; ybit = y % 10; y /= 10; if (xbit > 1 || ybit > 1) { goto invalid_operation; } z += (xbit|ybit) ? mpd_pow10[k] : 0; } result->data[i] = z; } /* most significant word of small */ x = small->data[i]; y = big->data[i]; z = 0; mswdigits = mpd_word_digits(x); for (k = 0; k < mswdigits; k++) { xbit = x % 10; x /= 10; ybit = y % 10; y /= 10; if (xbit > 1 || ybit > 1) { goto invalid_operation; } z += (xbit|ybit) ? mpd_pow10[k] : 0; } /* scan for digits > 1 and copy the rest of y */ for (; k < MPD_RDIGITS; k++) { ybit = y % 10; y /= 10; if (ybit > 1) { goto invalid_operation; } z += ybit*mpd_pow10[k]; } result->data[i++] = z; /* scan for digits > 1 and copy the rest of big */ for (; i < big->len; i++) { y = big->data[i]; for (k = 0; k < MPD_RDIGITS; k++) { ybit = y % 10; y /= 10; if (ybit > 1) { goto invalid_operation; } } result->data[i] = big->data[i]; } mpd_clear_flags(result); result->exp = 0; result->len = _mpd_real_size(result->data, big->len); mpd_qresize(result, result->len, status); mpd_setdigits(result); _mpd_cap(result, ctx); return; invalid_operation: mpd_seterror(result, MPD_Invalid_operation, status); } /* * Rotate the coefficient of 'a' by 'b' digits. 'b' must be an integer with * exponent 0. */ void mpd_qrotate(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { uint32_t workstatus = 0; MPD_NEW_STATIC(tmp,0,0,0,0); MPD_NEW_STATIC(big,0,0,0,0); MPD_NEW_STATIC(small,0,0,0,0); mpd_ssize_t n, lshift, rshift; if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(result, a, b, ctx, status)) { return; } } if (b->exp != 0 || mpd_isinfinite(b)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } n = mpd_qget_ssize(b, &workstatus); if (workstatus&MPD_Invalid_operation) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (n > ctx->prec || n < -ctx->prec) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (mpd_isinfinite(a)) { mpd_qcopy(result, a, status); return; } if (n >= 0) { lshift = n; rshift = ctx->prec-n; } else { lshift = ctx->prec+n; rshift = -n; } if (a->digits > ctx->prec) { if (!mpd_qcopy(&tmp, a, status)) { mpd_seterror(result, MPD_Malloc_error, status); goto finish; } _mpd_cap(&tmp, ctx); a = &tmp; } if (!mpd_qshiftl(&big, a, lshift, status)) { mpd_seterror(result, MPD_Malloc_error, status); goto finish; } _mpd_cap(&big, ctx); if (mpd_qshiftr(&small, a, rshift, status) == MPD_UINT_MAX) { mpd_seterror(result, MPD_Malloc_error, status); goto finish; } _mpd_qadd(result, &big, &small, ctx, status); finish: mpd_del(&tmp); mpd_del(&big); mpd_del(&small); } /* * b must be an integer with exponent 0 and in the range +-2*(emax + prec). * XXX: In my opinion +-(2*emax + prec) would be more sensible. * The result is a with the value of b added to its exponent. */ void mpd_qscaleb(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { uint32_t workstatus = 0; mpd_uint_t n, maxjump; #ifndef LEGACY_COMPILER int64_t exp; #else mpd_uint_t x; int x_sign, n_sign; mpd_ssize_t exp; #endif if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(result, a, b, ctx, status)) { return; } } if (b->exp != 0 || mpd_isinfinite(b)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } n = mpd_qabs_uint(b, &workstatus); /* the spec demands this */ maxjump = 2 * (mpd_uint_t)(ctx->emax + ctx->prec); if (n > maxjump || workstatus&MPD_Invalid_operation) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (mpd_isinfinite(a)) { mpd_qcopy(result, a, status); return; } #ifndef LEGACY_COMPILER exp = a->exp + (int64_t)n * mpd_arith_sign(b); exp = (exp > MPD_EXP_INF) ? MPD_EXP_INF : exp; exp = (exp < MPD_EXP_CLAMP) ? MPD_EXP_CLAMP : exp; #else x = (a->exp < 0) ? -a->exp : a->exp; x_sign = (a->exp < 0) ? 1 : 0; n_sign = mpd_isnegative(b) ? 1 : 0; if (x_sign == n_sign) { x = x + n; if (x < n) x = MPD_UINT_MAX; } else { x_sign = (x >= n) ? x_sign : n_sign; x = (x >= n) ? x - n : n - x; } if (!x_sign && x > MPD_EXP_INF) x = MPD_EXP_INF; if (x_sign && x > -MPD_EXP_CLAMP) x = -MPD_EXP_CLAMP; exp = x_sign ? -((mpd_ssize_t)x) : (mpd_ssize_t)x; #endif mpd_qcopy(result, a, status); result->exp = (mpd_ssize_t)exp; mpd_qfinalize(result, ctx, status); } /* * Shift the coefficient by n digits, positive n is a left shift. In the case * of a left shift, the result is decapitated to fit the context precision. If * you don't want that, use mpd_shiftl(). */ void mpd_qshiftn(mpd_t *result, const mpd_t *a, mpd_ssize_t n, const mpd_context_t *ctx, uint32_t *status) { if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } mpd_qcopy(result, a, status); return; } if (n >= 0 && n <= ctx->prec) { mpd_qshiftl(result, a, n, status); _mpd_cap(result, ctx); } else if (n < 0 && n >= -ctx->prec) { if (!mpd_qcopy(result, a, status)) { return; } _mpd_cap(result, ctx); mpd_qshiftr_inplace(result, -n); } else { mpd_seterror(result, MPD_Invalid_operation, status); } } /* * Same as mpd_shiftn(), but the shift is specified by the decimal b, which * must be an integer with a zero exponent. Infinities remain infinities. */ void mpd_qshift(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { uint32_t workstatus = 0; mpd_ssize_t n; if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(result, a, b, ctx, status)) { return; } } if (b->exp != 0 || mpd_isinfinite(b)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } n = mpd_qget_ssize(b, &workstatus); if (workstatus&MPD_Invalid_operation) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (n > ctx->prec || n < -ctx->prec) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (mpd_isinfinite(a)) { mpd_qcopy(result, a, status); return; } if (n >= 0) { mpd_qshiftl(result, a, n, status); _mpd_cap(result, ctx); } else { if (!mpd_qcopy(result, a, status)) { return; } _mpd_cap(result, ctx); mpd_qshiftr_inplace(result, -n); } } /* Logical Xor */ void mpd_qxor(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { const mpd_t *big = a, *small = b; mpd_uint_t x, y, z, xbit, ybit; int k, mswdigits; mpd_ssize_t i; if (mpd_isspecial(a) || mpd_isspecial(b) || mpd_isnegative(a) || mpd_isnegative(b) || a->exp != 0 || b->exp != 0) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (b->digits > a->digits) { big = b; small = a; } if (!mpd_qresize(result, big->len, status)) { return; } /* full words */ for (i = 0; i < small->len-1; i++) { x = small->data[i]; y = big->data[i]; z = 0; for (k = 0; k < MPD_RDIGITS; k++) { xbit = x % 10; x /= 10; ybit = y % 10; y /= 10; if (xbit > 1 || ybit > 1) { goto invalid_operation; } z += (xbit^ybit) ? mpd_pow10[k] : 0; } result->data[i] = z; } /* most significant word of small */ x = small->data[i]; y = big->data[i]; z = 0; mswdigits = mpd_word_digits(x); for (k = 0; k < mswdigits; k++) { xbit = x % 10; x /= 10; ybit = y % 10; y /= 10; if (xbit > 1 || ybit > 1) { goto invalid_operation; } z += (xbit^ybit) ? mpd_pow10[k] : 0; } /* scan for digits > 1 and copy the rest of y */ for (; k < MPD_RDIGITS; k++) { ybit = y % 10; y /= 10; if (ybit > 1) { goto invalid_operation; } z += ybit*mpd_pow10[k]; } result->data[i++] = z; /* scan for digits > 1 and copy the rest of big */ for (; i < big->len; i++) { y = big->data[i]; for (k = 0; k < MPD_RDIGITS; k++) { ybit = y % 10; y /= 10; if (ybit > 1) { goto invalid_operation; } } result->data[i] = big->data[i]; } mpd_clear_flags(result); result->exp = 0; result->len = _mpd_real_size(result->data, big->len); mpd_qresize(result, result->len, status); mpd_setdigits(result); _mpd_cap(result, ctx); return; invalid_operation: mpd_seterror(result, MPD_Invalid_operation, status); } /******************************************************************************/ /* Arithmetic operations */ /******************************************************************************/ /* * The absolute value of a. If a is negative, the result is the same * as the result of the minus operation. Otherwise, the result is the * result of the plus operation. */ void mpd_qabs(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } } if (mpd_isnegative(a)) { mpd_qminus(result, a, ctx, status); } else { mpd_qplus(result, a, ctx, status); } } static inline void _mpd_ptrswap(mpd_t **a, mpd_t **b) { mpd_t *t = *a; *a = *b; *b = t; } /* Add or subtract infinities. */ static void _mpd_qaddsub_inf(mpd_t *result, const mpd_t *a, const mpd_t *b, uint8_t sign_b, uint32_t *status) { if (mpd_isinfinite(a)) { if (mpd_sign(a) != sign_b && mpd_isinfinite(b)) { mpd_seterror(result, MPD_Invalid_operation, status); } else { mpd_setspecial(result, mpd_sign(a), MPD_INF); } return; } assert(mpd_isinfinite(b)); mpd_setspecial(result, sign_b, MPD_INF); } /* Add or subtract non-special numbers. */ static void _mpd_qaddsub(mpd_t *result, const mpd_t *a, const mpd_t *b, uint8_t sign_b, const mpd_context_t *ctx, uint32_t *status) { mpd_t *big, *small; MPD_NEW_STATIC(big_aligned,0,0,0,0); MPD_NEW_CONST(tiny,0,0,1,1,1,1); mpd_uint_t carry; mpd_ssize_t newsize, shift; mpd_ssize_t exp, i; int swap = 0; /* compare exponents */ big = (mpd_t *)a; small = (mpd_t *)b; if (big->exp != small->exp) { if (small->exp > big->exp) { _mpd_ptrswap(&big, &small); swap++; } /* align the coefficients */ if (!mpd_iszerocoeff(big)) { exp = big->exp - 1; exp += (big->digits > ctx->prec) ? 0 : big->digits-ctx->prec-1; if (mpd_adjexp(small) < exp) { /* * Avoid huge shifts by substituting a value for small that is * guaranteed to produce the same results. * * adjexp(small) < exp if and only if: * * bdigits <= prec AND * bdigits+shift >= prec+2+sdigits AND * exp = bexp+bdigits-prec-2 * * 1234567000000000 -> bdigits + shift * ----------XX1234 -> sdigits * ----------X1 -> tiny-digits * |- prec -| * * OR * * bdigits > prec AND * shift > sdigits AND * exp = bexp-1 * * 1234567892100000 -> bdigits + shift * ----------XX1234 -> sdigits * ----------X1 -> tiny-digits * |- prec -| * * If tiny is zero, adding or subtracting is a no-op. * Otherwise, adding tiny generates a non-zero digit either * below the rounding digit or the least significant digit * of big. When subtracting, tiny is in the same position as * the carry that would be generated by subtracting sdigits. */ mpd_copy_flags(&tiny, small); tiny.exp = exp; tiny.digits = 1; tiny.len = 1; tiny.data[0] = mpd_iszerocoeff(small) ? 0 : 1; small = &tiny; } /* This cannot wrap: the difference is positive and <= maxprec */ shift = big->exp - small->exp; if (!mpd_qshiftl(&big_aligned, big, shift, status)) { mpd_seterror(result, MPD_Malloc_error, status); goto finish; } big = &big_aligned; } } result->exp = small->exp; /* compare length of coefficients */ if (big->len < small->len) { _mpd_ptrswap(&big, &small); swap++; } newsize = big->len; if (!mpd_qresize(result, newsize, status)) { goto finish; } if (mpd_sign(a) == sign_b) { carry = _mpd_baseadd(result->data, big->data, small->data, big->len, small->len); if (carry) { newsize = big->len + 1; if (!mpd_qresize(result, newsize, status)) { goto finish; } result->data[newsize-1] = carry; } result->len = newsize; mpd_set_flags(result, sign_b); } else { if (big->len == small->len) { for (i=big->len-1; i >= 0; --i) { if (big->data[i] != small->data[i]) { if (big->data[i] < small->data[i]) { _mpd_ptrswap(&big, &small); swap++; } break; } } } _mpd_basesub(result->data, big->data, small->data, big->len, small->len); newsize = _mpd_real_size(result->data, big->len); /* resize to smaller cannot fail */ (void)mpd_qresize(result, newsize, status); result->len = newsize; sign_b = (swap & 1) ? sign_b : mpd_sign(a); mpd_set_flags(result, sign_b); if (mpd_iszerocoeff(result)) { mpd_set_positive(result); if (ctx->round == MPD_ROUND_FLOOR) { mpd_set_negative(result); } } } mpd_setdigits(result); finish: mpd_del(&big_aligned); } /* Add a and b. No specials, no finalizing. */ static void _mpd_qadd(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { _mpd_qaddsub(result, a, b, mpd_sign(b), ctx, status); } /* Subtract b from a. No specials, no finalizing. */ static void _mpd_qsub(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { _mpd_qaddsub(result, a, b, !mpd_sign(b), ctx, status); } /* Add a and b. */ void mpd_qadd(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(result, a, b, ctx, status)) { return; } _mpd_qaddsub_inf(result, a, b, mpd_sign(b), status); return; } _mpd_qaddsub(result, a, b, mpd_sign(b), ctx, status); mpd_qfinalize(result, ctx, status); } /* Add a and b. Set NaN/Invalid_operation if the result is inexact. */ static void _mpd_qadd_exact(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { uint32_t workstatus = 0; mpd_qadd(result, a, b, ctx, &workstatus); *status |= workstatus; if (workstatus & (MPD_Inexact|MPD_Rounded|MPD_Clamped)) { mpd_seterror(result, MPD_Invalid_operation, status); } } /* Subtract b from a. */ void mpd_qsub(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(result, a, b, ctx, status)) { return; } _mpd_qaddsub_inf(result, a, b, !mpd_sign(b), status); return; } _mpd_qaddsub(result, a, b, !mpd_sign(b), ctx, status); mpd_qfinalize(result, ctx, status); } /* Subtract b from a. Set NaN/Invalid_operation if the result is inexact. */ static void _mpd_qsub_exact(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { uint32_t workstatus = 0; mpd_qsub(result, a, b, ctx, &workstatus); *status |= workstatus; if (workstatus & (MPD_Inexact|MPD_Rounded|MPD_Clamped)) { mpd_seterror(result, MPD_Invalid_operation, status); } } /* Add decimal and mpd_ssize_t. */ void mpd_qadd_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t maxcontext; MPD_NEW_STATIC(bb,0,0,0,0); mpd_maxcontext(&maxcontext); mpd_qsset_ssize(&bb, b, &maxcontext, status); mpd_qadd(result, a, &bb, ctx, status); mpd_del(&bb); } /* Add decimal and mpd_uint_t. */ void mpd_qadd_uint(mpd_t *result, const mpd_t *a, mpd_uint_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t maxcontext; MPD_NEW_STATIC(bb,0,0,0,0); mpd_maxcontext(&maxcontext); mpd_qsset_uint(&bb, b, &maxcontext, status); mpd_qadd(result, a, &bb, ctx, status); mpd_del(&bb); } /* Subtract mpd_ssize_t from decimal. */ void mpd_qsub_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t maxcontext; MPD_NEW_STATIC(bb,0,0,0,0); mpd_maxcontext(&maxcontext); mpd_qsset_ssize(&bb, b, &maxcontext, status); mpd_qsub(result, a, &bb, ctx, status); mpd_del(&bb); } /* Subtract mpd_uint_t from decimal. */ void mpd_qsub_uint(mpd_t *result, const mpd_t *a, mpd_uint_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t maxcontext; MPD_NEW_STATIC(bb,0,0,0,0); mpd_maxcontext(&maxcontext); mpd_qsset_uint(&bb, b, &maxcontext, status); mpd_qsub(result, a, &bb, ctx, status); mpd_del(&bb); } /* Add decimal and int32_t. */ void mpd_qadd_i32(mpd_t *result, const mpd_t *a, int32_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qadd_ssize(result, a, b, ctx, status); } /* Add decimal and uint32_t. */ void mpd_qadd_u32(mpd_t *result, const mpd_t *a, uint32_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qadd_uint(result, a, b, ctx, status); } #ifdef CONFIG_64 /* Add decimal and int64_t. */ void mpd_qadd_i64(mpd_t *result, const mpd_t *a, int64_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qadd_ssize(result, a, b, ctx, status); } /* Add decimal and uint64_t. */ void mpd_qadd_u64(mpd_t *result, const mpd_t *a, uint64_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qadd_uint(result, a, b, ctx, status); } #endif /* Subtract int32_t from decimal. */ void mpd_qsub_i32(mpd_t *result, const mpd_t *a, int32_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qsub_ssize(result, a, b, ctx, status); } /* Subtract uint32_t from decimal. */ void mpd_qsub_u32(mpd_t *result, const mpd_t *a, uint32_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qsub_uint(result, a, b, ctx, status); } #ifdef CONFIG_64 /* Subtract int64_t from decimal. */ void mpd_qsub_i64(mpd_t *result, const mpd_t *a, int64_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qsub_ssize(result, a, b, ctx, status); } /* Subtract uint64_t from decimal. */ void mpd_qsub_u64(mpd_t *result, const mpd_t *a, uint64_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qsub_uint(result, a, b, ctx, status); } #endif /* Divide infinities. */ static void _mpd_qdiv_inf(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { if (mpd_isinfinite(a)) { if (mpd_isinfinite(b)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } mpd_setspecial(result, mpd_sign(a)^mpd_sign(b), MPD_INF); return; } assert(mpd_isinfinite(b)); _settriple(result, mpd_sign(a)^mpd_sign(b), 0, mpd_etiny(ctx)); *status |= MPD_Clamped; } enum {NO_IDEAL_EXP, SET_IDEAL_EXP}; /* Divide a by b. */ static void _mpd_qdiv(int action, mpd_t *q, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { MPD_NEW_STATIC(aligned,0,0,0,0); mpd_uint_t ld; mpd_ssize_t shift, exp, tz; mpd_ssize_t newsize; mpd_ssize_t ideal_exp; mpd_uint_t rem; uint8_t sign_a = mpd_sign(a); uint8_t sign_b = mpd_sign(b); if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(q, a, b, ctx, status)) { return; } _mpd_qdiv_inf(q, a, b, ctx, status); return; } if (mpd_iszerocoeff(b)) { if (mpd_iszerocoeff(a)) { mpd_seterror(q, MPD_Division_undefined, status); } else { mpd_setspecial(q, sign_a^sign_b, MPD_INF); *status |= MPD_Division_by_zero; } return; } if (mpd_iszerocoeff(a)) { exp = a->exp - b->exp; _settriple(q, sign_a^sign_b, 0, exp); mpd_qfinalize(q, ctx, status); return; } shift = (b->digits - a->digits) + ctx->prec + 1; ideal_exp = a->exp - b->exp; exp = ideal_exp - shift; if (shift > 0) { if (!mpd_qshiftl(&aligned, a, shift, status)) { mpd_seterror(q, MPD_Malloc_error, status); goto finish; } a = &aligned; } else if (shift < 0) { shift = -shift; if (!mpd_qshiftl(&aligned, b, shift, status)) { mpd_seterror(q, MPD_Malloc_error, status); goto finish; } b = &aligned; } newsize = a->len - b->len + 1; if ((q != b && q != a) || (q == b && newsize > b->len)) { if (!mpd_qresize(q, newsize, status)) { mpd_seterror(q, MPD_Malloc_error, status); goto finish; } } if (b->len == 1) { rem = _mpd_shortdiv(q->data, a->data, a->len, b->data[0]); } else if (b->len <= MPD_NEWTONDIV_CUTOFF) { int ret = _mpd_basedivmod(q->data, NULL, a->data, b->data, a->len, b->len); if (ret < 0) { mpd_seterror(q, MPD_Malloc_error, status); goto finish; } rem = ret; } else { MPD_NEW_STATIC(r,0,0,0,0); _mpd_base_ndivmod(q, &r, a, b, status); if (mpd_isspecial(q) || mpd_isspecial(&r)) { mpd_setspecial(q, MPD_POS, MPD_NAN); mpd_del(&r); goto finish; } rem = !mpd_iszerocoeff(&r); mpd_del(&r); newsize = q->len; } newsize = _mpd_real_size(q->data, newsize); /* resize to smaller cannot fail */ mpd_qresize(q, newsize, status); mpd_set_flags(q, sign_a^sign_b); q->len = newsize; mpd_setdigits(q); shift = ideal_exp - exp; if (rem) { ld = mpd_lsd(q->data[0]); if (ld == 0 || ld == 5) { q->data[0] += 1; } } else if (action == SET_IDEAL_EXP && shift > 0) { tz = mpd_trail_zeros(q); shift = (tz > shift) ? shift : tz; mpd_qshiftr_inplace(q, shift); exp += shift; } q->exp = exp; finish: mpd_del(&aligned); mpd_qfinalize(q, ctx, status); } /* Divide a by b. */ void mpd_qdiv(mpd_t *q, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { _mpd_qdiv(SET_IDEAL_EXP, q, a, b, ctx, status); } /* Internal function. */ static void _mpd_qdivmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { MPD_NEW_STATIC(aligned,0,0,0,0); mpd_ssize_t qsize, rsize; mpd_ssize_t ideal_exp, expdiff, shift; uint8_t sign_a = mpd_sign(a); uint8_t sign_ab = mpd_sign(a)^mpd_sign(b); ideal_exp = (a->exp > b->exp) ? b->exp : a->exp; if (mpd_iszerocoeff(a)) { if (!mpd_qcopy(r, a, status)) { goto nanresult; /* GCOV_NOT_REACHED */ } r->exp = ideal_exp; _settriple(q, sign_ab, 0, 0); return; } expdiff = mpd_adjexp(a) - mpd_adjexp(b); if (expdiff < 0) { if (a->exp > b->exp) { /* positive and less than b->digits - a->digits */ shift = a->exp - b->exp; if (!mpd_qshiftl(r, a, shift, status)) { goto nanresult; } r->exp = ideal_exp; } else { if (!mpd_qcopy(r, a, status)) { goto nanresult; } } _settriple(q, sign_ab, 0, 0); return; } if (expdiff > ctx->prec) { *status |= MPD_Division_impossible; goto nanresult; } /* * At this point we have: * (1) 0 <= a->exp + a->digits - b->exp - b->digits <= prec * (2) a->exp - b->exp >= b->digits - a->digits * (3) a->exp - b->exp <= prec + b->digits - a->digits */ if (a->exp != b->exp) { shift = a->exp - b->exp; if (shift > 0) { /* by (3), after the shift a->digits <= prec + b->digits */ if (!mpd_qshiftl(&aligned, a, shift, status)) { goto nanresult; } a = &aligned; } else { shift = -shift; /* by (2), after the shift b->digits <= a->digits */ if (!mpd_qshiftl(&aligned, b, shift, status)) { goto nanresult; } b = &aligned; } } qsize = a->len - b->len + 1; if (!(q == a && qsize < a->len) && !(q == b && qsize < b->len)) { if (!mpd_qresize(q, qsize, status)) { goto nanresult; } } rsize = b->len; if (!(r == a && rsize < a->len)) { if (!mpd_qresize(r, rsize, status)) { goto nanresult; } } if (b->len == 1) { if (a->len == 1) { _mpd_div_word(&q->data[0], &r->data[0], a->data[0], b->data[0]); } else { r->data[0] = _mpd_shortdiv(q->data, a->data, a->len, b->data[0]); } } else if (b->len <= MPD_NEWTONDIV_CUTOFF) { int ret; ret = _mpd_basedivmod(q->data, r->data, a->data, b->data, a->len, b->len); if (ret == -1) { *status |= MPD_Malloc_error; goto nanresult; } } else { _mpd_base_ndivmod(q, r, a, b, status); if (mpd_isspecial(q) || mpd_isspecial(r)) { goto nanresult; } qsize = q->len; rsize = r->len; } qsize = _mpd_real_size(q->data, qsize); /* resize to smaller cannot fail */ mpd_qresize(q, qsize, status); q->len = qsize; mpd_setdigits(q); mpd_set_flags(q, sign_ab); q->exp = 0; if (q->digits > ctx->prec) { *status |= MPD_Division_impossible; goto nanresult; } rsize = _mpd_real_size(r->data, rsize); /* resize to smaller cannot fail */ mpd_qresize(r, rsize, status); r->len = rsize; mpd_setdigits(r); mpd_set_flags(r, sign_a); r->exp = ideal_exp; out: mpd_del(&aligned); return; nanresult: mpd_setspecial(q, MPD_POS, MPD_NAN); mpd_setspecial(r, MPD_POS, MPD_NAN); goto out; } /* Integer division with remainder. */ void mpd_qdivmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { uint8_t sign = mpd_sign(a)^mpd_sign(b); if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(q, a, b, ctx, status)) { mpd_qcopy(r, q, status); return; } if (mpd_isinfinite(a)) { if (mpd_isinfinite(b)) { mpd_setspecial(q, MPD_POS, MPD_NAN); } else { mpd_setspecial(q, sign, MPD_INF); } mpd_setspecial(r, MPD_POS, MPD_NAN); *status |= MPD_Invalid_operation; return; } if (mpd_isinfinite(b)) { if (!mpd_qcopy(r, a, status)) { mpd_seterror(q, MPD_Malloc_error, status); return; } mpd_qfinalize(r, ctx, status); _settriple(q, sign, 0, 0); return; } /* debug */ abort(); /* GCOV_NOT_REACHED */ } if (mpd_iszerocoeff(b)) { if (mpd_iszerocoeff(a)) { mpd_setspecial(q, MPD_POS, MPD_NAN); mpd_setspecial(r, MPD_POS, MPD_NAN); *status |= MPD_Division_undefined; } else { mpd_setspecial(q, sign, MPD_INF); mpd_setspecial(r, MPD_POS, MPD_NAN); *status |= (MPD_Division_by_zero|MPD_Invalid_operation); } return; } _mpd_qdivmod(q, r, a, b, ctx, status); mpd_qfinalize(q, ctx, status); mpd_qfinalize(r, ctx, status); } void mpd_qdivint(mpd_t *q, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { MPD_NEW_STATIC(r,0,0,0,0); uint8_t sign = mpd_sign(a)^mpd_sign(b); if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(q, a, b, ctx, status)) { return; } if (mpd_isinfinite(a) && mpd_isinfinite(b)) { mpd_seterror(q, MPD_Invalid_operation, status); return; } if (mpd_isinfinite(a)) { mpd_setspecial(q, sign, MPD_INF); return; } if (mpd_isinfinite(b)) { _settriple(q, sign, 0, 0); return; } /* debug */ abort(); /* GCOV_NOT_REACHED */ } if (mpd_iszerocoeff(b)) { if (mpd_iszerocoeff(a)) { mpd_seterror(q, MPD_Division_undefined, status); } else { mpd_setspecial(q, sign, MPD_INF); *status |= MPD_Division_by_zero; } return; } _mpd_qdivmod(q, &r, a, b, ctx, status); mpd_del(&r); mpd_qfinalize(q, ctx, status); } /* Divide decimal by mpd_ssize_t. */ void mpd_qdiv_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t maxcontext; MPD_NEW_STATIC(bb,0,0,0,0); mpd_maxcontext(&maxcontext); mpd_qsset_ssize(&bb, b, &maxcontext, status); mpd_qdiv(result, a, &bb, ctx, status); mpd_del(&bb); } /* Divide decimal by mpd_uint_t. */ void mpd_qdiv_uint(mpd_t *result, const mpd_t *a, mpd_uint_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t maxcontext; MPD_NEW_STATIC(bb,0,0,0,0); mpd_maxcontext(&maxcontext); mpd_qsset_uint(&bb, b, &maxcontext, status); mpd_qdiv(result, a, &bb, ctx, status); mpd_del(&bb); } /* Divide decimal by int32_t. */ void mpd_qdiv_i32(mpd_t *result, const mpd_t *a, int32_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qdiv_ssize(result, a, b, ctx, status); } /* Divide decimal by uint32_t. */ void mpd_qdiv_u32(mpd_t *result, const mpd_t *a, uint32_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qdiv_uint(result, a, b, ctx, status); } #ifdef CONFIG_64 /* Divide decimal by int64_t. */ void mpd_qdiv_i64(mpd_t *result, const mpd_t *a, int64_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qdiv_ssize(result, a, b, ctx, status); } /* Divide decimal by uint64_t. */ void mpd_qdiv_u64(mpd_t *result, const mpd_t *a, uint64_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qdiv_uint(result, a, b, ctx, status); } #endif /* Pad the result with trailing zeros if it has fewer digits than prec. */ static void _mpd_zeropad(mpd_t *result, const mpd_context_t *ctx, uint32_t *status) { if (!mpd_isspecial(result) && !mpd_iszero(result) && result->digits < ctx->prec) { mpd_ssize_t shift = ctx->prec - result->digits; mpd_qshiftl(result, result, shift, status); result->exp -= shift; } } /* Check if the result is guaranteed to be one. */ static int _mpd_qexp_check_one(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { MPD_NEW_CONST(lim,0,-(ctx->prec+1),1,1,1,9); MPD_NEW_SHARED(aa, a); mpd_set_positive(&aa); /* abs(a) <= 9 * 10**(-prec-1) */ if (_mpd_cmp(&aa, &lim) <= 0) { _settriple(result, 0, 1, 0); *status |= MPD_Rounded|MPD_Inexact; return 1; } return 0; } /* * Get the number of iterations for the Horner scheme in _mpd_qexp(). */ static inline mpd_ssize_t _mpd_get_exp_iterations(const mpd_t *r, mpd_ssize_t p) { mpd_ssize_t log10pbyr; /* lower bound for log10(p / abs(r)) */ mpd_ssize_t n; assert(p >= 10); assert(!mpd_iszero(r)); assert(-p < mpd_adjexp(r) && mpd_adjexp(r) <= -1); #ifdef CONFIG_64 if (p > (mpd_ssize_t)(1ULL<<52)) { return MPD_SSIZE_MAX; } #endif /* * Lower bound for log10(p / abs(r)): adjexp(p) - (adjexp(r) + 1) * At this point (for CONFIG_64, CONFIG_32 is not problematic): * 1) 10 <= p <= 2**52 * 2) -p < adjexp(r) <= -1 * 3) 1 <= log10pbyr <= 2**52 + 14 */ log10pbyr = (mpd_word_digits(p)-1) - (mpd_adjexp(r)+1); /* * The numerator in the paper is 1.435 * p - 1.182, calculated * exactly. We compensate for rounding errors by using 1.43503. * ACL2 proofs: * 1) exp-iter-approx-lower-bound: The term below evaluated * in 53-bit floating point arithmetic is greater than or * equal to the exact term used in the paper. * 2) exp-iter-approx-upper-bound: The term below is less than * or equal to 3/2 * p <= 3/2 * 2**52. */ n = (mpd_ssize_t)ceil((1.43503*(double)p - 1.182) / (double)log10pbyr); return n >= 3 ? n : 3; } /* * Internal function, specials have been dealt with. Apart from Overflow * and Underflow, two cases must be considered for the error of the result: * * 1) abs(a) <= 9 * 10**(-prec-1) ==> result == 1 * * Absolute error: abs(1 - e**x) < 10**(-prec) * ------------------------------------------- * * 2) abs(a) > 9 * 10**(-prec-1) * * Relative error: abs(result - e**x) < 0.5 * 10**(-prec) * e**x * ------------------------------------------------------------- * * The algorithm is from Hull&Abrham, Variable Precision Exponential Function, * ACM Transactions on Mathematical Software, Vol. 12, No. 2, June 1986. * * Main differences: * * - The number of iterations for the Horner scheme is calculated using * 53-bit floating point arithmetic. * * - In the error analysis for ER (relative error accumulated in the * evaluation of the truncated series) the reduced operand r may * have any number of digits. * ACL2 proof: exponent-relative-error * * - The analysis for early abortion has been adapted for the mpd_t * ranges. */ static void _mpd_qexp(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t workctx; MPD_NEW_STATIC(tmp,0,0,0,0); MPD_NEW_STATIC(sum,0,0,0,0); MPD_NEW_CONST(word,0,0,1,1,1,1); mpd_ssize_t j, n, t; assert(!mpd_isspecial(a)); if (mpd_iszerocoeff(a)) { _settriple(result, MPD_POS, 1, 0); return; } /* * We are calculating e^x = e^(r*10^t) = (e^r)^(10^t), where abs(r) < 1 and t >= 0. * * If t > 0, we have: * * (1) 0.1 <= r < 1, so e^0.1 <= e^r. If t > MAX_T, overflow occurs: * * MAX-EMAX+1 < log10(e^(0.1*10*t)) <= log10(e^(r*10^t)) < adjexp(e^(r*10^t))+1 * * (2) -1 < r <= -0.1, so e^r <= e^-0.1. If t > MAX_T, underflow occurs: * * adjexp(e^(r*10^t)) <= log10(e^(r*10^t)) <= log10(e^(-0.1*10^t)) < MIN-ETINY */ #if defined(CONFIG_64) #define MPD_EXP_MAX_T 19 #elif defined(CONFIG_32) #define MPD_EXP_MAX_T 10 #endif t = a->digits + a->exp; t = (t > 0) ? t : 0; if (t > MPD_EXP_MAX_T) { if (mpd_ispositive(a)) { mpd_setspecial(result, MPD_POS, MPD_INF); *status |= MPD_Overflow|MPD_Inexact|MPD_Rounded; } else { _settriple(result, MPD_POS, 0, mpd_etiny(ctx)); *status |= (MPD_Inexact|MPD_Rounded|MPD_Subnormal| MPD_Underflow|MPD_Clamped); } return; } /* abs(a) <= 9 * 10**(-prec-1) */ if (_mpd_qexp_check_one(result, a, ctx, status)) { return; } mpd_maxcontext(&workctx); workctx.prec = ctx->prec + t + 2; workctx.prec = (workctx.prec < 10) ? 10 : workctx.prec; workctx.round = MPD_ROUND_HALF_EVEN; if (!mpd_qcopy(result, a, status)) { return; } result->exp -= t; /* * At this point: * 1) 9 * 10**(-prec-1) < abs(a) * 2) 9 * 10**(-prec-t-1) < abs(r) * 3) log10(9) - prec - t - 1 < log10(abs(r)) < adjexp(abs(r)) + 1 * 4) - prec - t - 2 < adjexp(abs(r)) <= -1 */ n = _mpd_get_exp_iterations(result, workctx.prec); if (n == MPD_SSIZE_MAX) { mpd_seterror(result, MPD_Invalid_operation, status); /* GCOV_UNLIKELY */ return; /* GCOV_UNLIKELY */ } _settriple(&sum, MPD_POS, 1, 0); for (j = n-1; j >= 1; j--) { word.data[0] = j; mpd_setdigits(&word); mpd_qdiv(&tmp, result, &word, &workctx, &workctx.status); mpd_qfma(&sum, &sum, &tmp, &one, &workctx, &workctx.status); } #ifdef CONFIG_64 _mpd_qpow_uint(result, &sum, mpd_pow10[t], MPD_POS, &workctx, status); #else if (t <= MPD_MAX_POW10) { _mpd_qpow_uint(result, &sum, mpd_pow10[t], MPD_POS, &workctx, status); } else { t -= MPD_MAX_POW10; _mpd_qpow_uint(&tmp, &sum, mpd_pow10[MPD_MAX_POW10], MPD_POS, &workctx, status); _mpd_qpow_uint(result, &tmp, mpd_pow10[t], MPD_POS, &workctx, status); } #endif mpd_del(&tmp); mpd_del(&sum); *status |= (workctx.status&MPD_Errors); *status |= (MPD_Inexact|MPD_Rounded); } /* exp(a) */ void mpd_qexp(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t workctx; if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } if (mpd_isnegative(a)) { _settriple(result, MPD_POS, 0, 0); } else { mpd_setspecial(result, MPD_POS, MPD_INF); } return; } if (mpd_iszerocoeff(a)) { _settriple(result, MPD_POS, 1, 0); return; } workctx = *ctx; workctx.round = MPD_ROUND_HALF_EVEN; if (ctx->allcr) { MPD_NEW_STATIC(t1, 0,0,0,0); MPD_NEW_STATIC(t2, 0,0,0,0); MPD_NEW_STATIC(ulp, 0,0,0,0); MPD_NEW_STATIC(aa, 0,0,0,0); mpd_ssize_t prec; mpd_ssize_t ulpexp; uint32_t workstatus; if (result == a) { if (!mpd_qcopy(&aa, a, status)) { mpd_seterror(result, MPD_Malloc_error, status); return; } a = &aa; } workctx.clamp = 0; prec = ctx->prec + 3; while (1) { workctx.prec = prec; workstatus = 0; _mpd_qexp(result, a, &workctx, &workstatus); *status |= workstatus; ulpexp = result->exp + result->digits - workctx.prec; if (workstatus & MPD_Underflow) { /* The effective work precision is result->digits. */ ulpexp = result->exp; } _ssettriple(&ulp, MPD_POS, 1, ulpexp); /* * At this point [1]: * 1) abs(result - e**x) < 0.5 * 10**(-prec) * e**x * 2) result - ulp < e**x < result + ulp * 3) result - ulp < result < result + ulp * * If round(result-ulp)==round(result+ulp), then * round(result)==round(e**x). Therefore the result * is correctly rounded. * * [1] If abs(a) <= 9 * 10**(-prec-1), use the absolute * error for a similar argument. */ workctx.prec = ctx->prec; mpd_qadd(&t1, result, &ulp, &workctx, &workctx.status); mpd_qsub(&t2, result, &ulp, &workctx, &workctx.status); if (mpd_isspecial(result) || mpd_iszerocoeff(result) || mpd_qcmp(&t1, &t2, status) == 0) { workctx.clamp = ctx->clamp; _mpd_zeropad(result, &workctx, status); mpd_check_underflow(result, &workctx, status); mpd_qfinalize(result, &workctx, status); break; } prec += MPD_RDIGITS; } mpd_del(&t1); mpd_del(&t2); mpd_del(&ulp); mpd_del(&aa); } else { _mpd_qexp(result, a, &workctx, status); _mpd_zeropad(result, &workctx, status); mpd_check_underflow(result, &workctx, status); mpd_qfinalize(result, &workctx, status); } } /* Fused multiply-add: (a * b) + c, with a single final rounding. */ void mpd_qfma(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_t *c, const mpd_context_t *ctx, uint32_t *status) { uint32_t workstatus = 0; mpd_t *cc = (mpd_t *)c; if (result == c) { if ((cc = mpd_qncopy(c)) == NULL) { mpd_seterror(result, MPD_Malloc_error, status); return; } } _mpd_qmul(result, a, b, ctx, &workstatus); if (!(workstatus&MPD_Invalid_operation)) { mpd_qadd(result, result, cc, ctx, &workstatus); } if (cc != c) mpd_del(cc); *status |= workstatus; } /* * Schedule the optimal precision increase for the Newton iteration. * v := input operand * z_0 := initial approximation * initprec := natural number such that abs(log(v) - z_0) < 10**-initprec * maxprec := target precision * * For convenience the output klist contains the elements in reverse order: * klist := [k_n-1, ..., k_0], where * 1) k_0 <= initprec and * 2) abs(log(v) - result) < 10**(-2*k_n-1 + 1) <= 10**-maxprec. */ static inline int ln_schedule_prec(mpd_ssize_t klist[MPD_MAX_PREC_LOG2], mpd_ssize_t maxprec, mpd_ssize_t initprec) { mpd_ssize_t k; int i; assert(maxprec >= 2 && initprec >= 2); if (maxprec <= initprec) return -1; i = 0; k = maxprec; do { k = (k+2) / 2; klist[i++] = k; } while (k > initprec); return i-1; } /* The constants have been verified with both decimal.py and mpfr. */ #ifdef CONFIG_64 #if MPD_RDIGITS != 19 #error "mpdecimal.c: MPD_RDIGITS must be 19." #endif static const mpd_uint_t mpd_ln10_data[MPD_MINALLOC_MAX] = { 6983716328982174407ULL, 9089704281976336583ULL, 1515961135648465461ULL, 4416816335727555703ULL, 2900988039194170265ULL, 2307925037472986509ULL, 107598438319191292ULL, 3466624107184669231ULL, 4450099781311469159ULL, 9807828059751193854ULL, 7713456862091670584ULL, 1492198849978748873ULL, 6528728696511086257ULL, 2385392051446341972ULL, 8692180205189339507ULL, 6518769751037497088ULL, 2375253577097505395ULL, 9095610299291824318ULL, 982748238504564801ULL, 5438635917781170543ULL, 7547331541421808427ULL, 752371033310119785ULL, 3171643095059950878ULL, 9785265383207606726ULL, 2932258279850258550ULL, 5497347726624257094ULL, 2976979522110718264ULL, 9221477656763693866ULL, 1979650047149510504ULL, 6674183485704422507ULL, 9702766860595249671ULL, 9278096762712757753ULL, 9314848524948644871ULL, 6826928280848118428ULL, 754403708474699401ULL, 230105703089634572ULL, 1929203337658714166ULL, 7589402567763113569ULL, 4208241314695689016ULL, 2922455440575892572ULL, 9356734206705811364ULL, 2684916746550586856ULL, 644507064800027750ULL, 9476834636167921018ULL, 5659121373450747856ULL, 2835522011480466371ULL, 6470806855677432162ULL, 7141748003688084012ULL, 9619404400222105101ULL, 5504893431493939147ULL, 6674744042432743651ULL, 2287698219886746543ULL, 7773262884616336622ULL, 1985283935053089653ULL, 4680843799894826233ULL, 8168948290720832555ULL, 8067566662873690987ULL, 6248633409525465082ULL, 9829834196778404228ULL, 3524802359972050895ULL, 3327900967572609677ULL, 110148862877297603ULL, 179914546843642076ULL, 2302585092994045684ULL }; #else #if MPD_RDIGITS != 9 #error "mpdecimal.c: MPD_RDIGITS must be 9." #endif static const mpd_uint_t mpd_ln10_data[MPD_MINALLOC_MAX] = { 401682692UL, 708474699UL, 720754403UL, 30896345UL, 602301057UL, 765871416UL, 192920333UL, 763113569UL, 589402567UL, 956890167UL, 82413146UL, 589257242UL, 245544057UL, 811364292UL, 734206705UL, 868569356UL, 167465505UL, 775026849UL, 706480002UL, 18064450UL, 636167921UL, 569476834UL, 734507478UL, 156591213UL, 148046637UL, 283552201UL, 677432162UL, 470806855UL, 880840126UL, 417480036UL, 210510171UL, 940440022UL, 939147961UL, 893431493UL, 436515504UL, 440424327UL, 654366747UL, 821988674UL, 622228769UL, 884616336UL, 537773262UL, 350530896UL, 319852839UL, 989482623UL, 468084379UL, 720832555UL, 168948290UL, 736909878UL, 675666628UL, 546508280UL, 863340952UL, 404228624UL, 834196778UL, 508959829UL, 23599720UL, 967735248UL, 96757260UL, 603332790UL, 862877297UL, 760110148UL, 468436420UL, 401799145UL, 299404568UL, 230258509UL }; #endif /* _mpd_ln10 is used directly for precisions smaller than MINALLOC_MAX*RDIGITS. Otherwise, it serves as the initial approximation for calculating ln(10). */ static const mpd_t _mpd_ln10 = { MPD_STATIC|MPD_CONST_DATA, -(MPD_MINALLOC_MAX*MPD_RDIGITS-1), MPD_MINALLOC_MAX*MPD_RDIGITS, MPD_MINALLOC_MAX, MPD_MINALLOC_MAX, (mpd_uint_t *)mpd_ln10_data }; /* * Set 'result' to log(10). * Ulp error: abs(result - log(10)) < ulp(log(10)) * Relative error: abs(result - log(10)) < 5 * 10**-prec * log(10) * * NOTE: The relative error is not derived from the ulp error, but * calculated separately using the fact that 23/10 < log(10) < 24/10. */ void mpd_qln10(mpd_t *result, mpd_ssize_t prec, uint32_t *status) { mpd_context_t varcontext, maxcontext; MPD_NEW_STATIC(tmp, 0,0,0,0); MPD_NEW_CONST(static10, 0,0,2,1,1,10); mpd_ssize_t klist[MPD_MAX_PREC_LOG2]; mpd_uint_t rnd; mpd_ssize_t shift; int i; assert(prec >= 1); shift = MPD_MINALLOC_MAX*MPD_RDIGITS-prec; shift = shift < 0 ? 0 : shift; rnd = mpd_qshiftr(result, &_mpd_ln10, shift, status); if (rnd == MPD_UINT_MAX) { mpd_seterror(result, MPD_Malloc_error, status); return; } result->exp = -(result->digits-1); mpd_maxcontext(&maxcontext); if (prec < MPD_MINALLOC_MAX*MPD_RDIGITS) { maxcontext.prec = prec; _mpd_apply_round_excess(result, rnd, &maxcontext, status); *status |= (MPD_Inexact|MPD_Rounded); return; } mpd_maxcontext(&varcontext); varcontext.round = MPD_ROUND_TRUNC; i = ln_schedule_prec(klist, prec+2, -result->exp); for (; i >= 0; i--) { varcontext.prec = 2*klist[i]+3; result->flags ^= MPD_NEG; _mpd_qexp(&tmp, result, &varcontext, status); result->flags ^= MPD_NEG; mpd_qmul(&tmp, &static10, &tmp, &varcontext, status); mpd_qsub(&tmp, &tmp, &one, &maxcontext, status); mpd_qadd(result, result, &tmp, &maxcontext, status); if (mpd_isspecial(result)) { break; } } mpd_del(&tmp); maxcontext.prec = prec; mpd_qfinalize(result, &maxcontext, status); } /* * Initial approximations for the ln() iteration. The values have the * following properties (established with both decimal.py and mpfr): * * Index 0 - 400, logarithms of x in [1.00, 5.00]: * abs(lnapprox[i] * 10**-3 - log((i+100)/100)) < 10**-2 * abs(lnapprox[i] * 10**-3 - log((i+1+100)/100)) < 10**-2 * * Index 401 - 899, logarithms of x in (0.500, 0.999]: * abs(-lnapprox[i] * 10**-3 - log((i+100)/1000)) < 10**-2 * abs(-lnapprox[i] * 10**-3 - log((i+1+100)/1000)) < 10**-2 */ static const uint16_t lnapprox[900] = { /* index 0 - 400: log((i+100)/100) * 1000 */ 0, 10, 20, 30, 39, 49, 58, 68, 77, 86, 95, 104, 113, 122, 131, 140, 148, 157, 166, 174, 182, 191, 199, 207, 215, 223, 231, 239, 247, 255, 262, 270, 278, 285, 293, 300, 308, 315, 322, 329, 336, 344, 351, 358, 365, 372, 378, 385, 392, 399, 406, 412, 419, 425, 432, 438, 445, 451, 457, 464, 470, 476, 482, 489, 495, 501, 507, 513, 519, 525, 531, 536, 542, 548, 554, 560, 565, 571, 577, 582, 588, 593, 599, 604, 610, 615, 621, 626, 631, 637, 642, 647, 652, 658, 663, 668, 673, 678, 683, 688, 693, 698, 703, 708, 713, 718, 723, 728, 732, 737, 742, 747, 751, 756, 761, 766, 770, 775, 779, 784, 788, 793, 798, 802, 806, 811, 815, 820, 824, 829, 833, 837, 842, 846, 850, 854, 859, 863, 867, 871, 876, 880, 884, 888, 892, 896, 900, 904, 908, 912, 916, 920, 924, 928, 932, 936, 940, 944, 948, 952, 956, 959, 963, 967, 971, 975, 978, 982, 986, 990, 993, 997, 1001, 1004, 1008, 1012, 1015, 1019, 1022, 1026, 1030, 1033, 1037, 1040, 1044, 1047, 1051, 1054, 1058, 1061, 1065, 1068, 1072, 1075, 1078, 1082, 1085, 1089, 1092, 1095, 1099, 1102, 1105, 1109, 1112, 1115, 1118, 1122, 1125, 1128, 1131, 1135, 1138, 1141, 1144, 1147, 1151, 1154, 1157, 1160, 1163, 1166, 1169, 1172, 1176, 1179, 1182, 1185, 1188, 1191, 1194, 1197, 1200, 1203, 1206, 1209, 1212, 1215, 1218, 1221, 1224, 1227, 1230, 1233, 1235, 1238, 1241, 1244, 1247, 1250, 1253, 1256, 1258, 1261, 1264, 1267, 1270, 1273, 1275, 1278, 1281, 1284, 1286, 1289, 1292, 1295, 1297, 1300, 1303, 1306, 1308, 1311, 1314, 1316, 1319, 1322, 1324, 1327, 1330, 1332, 1335, 1338, 1340, 1343, 1345, 1348, 1351, 1353, 1356, 1358, 1361, 1364, 1366, 1369, 1371, 1374, 1376, 1379, 1381, 1384, 1386, 1389, 1391, 1394, 1396, 1399, 1401, 1404, 1406, 1409, 1411, 1413, 1416, 1418, 1421, 1423, 1426, 1428, 1430, 1433, 1435, 1437, 1440, 1442, 1445, 1447, 1449, 1452, 1454, 1456, 1459, 1461, 1463, 1466, 1468, 1470, 1472, 1475, 1477, 1479, 1482, 1484, 1486, 1488, 1491, 1493, 1495, 1497, 1500, 1502, 1504, 1506, 1509, 1511, 1513, 1515, 1517, 1520, 1522, 1524, 1526, 1528, 1530, 1533, 1535, 1537, 1539, 1541, 1543, 1545, 1548, 1550, 1552, 1554, 1556, 1558, 1560, 1562, 1564, 1567, 1569, 1571, 1573, 1575, 1577, 1579, 1581, 1583, 1585, 1587, 1589, 1591, 1593, 1595, 1597, 1599, 1601, 1603, 1605, 1607, 1609, /* index 401 - 899: -log((i+100)/1000) * 1000 */ 691, 689, 687, 685, 683, 681, 679, 677, 675, 673, 671, 669, 668, 666, 664, 662, 660, 658, 656, 654, 652, 650, 648, 646, 644, 642, 641, 639, 637, 635, 633, 631, 629, 627, 626, 624, 622, 620, 618, 616, 614, 612, 611, 609, 607, 605, 603, 602, 600, 598, 596, 594, 592, 591, 589, 587, 585, 583, 582, 580, 578, 576, 574, 573, 571, 569, 567, 566, 564, 562, 560, 559, 557, 555, 553, 552, 550, 548, 546, 545, 543, 541, 540, 538, 536, 534, 533, 531, 529, 528, 526, 524, 523, 521, 519, 518, 516, 514, 512, 511, 509, 508, 506, 504, 502, 501, 499, 498, 496, 494, 493, 491, 489, 488, 486, 484, 483, 481, 480, 478, 476, 475, 473, 472, 470, 468, 467, 465, 464, 462, 460, 459, 457, 456, 454, 453, 451, 449, 448, 446, 445, 443, 442, 440, 438, 437, 435, 434, 432, 431, 429, 428, 426, 425, 423, 422, 420, 419, 417, 416, 414, 412, 411, 410, 408, 406, 405, 404, 402, 400, 399, 398, 396, 394, 393, 392, 390, 389, 387, 386, 384, 383, 381, 380, 378, 377, 375, 374, 372, 371, 370, 368, 367, 365, 364, 362, 361, 360, 358, 357, 355, 354, 352, 351, 350, 348, 347, 345, 344, 342, 341, 340, 338, 337, 336, 334, 333, 331, 330, 328, 327, 326, 324, 323, 322, 320, 319, 318, 316, 315, 313, 312, 311, 309, 308, 306, 305, 304, 302, 301, 300, 298, 297, 296, 294, 293, 292, 290, 289, 288, 286, 285, 284, 282, 281, 280, 278, 277, 276, 274, 273, 272, 270, 269, 268, 267, 265, 264, 263, 261, 260, 259, 258, 256, 255, 254, 252, 251, 250, 248, 247, 246, 245, 243, 242, 241, 240, 238, 237, 236, 234, 233, 232, 231, 229, 228, 227, 226, 224, 223, 222, 221, 219, 218, 217, 216, 214, 213, 212, 211, 210, 208, 207, 206, 205, 203, 202, 201, 200, 198, 197, 196, 195, 194, 192, 191, 190, 189, 188, 186, 185, 184, 183, 182, 180, 179, 178, 177, 176, 174, 173, 172, 171, 170, 168, 167, 166, 165, 164, 162, 161, 160, 159, 158, 157, 156, 154, 153, 152, 151, 150, 148, 147, 146, 145, 144, 143, 142, 140, 139, 138, 137, 136, 135, 134, 132, 131, 130, 129, 128, 127, 126, 124, 123, 122, 121, 120, 119, 118, 116, 115, 114, 113, 112, 111, 110, 109, 108, 106, 105, 104, 103, 102, 101, 100, 99, 98, 97, 95, 94, 93, 92, 91, 90, 89, 88, 87, 86, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 }; /* * Internal ln() function that does not check for specials, zero or one. * Relative error: abs(result - log(a)) < 0.1 * 10**-prec * abs(log(a)) */ static void _mpd_qln(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t varcontext, maxcontext; mpd_t *z = (mpd_t *) result; MPD_NEW_STATIC(v,0,0,0,0); MPD_NEW_STATIC(vtmp,0,0,0,0); MPD_NEW_STATIC(tmp,0,0,0,0); mpd_ssize_t klist[MPD_MAX_PREC_LOG2]; mpd_ssize_t maxprec, shift, t; mpd_ssize_t a_digits, a_exp; mpd_uint_t dummy, x; int i; assert(!mpd_isspecial(a) && !mpd_iszerocoeff(a)); /* * We are calculating ln(a) = ln(v * 10^t) = ln(v) + t*ln(10), * where 0.5 < v <= 5. */ if (!mpd_qcopy(&v, a, status)) { mpd_seterror(result, MPD_Malloc_error, status); goto finish; } /* Initial approximation: we have at least one non-zero digit */ _mpd_get_msdigits(&dummy, &x, &v, 3); if (x < 10) x *= 10; if (x < 100) x *= 10; x -= 100; /* a may equal z */ a_digits = a->digits; a_exp = a->exp; mpd_minalloc(z); mpd_clear_flags(z); z->data[0] = lnapprox[x]; z->len = 1; z->exp = -3; mpd_setdigits(z); if (x <= 400) { /* Reduce the input operand to 1.00 <= v <= 5.00. Let y = x + 100, * so 100 <= y <= 500. Since y contains the most significant digits * of v, y/100 <= v < (y+1)/100 and abs(z - log(v)) < 10**-2. */ v.exp = -(a_digits - 1); t = a_exp + a_digits - 1; } else { /* Reduce the input operand to 0.500 < v <= 0.999. Let y = x + 100, * so 500 < y <= 999. Since y contains the most significant digits * of v, y/1000 <= v < (y+1)/1000 and abs(z - log(v)) < 10**-2. */ v.exp = -a_digits; t = a_exp + a_digits; mpd_set_negative(z); } mpd_maxcontext(&maxcontext); mpd_maxcontext(&varcontext); varcontext.round = MPD_ROUND_TRUNC; maxprec = ctx->prec + 2; if (t == 0 && (x <= 15 || x >= 800)) { /* 0.900 <= v <= 1.15: Estimate the magnitude of the logarithm. * If ln(v) will underflow, skip the loop. Otherwise, adjust the * precision upwards in order to obtain a sufficient number of * significant digits. * * Case v > 1: * abs((v-1)/10) < abs((v-1)/v) < abs(ln(v)) < abs(v-1) * Case v < 1: * abs(v-1) < abs(ln(v)) < abs((v-1)/v) < abs((v-1)*10) */ int cmp = _mpd_cmp(&v, &one); /* Upper bound (assume v > 1): abs(v-1), unrounded */ _mpd_qsub(&tmp, &v, &one, &maxcontext, &maxcontext.status); if (maxcontext.status & MPD_Errors) { mpd_seterror(result, MPD_Malloc_error, status); goto finish; } if (cmp < 0) { /* v < 1: abs((v-1)*10) */ tmp.exp += 1; } if (mpd_adjexp(&tmp) < mpd_etiny(ctx)) { /* The upper bound is less than etiny: Underflow to zero */ _settriple(result, (cmp<0), 1, mpd_etiny(ctx)-1); goto finish; } /* Lower bound: abs((v-1)/10) or abs(v-1) */ tmp.exp -= 1; if (mpd_adjexp(&tmp) < 0) { /* Absolute error of the loop: abs(z - log(v)) < 10**-p. If * p = ctx->prec+2-adjexp(lower), then the relative error of * the result is (using 10**adjexp(x) <= abs(x)): * * abs(z - log(v)) / abs(log(v)) < 10**-p / abs(log(v)) * <= 10**(-ctx->prec-2) */ maxprec = maxprec - mpd_adjexp(&tmp); } } i = ln_schedule_prec(klist, maxprec, 2); for (; i >= 0; i--) { varcontext.prec = 2*klist[i]+3; z->flags ^= MPD_NEG; _mpd_qexp(&tmp, z, &varcontext, status); z->flags ^= MPD_NEG; if (v.digits > varcontext.prec) { shift = v.digits - varcontext.prec; mpd_qshiftr(&vtmp, &v, shift, status); vtmp.exp += shift; mpd_qmul(&tmp, &vtmp, &tmp, &varcontext, status); } else { mpd_qmul(&tmp, &v, &tmp, &varcontext, status); } mpd_qsub(&tmp, &tmp, &one, &maxcontext, status); mpd_qadd(z, z, &tmp, &maxcontext, status); if (mpd_isspecial(z)) { break; } } /* * Case t == 0: * t * log(10) == 0, the result does not change and the analysis * above applies. If v < 0.900 or v > 1.15, the relative error is * less than 10**(-ctx.prec-1). * Case t != 0: * z := approx(log(v)) * y := approx(log(10)) * p := maxprec = ctx->prec + 2 * Absolute errors: * 1) abs(z - log(v)) < 10**-p * 2) abs(y - log(10)) < 10**-p * The multiplication is exact, so: * 3) abs(t*y - t*log(10)) < t*10**-p * The sum is exact, so: * 4) abs((z + t*y) - (log(v) + t*log(10))) < (abs(t) + 1) * 10**-p * Bounds for log(v) and log(10): * 5) -7/10 < log(v) < 17/10 * 6) 23/10 < log(10) < 24/10 * Using 4), 5), 6) and t != 0, the relative error is: * * 7) relerr < ((abs(t) + 1)*10**-p) / abs(log(v) + t*log(10)) * < 0.5 * 10**(-p + 1) = 0.5 * 10**(-ctx->prec-1) */ mpd_qln10(&v, maxprec+1, status); mpd_qmul_ssize(&tmp, &v, t, &maxcontext, status); mpd_qadd(result, &tmp, z, &maxcontext, status); finish: *status |= (MPD_Inexact|MPD_Rounded); mpd_del(&v); mpd_del(&vtmp); mpd_del(&tmp); } /* ln(a) */ void mpd_qln(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t workctx; mpd_ssize_t adjexp, t; if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } if (mpd_isnegative(a)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } mpd_setspecial(result, MPD_POS, MPD_INF); return; } if (mpd_iszerocoeff(a)) { mpd_setspecial(result, MPD_NEG, MPD_INF); return; } if (mpd_isnegative(a)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (_mpd_cmp(a, &one) == 0) { _settriple(result, MPD_POS, 0, 0); return; } /* * Check if the result will overflow (0 < x, x != 1): * 1) log10(x) < 0 iff adjexp(x) < 0 * 2) 0 < x /\ x <= y ==> adjexp(x) <= adjexp(y) * 3) 0 < x /\ x != 1 ==> 2 * abs(log10(x)) < abs(log(x)) * 4) adjexp(x) <= log10(x) < adjexp(x) + 1 * * Case adjexp(x) >= 0: * 5) 2 * adjexp(x) < abs(log(x)) * Case adjexp(x) > 0: * 6) adjexp(2 * adjexp(x)) <= adjexp(abs(log(x))) * Case adjexp(x) == 0: * mpd_exp_digits(t)-1 == 0 <= emax (the shortcut is not triggered) * * Case adjexp(x) < 0: * 7) 2 * (-adjexp(x) - 1) < abs(log(x)) * Case adjexp(x) < -1: * 8) adjexp(2 * (-adjexp(x) - 1)) <= adjexp(abs(log(x))) * Case adjexp(x) == -1: * mpd_exp_digits(t)-1 == 0 <= emax (the shortcut is not triggered) */ adjexp = mpd_adjexp(a); t = (adjexp < 0) ? -adjexp-1 : adjexp; t *= 2; if (mpd_exp_digits(t)-1 > ctx->emax) { *status |= MPD_Overflow|MPD_Inexact|MPD_Rounded; mpd_setspecial(result, (adjexp<0), MPD_INF); return; } workctx = *ctx; workctx.round = MPD_ROUND_HALF_EVEN; if (ctx->allcr) { MPD_NEW_STATIC(t1, 0,0,0,0); MPD_NEW_STATIC(t2, 0,0,0,0); MPD_NEW_STATIC(ulp, 0,0,0,0); MPD_NEW_STATIC(aa, 0,0,0,0); mpd_ssize_t prec; if (result == a) { if (!mpd_qcopy(&aa, a, status)) { mpd_seterror(result, MPD_Malloc_error, status); return; } a = &aa; } workctx.clamp = 0; prec = ctx->prec + 3; while (1) { workctx.prec = prec; _mpd_qln(result, a, &workctx, status); _ssettriple(&ulp, MPD_POS, 1, result->exp + result->digits-workctx.prec); workctx.prec = ctx->prec; mpd_qadd(&t1, result, &ulp, &workctx, &workctx.status); mpd_qsub(&t2, result, &ulp, &workctx, &workctx.status); if (mpd_isspecial(result) || mpd_iszerocoeff(result) || mpd_qcmp(&t1, &t2, status) == 0) { workctx.clamp = ctx->clamp; mpd_check_underflow(result, &workctx, status); mpd_qfinalize(result, &workctx, status); break; } prec += MPD_RDIGITS; } mpd_del(&t1); mpd_del(&t2); mpd_del(&ulp); mpd_del(&aa); } else { _mpd_qln(result, a, &workctx, status); mpd_check_underflow(result, &workctx, status); mpd_qfinalize(result, &workctx, status); } } /* * Internal log10() function that does not check for specials, zero or one. * Case SKIP_FINALIZE: * Relative error: abs(result - log10(a)) < 0.1 * 10**-prec * abs(log10(a)) * Case DO_FINALIZE: * Ulp error: abs(result - log10(a)) < ulp(log10(a)) */ enum {SKIP_FINALIZE, DO_FINALIZE}; static void _mpd_qlog10(int action, mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t workctx; MPD_NEW_STATIC(ln10,0,0,0,0); mpd_maxcontext(&workctx); workctx.prec = ctx->prec + 3; /* relative error: 0.1 * 10**(-p-3). The specific underflow shortcut * in _mpd_qln() does not change the final result. */ _mpd_qln(result, a, &workctx, status); /* relative error: 5 * 10**(-p-3) */ mpd_qln10(&ln10, workctx.prec, status); if (action == DO_FINALIZE) { workctx = *ctx; workctx.round = MPD_ROUND_HALF_EVEN; } /* SKIP_FINALIZE: relative error: 5 * 10**(-p-3) */ _mpd_qdiv(NO_IDEAL_EXP, result, result, &ln10, &workctx, status); mpd_del(&ln10); } /* log10(a) */ void mpd_qlog10(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t workctx; mpd_ssize_t adjexp, t; workctx = *ctx; workctx.round = MPD_ROUND_HALF_EVEN; if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } if (mpd_isnegative(a)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } mpd_setspecial(result, MPD_POS, MPD_INF); return; } if (mpd_iszerocoeff(a)) { mpd_setspecial(result, MPD_NEG, MPD_INF); return; } if (mpd_isnegative(a)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (mpd_coeff_ispow10(a)) { uint8_t sign = 0; adjexp = mpd_adjexp(a); if (adjexp < 0) { sign = 1; adjexp = -adjexp; } _settriple(result, sign, adjexp, 0); mpd_qfinalize(result, &workctx, status); return; } /* * Check if the result will overflow (0 < x, x != 1): * 1) log10(x) < 0 iff adjexp(x) < 0 * 2) 0 < x /\ x <= y ==> adjexp(x) <= adjexp(y) * 3) adjexp(x) <= log10(x) < adjexp(x) + 1 * * Case adjexp(x) >= 0: * 4) adjexp(x) <= abs(log10(x)) * Case adjexp(x) > 0: * 5) adjexp(adjexp(x)) <= adjexp(abs(log10(x))) * Case adjexp(x) == 0: * mpd_exp_digits(t)-1 == 0 <= emax (the shortcut is not triggered) * * Case adjexp(x) < 0: * 6) -adjexp(x) - 1 < abs(log10(x)) * Case adjexp(x) < -1: * 7) adjexp(-adjexp(x) - 1) <= adjexp(abs(log(x))) * Case adjexp(x) == -1: * mpd_exp_digits(t)-1 == 0 <= emax (the shortcut is not triggered) */ adjexp = mpd_adjexp(a); t = (adjexp < 0) ? -adjexp-1 : adjexp; if (mpd_exp_digits(t)-1 > ctx->emax) { *status |= MPD_Overflow|MPD_Inexact|MPD_Rounded; mpd_setspecial(result, (adjexp<0), MPD_INF); return; } if (ctx->allcr) { MPD_NEW_STATIC(t1, 0,0,0,0); MPD_NEW_STATIC(t2, 0,0,0,0); MPD_NEW_STATIC(ulp, 0,0,0,0); MPD_NEW_STATIC(aa, 0,0,0,0); mpd_ssize_t prec; if (result == a) { if (!mpd_qcopy(&aa, a, status)) { mpd_seterror(result, MPD_Malloc_error, status); return; } a = &aa; } workctx.clamp = 0; prec = ctx->prec + 3; while (1) { workctx.prec = prec; _mpd_qlog10(SKIP_FINALIZE, result, a, &workctx, status); _ssettriple(&ulp, MPD_POS, 1, result->exp + result->digits-workctx.prec); workctx.prec = ctx->prec; mpd_qadd(&t1, result, &ulp, &workctx, &workctx.status); mpd_qsub(&t2, result, &ulp, &workctx, &workctx.status); if (mpd_isspecial(result) || mpd_iszerocoeff(result) || mpd_qcmp(&t1, &t2, status) == 0) { workctx.clamp = ctx->clamp; mpd_check_underflow(result, &workctx, status); mpd_qfinalize(result, &workctx, status); break; } prec += MPD_RDIGITS; } mpd_del(&t1); mpd_del(&t2); mpd_del(&ulp); mpd_del(&aa); } else { _mpd_qlog10(DO_FINALIZE, result, a, &workctx, status); mpd_check_underflow(result, &workctx, status); } } /* * Maximum of the two operands. Attention: If one operand is a quiet NaN and the * other is numeric, the numeric operand is returned. This may not be what one * expects. */ void mpd_qmax(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { int c; if (mpd_isqnan(a) && !mpd_isnan(b)) { mpd_qcopy(result, b, status); } else if (mpd_isqnan(b) && !mpd_isnan(a)) { mpd_qcopy(result, a, status); } else if (mpd_qcheck_nans(result, a, b, ctx, status)) { return; } else { c = _mpd_cmp(a, b); if (c == 0) { c = _mpd_cmp_numequal(a, b); } if (c < 0) { mpd_qcopy(result, b, status); } else { mpd_qcopy(result, a, status); } } mpd_qfinalize(result, ctx, status); } /* * Maximum magnitude: Same as mpd_max(), but compares the operands with their * sign ignored. */ void mpd_qmax_mag(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { int c; if (mpd_isqnan(a) && !mpd_isnan(b)) { mpd_qcopy(result, b, status); } else if (mpd_isqnan(b) && !mpd_isnan(a)) { mpd_qcopy(result, a, status); } else if (mpd_qcheck_nans(result, a, b, ctx, status)) { return; } else { c = _mpd_cmp_abs(a, b); if (c == 0) { c = _mpd_cmp_numequal(a, b); } if (c < 0) { mpd_qcopy(result, b, status); } else { mpd_qcopy(result, a, status); } } mpd_qfinalize(result, ctx, status); } /* * Minimum of the two operands. Attention: If one operand is a quiet NaN and the * other is numeric, the numeric operand is returned. This may not be what one * expects. */ void mpd_qmin(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { int c; if (mpd_isqnan(a) && !mpd_isnan(b)) { mpd_qcopy(result, b, status); } else if (mpd_isqnan(b) && !mpd_isnan(a)) { mpd_qcopy(result, a, status); } else if (mpd_qcheck_nans(result, a, b, ctx, status)) { return; } else { c = _mpd_cmp(a, b); if (c == 0) { c = _mpd_cmp_numequal(a, b); } if (c < 0) { mpd_qcopy(result, a, status); } else { mpd_qcopy(result, b, status); } } mpd_qfinalize(result, ctx, status); } /* * Minimum magnitude: Same as mpd_min(), but compares the operands with their * sign ignored. */ void mpd_qmin_mag(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { int c; if (mpd_isqnan(a) && !mpd_isnan(b)) { mpd_qcopy(result, b, status); } else if (mpd_isqnan(b) && !mpd_isnan(a)) { mpd_qcopy(result, a, status); } else if (mpd_qcheck_nans(result, a, b, ctx, status)) { return; } else { c = _mpd_cmp_abs(a, b); if (c == 0) { c = _mpd_cmp_numequal(a, b); } if (c < 0) { mpd_qcopy(result, a, status); } else { mpd_qcopy(result, b, status); } } mpd_qfinalize(result, ctx, status); } /* Minimum space needed for the result array in _karatsuba_rec(). */ static inline mpd_size_t _kmul_resultsize(mpd_size_t la, mpd_size_t lb) { mpd_size_t n, m; n = add_size_t(la, lb); n = add_size_t(n, 1); m = (la+1)/2 + 1; m = mul_size_t(m, 3); return (m > n) ? m : n; } /* Work space needed in _karatsuba_rec(). lim >= 4 */ static inline mpd_size_t _kmul_worksize(mpd_size_t n, mpd_size_t lim) { mpd_size_t m; if (n <= lim) { return 0; } m = (n+1)/2 + 1; return add_size_t(mul_size_t(m, 2), _kmul_worksize(m, lim)); } #define MPD_KARATSUBA_BASECASE 16 /* must be >= 4 */ /* * Add the product of a and b to c. * c must be _kmul_resultsize(la, lb) in size. * w is used as a work array and must be _kmul_worksize(a, lim) in size. * Roman E. Maeder, Storage Allocation for the Karatsuba Integer Multiplication * Algorithm. In "Design and implementation of symbolic computation systems", * Springer, 1993, ISBN 354057235X, 9783540572350. */ static void _karatsuba_rec(mpd_uint_t *c, const mpd_uint_t *a, const mpd_uint_t *b, mpd_uint_t *w, mpd_size_t la, mpd_size_t lb) { mpd_size_t m, lt; assert(la >= lb && lb > 0); assert(la <= MPD_KARATSUBA_BASECASE || w != NULL); if (la <= MPD_KARATSUBA_BASECASE) { _mpd_basemul(c, a, b, la, lb); return; } m = (la+1)/2; /* ceil(la/2) */ /* lb <= m < la */ if (lb <= m) { /* lb can now be larger than la-m */ if (lb > la-m) { lt = lb + lb + 1; /* space needed for result array */ mpd_uint_zero(w, lt); /* clear result array */ _karatsuba_rec(w, b, a+m, w+lt, lb, la-m); /* b*ah */ } else { lt = (la-m) + (la-m) + 1; /* space needed for result array */ mpd_uint_zero(w, lt); /* clear result array */ _karatsuba_rec(w, a+m, b, w+lt, la-m, lb); /* ah*b */ } _mpd_baseaddto(c+m, w, (la-m)+lb); /* add ah*b*B**m */ lt = m + m + 1; /* space needed for the result array */ mpd_uint_zero(w, lt); /* clear result array */ _karatsuba_rec(w, a, b, w+lt, m, lb); /* al*b */ _mpd_baseaddto(c, w, m+lb); /* add al*b */ return; } /* la >= lb > m */ memcpy(w, a, m * sizeof *w); w[m] = 0; _mpd_baseaddto(w, a+m, la-m); memcpy(w+(m+1), b, m * sizeof *w); w[m+1+m] = 0; _mpd_baseaddto(w+(m+1), b+m, lb-m); _karatsuba_rec(c+m, w, w+(m+1), w+2*(m+1), m+1, m+1); lt = (la-m) + (la-m) + 1; mpd_uint_zero(w, lt); _karatsuba_rec(w, a+m, b+m, w+lt, la-m, lb-m); _mpd_baseaddto(c+2*m, w, (la-m) + (lb-m)); _mpd_basesubfrom(c+m, w, (la-m) + (lb-m)); lt = m + m + 1; mpd_uint_zero(w, lt); _karatsuba_rec(w, a, b, w+lt, m, m); _mpd_baseaddto(c, w, m+m); _mpd_basesubfrom(c+m, w, m+m); return; } /* * Multiply u and v, using Karatsuba multiplication. Returns a pointer * to the result or NULL in case of failure (malloc error). * Conditions: ulen >= vlen, ulen >= 4 */ static mpd_uint_t * _mpd_kmul(const mpd_uint_t *u, const mpd_uint_t *v, mpd_size_t ulen, mpd_size_t vlen, mpd_size_t *rsize) { mpd_uint_t *result = NULL, *w = NULL; mpd_size_t m; assert(ulen >= 4); assert(ulen >= vlen); *rsize = _kmul_resultsize(ulen, vlen); if ((result = mpd_calloc(*rsize, sizeof *result)) == NULL) { return NULL; } m = _kmul_worksize(ulen, MPD_KARATSUBA_BASECASE); if (m && ((w = mpd_calloc(m, sizeof *w)) == NULL)) { mpd_free(result); return NULL; } _karatsuba_rec(result, u, v, w, ulen, vlen); if (w) mpd_free(w); return result; } /* * Determine the minimum length for the number theoretic transform. Valid * transform lengths are 2**n or 3*2**n, where 2**n <= MPD_MAXTRANSFORM_2N. * The function finds the shortest length m such that rsize <= m. */ static inline mpd_size_t _mpd_get_transform_len(mpd_size_t rsize) { mpd_size_t log2rsize; mpd_size_t x, step; assert(rsize >= 4); log2rsize = mpd_bsr(rsize); if (rsize <= 1024) { /* 2**n is faster in this range. */ x = ((mpd_size_t)1)<>1; x += step; return (rsize <= x) ? x : x + step; } else if (rsize <= MPD_MAXTRANSFORM_2N+MPD_MAXTRANSFORM_2N/2) { return MPD_MAXTRANSFORM_2N+MPD_MAXTRANSFORM_2N/2; } else if (rsize <= 3*MPD_MAXTRANSFORM_2N) { return 3*MPD_MAXTRANSFORM_2N; } else { return MPD_SIZE_MAX; } } #ifdef PPRO #ifndef _MSC_VER static inline unsigned short _mpd_get_control87(void) { unsigned short cw; __asm__ __volatile__ ("fnstcw %0" : "=m" (cw)); return cw; } static inline void _mpd_set_control87(unsigned short cw) { __asm__ __volatile__ ("fldcw %0" : : "m" (cw)); } #endif static unsigned int mpd_set_fenv(void) { unsigned int cw; #ifdef _MSC_VER unsigned int flags = _EM_INVALID|_EM_DENORMAL|_EM_ZERODIVIDE|_EM_OVERFLOW| _EM_UNDERFLOW|_EM_INEXACT|_RC_CHOP|_PC_64; unsigned int mask = _MCW_EM|_MCW_RC|_MCW_PC; unsigned int dummy; __control87_2(0, 0, &cw, NULL); __control87_2(flags, mask, &dummy, NULL); #else cw = _mpd_get_control87(); _mpd_set_control87(cw|0xF3F); #endif return cw; } static void mpd_restore_fenv(unsigned int cw) { #ifdef _MSC_VER unsigned int mask = _MCW_EM|_MCW_RC|_MCW_PC; unsigned int dummy; __control87_2(cw, mask, &dummy, NULL); #else _mpd_set_control87((unsigned short)cw); #endif } #endif /* PPRO */ /* * Multiply u and v, using the fast number theoretic transform. Returns * a pointer to the result or NULL in case of failure (malloc error). */ static mpd_uint_t * _mpd_fntmul(const mpd_uint_t *u, const mpd_uint_t *v, mpd_size_t ulen, mpd_size_t vlen, mpd_size_t *rsize) { mpd_uint_t *c1 = NULL, *c2 = NULL, *c3 = NULL, *vtmp = NULL; mpd_size_t n; #ifdef PPRO unsigned int cw; cw = mpd_set_fenv(); #endif *rsize = add_size_t(ulen, vlen); if ((n = _mpd_get_transform_len(*rsize)) == MPD_SIZE_MAX) { goto malloc_error; } if ((c1 = mpd_calloc(n, sizeof *c1)) == NULL) { goto malloc_error; } if ((c2 = mpd_calloc(n, sizeof *c2)) == NULL) { goto malloc_error; } if ((c3 = mpd_calloc(n, sizeof *c3)) == NULL) { goto malloc_error; } memcpy(c1, u, ulen * (sizeof *c1)); memcpy(c2, u, ulen * (sizeof *c2)); memcpy(c3, u, ulen * (sizeof *c3)); if (u == v) { if (!fnt_autoconvolute(c1, n, P1) || !fnt_autoconvolute(c2, n, P2) || !fnt_autoconvolute(c3, n, P3)) { goto malloc_error; } } else { if ((vtmp = mpd_calloc(n, sizeof *vtmp)) == NULL) { goto malloc_error; } memcpy(vtmp, v, vlen * (sizeof *vtmp)); if (!fnt_convolute(c1, vtmp, n, P1)) { mpd_free(vtmp); goto malloc_error; } memcpy(vtmp, v, vlen * (sizeof *vtmp)); mpd_uint_zero(vtmp+vlen, n-vlen); if (!fnt_convolute(c2, vtmp, n, P2)) { mpd_free(vtmp); goto malloc_error; } memcpy(vtmp, v, vlen * (sizeof *vtmp)); mpd_uint_zero(vtmp+vlen, n-vlen); if (!fnt_convolute(c3, vtmp, n, P3)) { mpd_free(vtmp); goto malloc_error; } mpd_free(vtmp); } crt3(c1, c2, c3, *rsize); out: #ifdef PPRO mpd_restore_fenv(cw); #endif if (c2) mpd_free(c2); if (c3) mpd_free(c3); return c1; malloc_error: if (c1) mpd_free(c1); c1 = NULL; goto out; } /* * Karatsuba multiplication with FNT/basemul as the base case. */ static int _karatsuba_rec_fnt(mpd_uint_t *c, const mpd_uint_t *a, const mpd_uint_t *b, mpd_uint_t *w, mpd_size_t la, mpd_size_t lb) { mpd_size_t m, lt; assert(la >= lb && lb > 0); assert(la <= 3*(MPD_MAXTRANSFORM_2N/2) || w != NULL); if (la <= 3*(MPD_MAXTRANSFORM_2N/2)) { if (lb <= 192) { _mpd_basemul(c, b, a, lb, la); } else { mpd_uint_t *result; mpd_size_t dummy; if ((result = _mpd_fntmul(a, b, la, lb, &dummy)) == NULL) { return 0; } memcpy(c, result, (la+lb) * (sizeof *result)); mpd_free(result); } return 1; } m = (la+1)/2; /* ceil(la/2) */ /* lb <= m < la */ if (lb <= m) { /* lb can now be larger than la-m */ if (lb > la-m) { lt = lb + lb + 1; /* space needed for result array */ mpd_uint_zero(w, lt); /* clear result array */ if (!_karatsuba_rec_fnt(w, b, a+m, w+lt, lb, la-m)) { /* b*ah */ return 0; /* GCOV_UNLIKELY */ } } else { lt = (la-m) + (la-m) + 1; /* space needed for result array */ mpd_uint_zero(w, lt); /* clear result array */ if (!_karatsuba_rec_fnt(w, a+m, b, w+lt, la-m, lb)) { /* ah*b */ return 0; /* GCOV_UNLIKELY */ } } _mpd_baseaddto(c+m, w, (la-m)+lb); /* add ah*b*B**m */ lt = m + m + 1; /* space needed for the result array */ mpd_uint_zero(w, lt); /* clear result array */ if (!_karatsuba_rec_fnt(w, a, b, w+lt, m, lb)) { /* al*b */ return 0; /* GCOV_UNLIKELY */ } _mpd_baseaddto(c, w, m+lb); /* add al*b */ return 1; } /* la >= lb > m */ memcpy(w, a, m * sizeof *w); w[m] = 0; _mpd_baseaddto(w, a+m, la-m); memcpy(w+(m+1), b, m * sizeof *w); w[m+1+m] = 0; _mpd_baseaddto(w+(m+1), b+m, lb-m); if (!_karatsuba_rec_fnt(c+m, w, w+(m+1), w+2*(m+1), m+1, m+1)) { return 0; /* GCOV_UNLIKELY */ } lt = (la-m) + (la-m) + 1; mpd_uint_zero(w, lt); if (!_karatsuba_rec_fnt(w, a+m, b+m, w+lt, la-m, lb-m)) { return 0; /* GCOV_UNLIKELY */ } _mpd_baseaddto(c+2*m, w, (la-m) + (lb-m)); _mpd_basesubfrom(c+m, w, (la-m) + (lb-m)); lt = m + m + 1; mpd_uint_zero(w, lt); if (!_karatsuba_rec_fnt(w, a, b, w+lt, m, m)) { return 0; /* GCOV_UNLIKELY */ } _mpd_baseaddto(c, w, m+m); _mpd_basesubfrom(c+m, w, m+m); return 1; } /* * Multiply u and v, using Karatsuba multiplication with the FNT as the * base case. Returns a pointer to the result or NULL in case of failure * (malloc error). Conditions: ulen >= vlen, ulen >= 4. */ static mpd_uint_t * _mpd_kmul_fnt(const mpd_uint_t *u, const mpd_uint_t *v, mpd_size_t ulen, mpd_size_t vlen, mpd_size_t *rsize) { mpd_uint_t *result = NULL, *w = NULL; mpd_size_t m; assert(ulen >= 4); assert(ulen >= vlen); *rsize = _kmul_resultsize(ulen, vlen); if ((result = mpd_calloc(*rsize, sizeof *result)) == NULL) { return NULL; } m = _kmul_worksize(ulen, 3*(MPD_MAXTRANSFORM_2N/2)); if (m && ((w = mpd_calloc(m, sizeof *w)) == NULL)) { mpd_free(result); /* GCOV_UNLIKELY */ return NULL; /* GCOV_UNLIKELY */ } if (!_karatsuba_rec_fnt(result, u, v, w, ulen, vlen)) { mpd_free(result); result = NULL; } if (w) mpd_free(w); return result; } /* Deal with the special cases of multiplying infinities. */ static void _mpd_qmul_inf(mpd_t *result, const mpd_t *a, const mpd_t *b, uint32_t *status) { if (mpd_isinfinite(a)) { if (mpd_iszero(b)) { mpd_seterror(result, MPD_Invalid_operation, status); } else { mpd_setspecial(result, mpd_sign(a)^mpd_sign(b), MPD_INF); } return; } assert(mpd_isinfinite(b)); if (mpd_iszero(a)) { mpd_seterror(result, MPD_Invalid_operation, status); } else { mpd_setspecial(result, mpd_sign(a)^mpd_sign(b), MPD_INF); } } /* * Internal function: Multiply a and b. _mpd_qmul deals with specials but * does NOT finalize the result. This is for use in mpd_fma(). */ static inline void _mpd_qmul(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { mpd_t *big = (mpd_t *)a, *small = (mpd_t *)b; mpd_uint_t *rdata = NULL; mpd_uint_t rbuf[MPD_MINALLOC_MAX]; mpd_size_t rsize, i; if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(result, a, b, ctx, status)) { return; } _mpd_qmul_inf(result, a, b, status); return; } if (small->len > big->len) { _mpd_ptrswap(&big, &small); } rsize = big->len + small->len; if (big->len == 1) { _mpd_singlemul(result->data, big->data[0], small->data[0]); goto finish; } if (rsize <= (mpd_size_t)MPD_MINALLOC_MAX) { if (big->len == 2) { _mpd_mul_2_le2(rbuf, big->data, small->data, small->len); } else { mpd_uint_zero(rbuf, rsize); if (small->len == 1) { _mpd_shortmul(rbuf, big->data, big->len, small->data[0]); } else { _mpd_basemul(rbuf, small->data, big->data, small->len, big->len); } } if (!mpd_qresize(result, rsize, status)) { return; } for(i = 0; i < rsize; i++) { result->data[i] = rbuf[i]; } goto finish; } if (small->len <= 256) { rdata = mpd_calloc(rsize, sizeof *rdata); if (rdata != NULL) { if (small->len == 1) { _mpd_shortmul(rdata, big->data, big->len, small->data[0]); } else { _mpd_basemul(rdata, small->data, big->data, small->len, big->len); } } } else if (rsize <= 1024) { rdata = _mpd_kmul(big->data, small->data, big->len, small->len, &rsize); } else if (rsize <= 3*MPD_MAXTRANSFORM_2N) { rdata = _mpd_fntmul(big->data, small->data, big->len, small->len, &rsize); } else { rdata = _mpd_kmul_fnt(big->data, small->data, big->len, small->len, &rsize); } if (rdata == NULL) { mpd_seterror(result, MPD_Malloc_error, status); return; } if (mpd_isdynamic_data(result)) { mpd_free(result->data); } result->data = rdata; result->alloc = rsize; mpd_set_dynamic_data(result); finish: mpd_set_flags(result, mpd_sign(a)^mpd_sign(b)); result->exp = big->exp + small->exp; result->len = _mpd_real_size(result->data, rsize); /* resize to smaller cannot fail */ mpd_qresize(result, result->len, status); mpd_setdigits(result); } /* Multiply a and b. */ void mpd_qmul(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { _mpd_qmul(result, a, b, ctx, status); mpd_qfinalize(result, ctx, status); } /* Multiply a and b. Set NaN/Invalid_operation if the result is inexact. */ static void _mpd_qmul_exact(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { uint32_t workstatus = 0; mpd_qmul(result, a, b, ctx, &workstatus); *status |= workstatus; if (workstatus & (MPD_Inexact|MPD_Rounded|MPD_Clamped)) { mpd_seterror(result, MPD_Invalid_operation, status); } } /* Multiply decimal and mpd_ssize_t. */ void mpd_qmul_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t maxcontext; MPD_NEW_STATIC(bb,0,0,0,0); mpd_maxcontext(&maxcontext); mpd_qsset_ssize(&bb, b, &maxcontext, status); mpd_qmul(result, a, &bb, ctx, status); mpd_del(&bb); } /* Multiply decimal and mpd_uint_t. */ void mpd_qmul_uint(mpd_t *result, const mpd_t *a, mpd_uint_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t maxcontext; MPD_NEW_STATIC(bb,0,0,0,0); mpd_maxcontext(&maxcontext); mpd_qsset_uint(&bb, b, &maxcontext, status); mpd_qmul(result, a, &bb, ctx, status); mpd_del(&bb); } void mpd_qmul_i32(mpd_t *result, const mpd_t *a, int32_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qmul_ssize(result, a, b, ctx, status); } void mpd_qmul_u32(mpd_t *result, const mpd_t *a, uint32_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qmul_uint(result, a, b, ctx, status); } #ifdef CONFIG_64 void mpd_qmul_i64(mpd_t *result, const mpd_t *a, int64_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qmul_ssize(result, a, b, ctx, status); } void mpd_qmul_u64(mpd_t *result, const mpd_t *a, uint64_t b, const mpd_context_t *ctx, uint32_t *status) { mpd_qmul_uint(result, a, b, ctx, status); } #endif /* Like the minus operator. */ void mpd_qminus(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } } if (mpd_iszero(a) && ctx->round != MPD_ROUND_FLOOR) { mpd_qcopy_abs(result, a, status); } else { mpd_qcopy_negate(result, a, status); } mpd_qfinalize(result, ctx, status); } /* Like the plus operator. */ void mpd_qplus(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } } if (mpd_iszero(a) && ctx->round != MPD_ROUND_FLOOR) { mpd_qcopy_abs(result, a, status); } else { mpd_qcopy(result, a, status); } mpd_qfinalize(result, ctx, status); } /* The largest representable number that is smaller than the operand. */ void mpd_qnext_minus(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t workctx; MPD_NEW_CONST(tiny,MPD_POS,mpd_etiny(ctx)-1,1,1,1,1); if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } assert(mpd_isinfinite(a)); if (mpd_isnegative(a)) { mpd_qcopy(result, a, status); return; } else { mpd_clear_flags(result); mpd_qmaxcoeff(result, ctx, status); if (mpd_isnan(result)) { return; } result->exp = mpd_etop(ctx); return; } } mpd_workcontext(&workctx, ctx); workctx.round = MPD_ROUND_FLOOR; if (!mpd_qcopy(result, a, status)) { return; } mpd_qfinalize(result, &workctx, &workctx.status); if (workctx.status&(MPD_Inexact|MPD_Errors)) { *status |= (workctx.status&MPD_Errors); return; } workctx.status = 0; mpd_qsub(result, a, &tiny, &workctx, &workctx.status); *status |= (workctx.status&MPD_Errors); } /* The smallest representable number that is larger than the operand. */ void mpd_qnext_plus(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t workctx; MPD_NEW_CONST(tiny,MPD_POS,mpd_etiny(ctx)-1,1,1,1,1); if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } assert(mpd_isinfinite(a)); if (mpd_ispositive(a)) { mpd_qcopy(result, a, status); } else { mpd_clear_flags(result); mpd_qmaxcoeff(result, ctx, status); if (mpd_isnan(result)) { return; } mpd_set_flags(result, MPD_NEG); result->exp = mpd_etop(ctx); } return; } mpd_workcontext(&workctx, ctx); workctx.round = MPD_ROUND_CEILING; if (!mpd_qcopy(result, a, status)) { return; } mpd_qfinalize(result, &workctx, &workctx.status); if (workctx.status & (MPD_Inexact|MPD_Errors)) { *status |= (workctx.status&MPD_Errors); return; } workctx.status = 0; mpd_qadd(result, a, &tiny, &workctx, &workctx.status); *status |= (workctx.status&MPD_Errors); } /* * The number closest to the first operand that is in the direction towards * the second operand. */ void mpd_qnext_toward(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { int c; if (mpd_qcheck_nans(result, a, b, ctx, status)) { return; } c = _mpd_cmp(a, b); if (c == 0) { mpd_qcopy_sign(result, a, b, status); return; } if (c < 0) { mpd_qnext_plus(result, a, ctx, status); } else { mpd_qnext_minus(result, a, ctx, status); } if (mpd_isinfinite(result)) { *status |= (MPD_Overflow|MPD_Rounded|MPD_Inexact); } else if (mpd_adjexp(result) < ctx->emin) { *status |= (MPD_Underflow|MPD_Subnormal|MPD_Rounded|MPD_Inexact); if (mpd_iszero(result)) { *status |= MPD_Clamped; } } } /* * Internal function: Integer power with mpd_uint_t exponent. The function * can fail with MPD_Malloc_error. * * The error is equal to the error incurred in k-1 multiplications. Assuming * the upper bound for the relative error in each operation: * * abs(err) = 5 * 10**-prec * result = x**k * (1 + err)**(k-1) */ static inline void _mpd_qpow_uint(mpd_t *result, const mpd_t *base, mpd_uint_t exp, uint8_t resultsign, const mpd_context_t *ctx, uint32_t *status) { uint32_t workstatus = 0; mpd_uint_t n; if (exp == 0) { _settriple(result, resultsign, 1, 0); /* GCOV_NOT_REACHED */ return; /* GCOV_NOT_REACHED */ } if (!mpd_qcopy(result, base, status)) { return; } n = mpd_bits[mpd_bsr(exp)]; while (n >>= 1) { mpd_qmul(result, result, result, ctx, &workstatus); if (exp & n) { mpd_qmul(result, result, base, ctx, &workstatus); } if (mpd_isspecial(result) || (mpd_iszerocoeff(result) && (workstatus & MPD_Clamped))) { break; } } *status |= workstatus; mpd_set_sign(result, resultsign); } /* * Internal function: Integer power with mpd_t exponent, tbase and texp * are modified!! Function can fail with MPD_Malloc_error. * * The error is equal to the error incurred in k multiplications. Assuming * the upper bound for the relative error in each operation: * * abs(err) = 5 * 10**-prec * result = x**k * (1 + err)**k */ static inline void _mpd_qpow_mpd(mpd_t *result, mpd_t *tbase, mpd_t *texp, uint8_t resultsign, const mpd_context_t *ctx, uint32_t *status) { uint32_t workstatus = 0; mpd_context_t maxctx; MPD_NEW_CONST(two,0,0,1,1,1,2); mpd_maxcontext(&maxctx); /* resize to smaller cannot fail */ mpd_qcopy(result, &one, status); while (!mpd_iszero(texp)) { if (mpd_isodd(texp)) { mpd_qmul(result, result, tbase, ctx, &workstatus); *status |= workstatus; if (mpd_isspecial(result) || (mpd_iszerocoeff(result) && (workstatus & MPD_Clamped))) { break; } } mpd_qmul(tbase, tbase, tbase, ctx, &workstatus); mpd_qdivint(texp, texp, &two, &maxctx, &workstatus); if (mpd_isnan(tbase) || mpd_isnan(texp)) { mpd_seterror(result, workstatus&MPD_Errors, status); return; } } mpd_set_sign(result, resultsign); } /* * The power function for integer exponents. Relative error _before_ the * final rounding to prec: * abs(result - base**exp) < 0.1 * 10**-prec * abs(base**exp) */ static void _mpd_qpow_int(mpd_t *result, const mpd_t *base, const mpd_t *exp, uint8_t resultsign, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t workctx; MPD_NEW_STATIC(tbase,0,0,0,0); MPD_NEW_STATIC(texp,0,0,0,0); mpd_ssize_t n; mpd_workcontext(&workctx, ctx); workctx.prec += (exp->digits + exp->exp + 2); workctx.round = MPD_ROUND_HALF_EVEN; workctx.clamp = 0; if (mpd_isnegative(exp)) { workctx.prec += 1; mpd_qdiv(&tbase, &one, base, &workctx, status); if (*status&MPD_Errors) { mpd_setspecial(result, MPD_POS, MPD_NAN); goto finish; } } else { if (!mpd_qcopy(&tbase, base, status)) { mpd_setspecial(result, MPD_POS, MPD_NAN); goto finish; } } n = mpd_qabs_uint(exp, &workctx.status); if (workctx.status&MPD_Invalid_operation) { if (!mpd_qcopy(&texp, exp, status)) { mpd_setspecial(result, MPD_POS, MPD_NAN); /* GCOV_UNLIKELY */ goto finish; /* GCOV_UNLIKELY */ } _mpd_qpow_mpd(result, &tbase, &texp, resultsign, &workctx, status); } else { _mpd_qpow_uint(result, &tbase, n, resultsign, &workctx, status); } if (mpd_isinfinite(result)) { /* for ROUND_DOWN, ROUND_FLOOR, etc. */ _settriple(result, resultsign, 1, MPD_EXP_INF); } finish: mpd_del(&tbase); mpd_del(&texp); mpd_qfinalize(result, ctx, status); } /* * If the exponent is infinite and base equals one, the result is one * with a coefficient of length prec. Otherwise, result is undefined. * Return the value of the comparison against one. */ static int _qcheck_pow_one_inf(mpd_t *result, const mpd_t *base, uint8_t resultsign, const mpd_context_t *ctx, uint32_t *status) { mpd_ssize_t shift; int cmp; if ((cmp = _mpd_cmp(base, &one)) == 0) { shift = ctx->prec-1; mpd_qshiftl(result, &one, shift, status); result->exp = -shift; mpd_set_flags(result, resultsign); *status |= (MPD_Inexact|MPD_Rounded); } return cmp; } /* * If abs(base) equals one, calculate the correct power of one result. * Otherwise, result is undefined. Return the value of the comparison * against 1. * * This is an internal function that does not check for specials. */ static int _qcheck_pow_one(mpd_t *result, const mpd_t *base, const mpd_t *exp, uint8_t resultsign, const mpd_context_t *ctx, uint32_t *status) { uint32_t workstatus = 0; mpd_ssize_t shift; int cmp; if ((cmp = _mpd_cmp_abs(base, &one)) == 0) { if (_mpd_isint(exp)) { if (mpd_isnegative(exp)) { _settriple(result, resultsign, 1, 0); return 0; } /* 1.000**3 = 1.000000000 */ mpd_qmul_ssize(result, exp, -base->exp, ctx, &workstatus); if (workstatus&MPD_Errors) { *status |= (workstatus&MPD_Errors); return 0; } /* digits-1 after exponentiation */ shift = mpd_qget_ssize(result, &workstatus); /* shift is MPD_SSIZE_MAX if result is too large */ if (shift > ctx->prec-1) { shift = ctx->prec-1; *status |= MPD_Rounded; } } else if (mpd_ispositive(base)) { shift = ctx->prec-1; *status |= (MPD_Inexact|MPD_Rounded); } else { return -2; /* GCOV_NOT_REACHED */ } if (!mpd_qshiftl(result, &one, shift, status)) { return 0; } result->exp = -shift; mpd_set_flags(result, resultsign); } return cmp; } /* * Detect certain over/underflow of x**y. * ACL2 proof: pow-bounds.lisp. * * Symbols: * * e: EXP_INF or EXP_CLAMP * x: base * y: exponent * * omega(e) = log10(abs(e)) * zeta(x) = log10(abs(log10(x))) * theta(y) = log10(abs(y)) * * Upper and lower bounds: * * ub_omega(e) = ceil(log10(abs(e))) * lb_theta(y) = floor(log10(abs(y))) * * | floor(log10(floor(abs(log10(x))))) if x < 1/10 or x >= 10 * lb_zeta(x) = | floor(log10(abs(x-1)/10)) if 1/10 <= x < 1 * | floor(log10(abs((x-1)/100))) if 1 < x < 10 * * ub_omega(e) and lb_theta(y) are obviously upper and lower bounds * for omega(e) and theta(y). * * lb_zeta is a lower bound for zeta(x): * * x < 1/10 or x >= 10: * * abs(log10(x)) >= 1, so the outer log10 is well defined. Since log10 * is strictly increasing, the end result is a lower bound. * * 1/10 <= x < 1: * * We use: log10(x) <= (x-1)/log(10) * abs(log10(x)) >= abs(x-1)/log(10) * abs(log10(x)) >= abs(x-1)/10 * * 1 < x < 10: * * We use: (x-1)/(x*log(10)) < log10(x) * abs((x-1)/100) < abs(log10(x)) * * XXX: abs((x-1)/10) would work, need ACL2 proof. * * * Let (0 < x < 1 and y < 0) or (x > 1 and y > 0). (H1) * Let ub_omega(exp_inf) < lb_zeta(x) + lb_theta(y) (H2) * * Then: * log10(abs(exp_inf)) < log10(abs(log10(x))) + log10(abs(y)). (1) * exp_inf < log10(x) * y (2) * 10**exp_inf < x**y (3) * * Let (0 < x < 1 and y > 0) or (x > 1 and y < 0). (H3) * Let ub_omega(exp_clamp) < lb_zeta(x) + lb_theta(y) (H4) * * Then: * log10(abs(exp_clamp)) < log10(abs(log10(x))) + log10(abs(y)). (4) * log10(x) * y < exp_clamp (5) * x**y < 10**exp_clamp (6) * */ static mpd_ssize_t _lower_bound_zeta(const mpd_t *x, uint32_t *status) { mpd_context_t maxctx; MPD_NEW_STATIC(scratch,0,0,0,0); mpd_ssize_t t, u; t = mpd_adjexp(x); if (t > 0) { /* x >= 10 -> floor(log10(floor(abs(log10(x))))) */ return mpd_exp_digits(t) - 1; } else if (t < -1) { /* x < 1/10 -> floor(log10(floor(abs(log10(x))))) */ return mpd_exp_digits(t+1) - 1; } else { mpd_maxcontext(&maxctx); mpd_qsub(&scratch, x, &one, &maxctx, status); if (mpd_isspecial(&scratch)) { mpd_del(&scratch); return MPD_SSIZE_MAX; } u = mpd_adjexp(&scratch); mpd_del(&scratch); /* t == -1, 1/10 <= x < 1 -> floor(log10(abs(x-1)/10)) * t == 0, 1 < x < 10 -> floor(log10(abs(x-1)/100)) */ return (t == 0) ? u-2 : u-1; } } /* * Detect cases of certain overflow/underflow in the power function. * Assumptions: x != 1, y != 0. The proof above is for positive x. * If x is negative and y is an odd integer, x**y == -(abs(x)**y), * so the analysis does not change. */ static int _qcheck_pow_bounds(mpd_t *result, const mpd_t *x, const mpd_t *y, uint8_t resultsign, const mpd_context_t *ctx, uint32_t *status) { MPD_NEW_SHARED(abs_x, x); mpd_ssize_t ub_omega, lb_zeta, lb_theta; uint8_t sign; mpd_set_positive(&abs_x); lb_theta = mpd_adjexp(y); lb_zeta = _lower_bound_zeta(&abs_x, status); if (lb_zeta == MPD_SSIZE_MAX) { mpd_seterror(result, MPD_Malloc_error, status); return 1; } sign = (mpd_adjexp(&abs_x) < 0) ^ mpd_sign(y); if (sign == 0) { /* (0 < |x| < 1 and y < 0) or (|x| > 1 and y > 0) */ ub_omega = mpd_exp_digits(ctx->emax); if (ub_omega < lb_zeta + lb_theta) { _settriple(result, resultsign, 1, MPD_EXP_INF); mpd_qfinalize(result, ctx, status); return 1; } } else { /* (0 < |x| < 1 and y > 0) or (|x| > 1 and y < 0). */ ub_omega = mpd_exp_digits(mpd_etiny(ctx)); if (ub_omega < lb_zeta + lb_theta) { _settriple(result, resultsign, 1, mpd_etiny(ctx)-1); mpd_qfinalize(result, ctx, status); return 1; } } return 0; } /* * TODO: Implement algorithm for computing exact powers from decimal.py. * In order to prevent infinite loops, this has to be called before * using Ziv's strategy for correct rounding. */ /* static int _mpd_qpow_exact(mpd_t *result, const mpd_t *base, const mpd_t *exp, const mpd_context_t *ctx, uint32_t *status) { return 0; } */ /* * The power function for real exponents. * Relative error: abs(result - e**y) < e**y * 1/5 * 10**(-prec - 1) */ static void _mpd_qpow_real(mpd_t *result, const mpd_t *base, const mpd_t *exp, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t workctx; MPD_NEW_STATIC(texp,0,0,0,0); if (!mpd_qcopy(&texp, exp, status)) { mpd_seterror(result, MPD_Malloc_error, status); return; } mpd_maxcontext(&workctx); workctx.prec = (base->digits > ctx->prec) ? base->digits : ctx->prec; workctx.prec += (4 + MPD_EXPDIGITS); workctx.round = MPD_ROUND_HALF_EVEN; workctx.allcr = ctx->allcr; /* * extra := MPD_EXPDIGITS = MPD_EXP_MAX_T * wp := prec + 4 + extra * abs(err) < 5 * 10**-wp * y := log(base) * exp * Calculate: * 1) e**(y * (1 + err)**2) * (1 + err) * = e**y * e**(y * (2*err + err**2)) * (1 + err) * ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ * Relative error of the underlined term: * 2) abs(e**(y * (2*err + err**2)) - 1) * Case abs(y) >= 10**extra: * 3) adjexp(y)+1 > log10(abs(y)) >= extra * This triggers the Overflow/Underflow shortcut in _mpd_qexp(), * so no further analysis is necessary. * Case abs(y) < 10**extra: * 4) abs(y * (2*err + err**2)) < 1/5 * 10**(-prec - 2) * Use (see _mpd_qexp): * 5) abs(x) <= 9/10 * 10**-p ==> abs(e**x - 1) < 10**-p * With 2), 4) and 5): * 6) abs(e**(y * (2*err + err**2)) - 1) < 10**(-prec - 2) * The complete relative error of 1) is: * 7) abs(result - e**y) < e**y * 1/5 * 10**(-prec - 1) */ mpd_qln(result, base, &workctx, &workctx.status); mpd_qmul(result, result, &texp, &workctx, &workctx.status); mpd_qexp(result, result, &workctx, status); mpd_del(&texp); *status |= (workctx.status&MPD_Errors); *status |= (MPD_Inexact|MPD_Rounded); } /* The power function: base**exp */ void mpd_qpow(mpd_t *result, const mpd_t *base, const mpd_t *exp, const mpd_context_t *ctx, uint32_t *status) { uint8_t resultsign = 0; int intexp = 0; int cmp; if (mpd_isspecial(base) || mpd_isspecial(exp)) { if (mpd_qcheck_nans(result, base, exp, ctx, status)) { return; } } if (mpd_isinteger(exp)) { intexp = 1; resultsign = mpd_isnegative(base) && mpd_isodd(exp); } if (mpd_iszero(base)) { if (mpd_iszero(exp)) { mpd_seterror(result, MPD_Invalid_operation, status); } else if (mpd_isnegative(exp)) { mpd_setspecial(result, resultsign, MPD_INF); } else { _settriple(result, resultsign, 0, 0); } return; } if (mpd_isnegative(base)) { if (!intexp || mpd_isinfinite(exp)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } } if (mpd_isinfinite(exp)) { /* power of one */ cmp = _qcheck_pow_one_inf(result, base, resultsign, ctx, status); if (cmp == 0) { return; } else { cmp *= mpd_arith_sign(exp); if (cmp < 0) { _settriple(result, resultsign, 0, 0); } else { mpd_setspecial(result, resultsign, MPD_INF); } } return; } if (mpd_isinfinite(base)) { if (mpd_iszero(exp)) { _settriple(result, resultsign, 1, 0); } else if (mpd_isnegative(exp)) { _settriple(result, resultsign, 0, 0); } else { mpd_setspecial(result, resultsign, MPD_INF); } return; } if (mpd_iszero(exp)) { _settriple(result, resultsign, 1, 0); return; } if (_qcheck_pow_one(result, base, exp, resultsign, ctx, status) == 0) { return; } if (_qcheck_pow_bounds(result, base, exp, resultsign, ctx, status)) { return; } if (intexp) { _mpd_qpow_int(result, base, exp, resultsign, ctx, status); } else { _mpd_qpow_real(result, base, exp, ctx, status); if (!mpd_isspecial(result) && _mpd_cmp(result, &one) == 0) { mpd_ssize_t shift = ctx->prec-1; mpd_qshiftl(result, &one, shift, status); result->exp = -shift; } if (mpd_isinfinite(result)) { /* for ROUND_DOWN, ROUND_FLOOR, etc. */ _settriple(result, MPD_POS, 1, MPD_EXP_INF); } mpd_qfinalize(result, ctx, status); } } /* * Internal function: Integer powmod with mpd_uint_t exponent, base is modified! * Function can fail with MPD_Malloc_error. */ static inline void _mpd_qpowmod_uint(mpd_t *result, mpd_t *base, mpd_uint_t exp, const mpd_t *mod, uint32_t *status) { mpd_context_t maxcontext; mpd_maxcontext(&maxcontext); /* resize to smaller cannot fail */ mpd_qcopy(result, &one, status); while (exp > 0) { if (exp & 1) { _mpd_qmul_exact(result, result, base, &maxcontext, status); mpd_qrem(result, result, mod, &maxcontext, status); } _mpd_qmul_exact(base, base, base, &maxcontext, status); mpd_qrem(base, base, mod, &maxcontext, status); exp >>= 1; } } /* The powmod function: (base**exp) % mod */ void mpd_qpowmod(mpd_t *result, const mpd_t *base, const mpd_t *exp, const mpd_t *mod, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t maxcontext; MPD_NEW_STATIC(tbase,0,0,0,0); MPD_NEW_STATIC(texp,0,0,0,0); MPD_NEW_STATIC(tmod,0,0,0,0); MPD_NEW_STATIC(tmp,0,0,0,0); MPD_NEW_CONST(two,0,0,1,1,1,2); mpd_ssize_t tbase_exp, texp_exp; mpd_ssize_t i; mpd_t t; mpd_uint_t r; uint8_t sign; if (mpd_isspecial(base) || mpd_isspecial(exp) || mpd_isspecial(mod)) { if (mpd_qcheck_3nans(result, base, exp, mod, ctx, status)) { return; } mpd_seterror(result, MPD_Invalid_operation, status); return; } if (!_mpd_isint(base) || !_mpd_isint(exp) || !_mpd_isint(mod)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (mpd_iszerocoeff(mod)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (mod->digits+mod->exp > ctx->prec) { mpd_seterror(result, MPD_Invalid_operation, status); return; } sign = (mpd_isnegative(base)) && (mpd_isodd(exp)); if (mpd_iszerocoeff(exp)) { if (mpd_iszerocoeff(base)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } r = (_mpd_cmp_abs(mod, &one)==0) ? 0 : 1; _settriple(result, sign, r, 0); return; } if (mpd_isnegative(exp)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (mpd_iszerocoeff(base)) { _settriple(result, sign, 0, 0); return; } mpd_maxcontext(&maxcontext); mpd_qrescale(&tmod, mod, 0, &maxcontext, &maxcontext.status); if (maxcontext.status&MPD_Errors) { mpd_seterror(result, maxcontext.status&MPD_Errors, status); goto out; } maxcontext.status = 0; mpd_set_positive(&tmod); mpd_qround_to_int(&tbase, base, &maxcontext, status); mpd_set_positive(&tbase); tbase_exp = tbase.exp; tbase.exp = 0; mpd_qround_to_int(&texp, exp, &maxcontext, status); texp_exp = texp.exp; texp.exp = 0; /* base = (base.int % modulo * pow(10, base.exp, modulo)) % modulo */ mpd_qrem(&tbase, &tbase, &tmod, &maxcontext, status); mpd_qshiftl(result, &one, tbase_exp, status); mpd_qrem(result, result, &tmod, &maxcontext, status); _mpd_qmul_exact(&tbase, &tbase, result, &maxcontext, status); mpd_qrem(&tbase, &tbase, &tmod, &maxcontext, status); if (mpd_isspecial(&tbase) || mpd_isspecial(&texp) || mpd_isspecial(&tmod)) { goto mpd_errors; } for (i = 0; i < texp_exp; i++) { _mpd_qpowmod_uint(&tmp, &tbase, 10, &tmod, status); t = tmp; tmp = tbase; tbase = t; } if (mpd_isspecial(&tbase)) { goto mpd_errors; /* GCOV_UNLIKELY */ } /* resize to smaller cannot fail */ mpd_qcopy(result, &one, status); while (mpd_isfinite(&texp) && !mpd_iszero(&texp)) { if (mpd_isodd(&texp)) { _mpd_qmul_exact(result, result, &tbase, &maxcontext, status); mpd_qrem(result, result, &tmod, &maxcontext, status); } _mpd_qmul_exact(&tbase, &tbase, &tbase, &maxcontext, status); mpd_qrem(&tbase, &tbase, &tmod, &maxcontext, status); mpd_qdivint(&texp, &texp, &two, &maxcontext, status); } if (mpd_isspecial(&texp) || mpd_isspecial(&tbase) || mpd_isspecial(&tmod) || mpd_isspecial(result)) { /* MPD_Malloc_error */ goto mpd_errors; } else { mpd_set_sign(result, sign); } out: mpd_del(&tbase); mpd_del(&texp); mpd_del(&tmod); mpd_del(&tmp); return; mpd_errors: mpd_setspecial(result, MPD_POS, MPD_NAN); goto out; } void mpd_qquantize(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { uint32_t workstatus = 0; mpd_ssize_t b_exp = b->exp; mpd_ssize_t expdiff, shift; mpd_uint_t rnd; if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(result, a, b, ctx, status)) { return; } if (mpd_isinfinite(a) && mpd_isinfinite(b)) { mpd_qcopy(result, a, status); return; } mpd_seterror(result, MPD_Invalid_operation, status); return; } if (b->exp > ctx->emax || b->exp < mpd_etiny(ctx)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (mpd_iszero(a)) { _settriple(result, mpd_sign(a), 0, b->exp); mpd_qfinalize(result, ctx, status); return; } expdiff = a->exp - b->exp; if (a->digits + expdiff > ctx->prec) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (expdiff >= 0) { shift = expdiff; if (!mpd_qshiftl(result, a, shift, status)) { return; } result->exp = b_exp; } else { /* At this point expdiff < 0 and a->digits+expdiff <= prec, * so the shift before an increment will fit in prec. */ shift = -expdiff; rnd = mpd_qshiftr(result, a, shift, status); if (rnd == MPD_UINT_MAX) { return; } result->exp = b_exp; if (!_mpd_apply_round_fit(result, rnd, ctx, status)) { return; } workstatus |= MPD_Rounded; if (rnd) { workstatus |= MPD_Inexact; } } if (mpd_adjexp(result) > ctx->emax || mpd_adjexp(result) < mpd_etiny(ctx)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } *status |= workstatus; mpd_qfinalize(result, ctx, status); } void mpd_qreduce(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_ssize_t shift, maxexp, maxshift; uint8_t sign_a = mpd_sign(a); if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } mpd_qcopy(result, a, status); return; } if (!mpd_qcopy(result, a, status)) { return; } mpd_qfinalize(result, ctx, status); if (mpd_isspecial(result)) { return; } if (mpd_iszero(result)) { _settriple(result, sign_a, 0, 0); return; } shift = mpd_trail_zeros(result); maxexp = (ctx->clamp) ? mpd_etop(ctx) : ctx->emax; /* After the finalizing above result->exp <= maxexp. */ maxshift = maxexp - result->exp; shift = (shift > maxshift) ? maxshift : shift; mpd_qshiftr_inplace(result, shift); result->exp += shift; } void mpd_qrem(mpd_t *r, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { MPD_NEW_STATIC(q,0,0,0,0); if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(r, a, b, ctx, status)) { return; } if (mpd_isinfinite(a)) { mpd_seterror(r, MPD_Invalid_operation, status); return; } if (mpd_isinfinite(b)) { mpd_qcopy(r, a, status); mpd_qfinalize(r, ctx, status); return; } /* debug */ abort(); /* GCOV_NOT_REACHED */ } if (mpd_iszerocoeff(b)) { if (mpd_iszerocoeff(a)) { mpd_seterror(r, MPD_Division_undefined, status); } else { mpd_seterror(r, MPD_Invalid_operation, status); } return; } _mpd_qdivmod(&q, r, a, b, ctx, status); mpd_del(&q); mpd_qfinalize(r, ctx, status); } void mpd_qrem_near(mpd_t *r, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t workctx; MPD_NEW_STATIC(btmp,0,0,0,0); MPD_NEW_STATIC(q,0,0,0,0); mpd_ssize_t expdiff, qdigits; int cmp, isodd, allnine; if (mpd_isspecial(a) || mpd_isspecial(b)) { if (mpd_qcheck_nans(r, a, b, ctx, status)) { return; } if (mpd_isinfinite(a)) { mpd_seterror(r, MPD_Invalid_operation, status); return; } if (mpd_isinfinite(b)) { mpd_qcopy(r, a, status); mpd_qfinalize(r, ctx, status); return; } /* debug */ abort(); /* GCOV_NOT_REACHED */ } if (mpd_iszerocoeff(b)) { if (mpd_iszerocoeff(a)) { mpd_seterror(r, MPD_Division_undefined, status); } else { mpd_seterror(r, MPD_Invalid_operation, status); } return; } if (r == b) { if (!mpd_qcopy(&btmp, b, status)) { mpd_seterror(r, MPD_Malloc_error, status); return; } b = &btmp; } _mpd_qdivmod(&q, r, a, b, ctx, status); if (mpd_isnan(&q) || mpd_isnan(r)) { goto finish; } if (mpd_iszerocoeff(r)) { goto finish; } expdiff = mpd_adjexp(b) - mpd_adjexp(r); if (-1 <= expdiff && expdiff <= 1) { allnine = mpd_coeff_isallnine(&q); qdigits = q.digits; isodd = mpd_isodd(&q); mpd_maxcontext(&workctx); if (mpd_sign(a) == mpd_sign(b)) { /* sign(r) == sign(b) */ _mpd_qsub(&q, r, b, &workctx, &workctx.status); } else { /* sign(r) != sign(b) */ _mpd_qadd(&q, r, b, &workctx, &workctx.status); } if (workctx.status&MPD_Errors) { mpd_seterror(r, workctx.status&MPD_Errors, status); goto finish; } cmp = _mpd_cmp_abs(&q, r); if (cmp < 0 || (cmp == 0 && isodd)) { /* abs(r) > abs(b)/2 or abs(r) == abs(b)/2 and isodd(quotient) */ if (allnine && qdigits == ctx->prec) { /* abs(quotient) + 1 == 10**prec */ mpd_seterror(r, MPD_Division_impossible, status); goto finish; } mpd_qcopy(r, &q, status); } } finish: mpd_del(&btmp); mpd_del(&q); mpd_qfinalize(r, ctx, status); } static void _mpd_qrescale(mpd_t *result, const mpd_t *a, mpd_ssize_t exp, const mpd_context_t *ctx, uint32_t *status) { mpd_ssize_t expdiff, shift; mpd_uint_t rnd; if (mpd_isspecial(a)) { mpd_qcopy(result, a, status); return; } if (mpd_iszero(a)) { _settriple(result, mpd_sign(a), 0, exp); return; } expdiff = a->exp - exp; if (expdiff >= 0) { shift = expdiff; if (a->digits + shift > MPD_MAX_PREC+1) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (!mpd_qshiftl(result, a, shift, status)) { return; } result->exp = exp; } else { shift = -expdiff; rnd = mpd_qshiftr(result, a, shift, status); if (rnd == MPD_UINT_MAX) { return; } result->exp = exp; _mpd_apply_round_excess(result, rnd, ctx, status); *status |= MPD_Rounded; if (rnd) { *status |= MPD_Inexact; } } if (mpd_issubnormal(result, ctx)) { *status |= MPD_Subnormal; } } /* * Rescale a number so that it has exponent 'exp'. Does not regard context * precision, emax, emin, but uses the rounding mode. Special numbers are * quietly copied. Restrictions: * * MPD_MIN_ETINY <= exp <= MPD_MAX_EMAX+1 * result->digits <= MPD_MAX_PREC+1 */ void mpd_qrescale(mpd_t *result, const mpd_t *a, mpd_ssize_t exp, const mpd_context_t *ctx, uint32_t *status) { if (exp > MPD_MAX_EMAX+1 || exp < MPD_MIN_ETINY) { mpd_seterror(result, MPD_Invalid_operation, status); return; } _mpd_qrescale(result, a, exp, ctx, status); } /* * Same as mpd_qrescale, but with relaxed restrictions. The result of this * function should only be used for formatting a number and never as input * for other operations. * * MPD_MIN_ETINY-MPD_MAX_PREC <= exp <= MPD_MAX_EMAX+1 * result->digits <= MPD_MAX_PREC+1 */ void mpd_qrescale_fmt(mpd_t *result, const mpd_t *a, mpd_ssize_t exp, const mpd_context_t *ctx, uint32_t *status) { if (exp > MPD_MAX_EMAX+1 || exp < MPD_MIN_ETINY-MPD_MAX_PREC) { mpd_seterror(result, MPD_Invalid_operation, status); return; } _mpd_qrescale(result, a, exp, ctx, status); } /* Round to an integer according to 'action' and ctx->round. */ enum {TO_INT_EXACT, TO_INT_SILENT, TO_INT_TRUNC}; static void _mpd_qround_to_integral(int action, mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_uint_t rnd; if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } mpd_qcopy(result, a, status); return; } if (a->exp >= 0) { mpd_qcopy(result, a, status); return; } if (mpd_iszerocoeff(a)) { _settriple(result, mpd_sign(a), 0, 0); return; } rnd = mpd_qshiftr(result, a, -a->exp, status); if (rnd == MPD_UINT_MAX) { return; } result->exp = 0; if (action == TO_INT_EXACT || action == TO_INT_SILENT) { _mpd_apply_round_excess(result, rnd, ctx, status); if (action == TO_INT_EXACT) { *status |= MPD_Rounded; if (rnd) { *status |= MPD_Inexact; } } } } void mpd_qround_to_intx(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { (void)_mpd_qround_to_integral(TO_INT_EXACT, result, a, ctx, status); } void mpd_qround_to_int(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { (void)_mpd_qround_to_integral(TO_INT_SILENT, result, a, ctx, status); } void mpd_qtrunc(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { (void)_mpd_qround_to_integral(TO_INT_TRUNC, result, a, ctx, status); } void mpd_qfloor(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t workctx = *ctx; workctx.round = MPD_ROUND_FLOOR; (void)_mpd_qround_to_integral(TO_INT_SILENT, result, a, &workctx, status); } void mpd_qceil(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t workctx = *ctx; workctx.round = MPD_ROUND_CEILING; (void)_mpd_qround_to_integral(TO_INT_SILENT, result, a, &workctx, status); } int mpd_same_quantum(const mpd_t *a, const mpd_t *b) { if (mpd_isspecial(a) || mpd_isspecial(b)) { return ((mpd_isnan(a) && mpd_isnan(b)) || (mpd_isinfinite(a) && mpd_isinfinite(b))); } return a->exp == b->exp; } /* Schedule the increase in precision for the Newton iteration. */ static inline int recpr_schedule_prec(mpd_ssize_t klist[MPD_MAX_PREC_LOG2], mpd_ssize_t maxprec, mpd_ssize_t initprec) { mpd_ssize_t k; int i; assert(maxprec > 0 && initprec > 0); if (maxprec <= initprec) return -1; i = 0; k = maxprec; do { k = (k+1) / 2; klist[i++] = k; } while (k > initprec); return i-1; } /* * Initial approximation for the reciprocal: * k_0 := MPD_RDIGITS-2 * z_0 := 10**(-k_0) * floor(10**(2*k_0 + 2) / floor(v * 10**(k_0 + 2))) * Absolute error: * |1/v - z_0| < 10**(-k_0) * ACL2 proof: maxerror-inverse-approx */ static void _mpd_qreciprocal_approx(mpd_t *z, const mpd_t *v, uint32_t *status) { mpd_uint_t p10data[2] = {0, mpd_pow10[MPD_RDIGITS-2]}; mpd_uint_t dummy, word; int n; assert(v->exp == -v->digits); _mpd_get_msdigits(&dummy, &word, v, MPD_RDIGITS); n = mpd_word_digits(word); word *= mpd_pow10[MPD_RDIGITS-n]; mpd_qresize(z, 2, status); (void)_mpd_shortdiv(z->data, p10data, 2, word); mpd_clear_flags(z); z->exp = -(MPD_RDIGITS-2); z->len = (z->data[1] == 0) ? 1 : 2; mpd_setdigits(z); } /* * Reciprocal, calculated with Newton's Method. Assumption: result != a. * NOTE: The comments in the function show that certain operations are * exact. The proof for the maximum error is too long to fit in here. * ACL2 proof: maxerror-inverse-complete */ static void _mpd_qreciprocal(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t varcontext, maxcontext; mpd_t *z = result; /* current approximation */ mpd_t *v; /* a, normalized to a number between 0.1 and 1 */ MPD_NEW_SHARED(vtmp, a); /* v shares data with a */ MPD_NEW_STATIC(s,0,0,0,0); /* temporary variable */ MPD_NEW_STATIC(t,0,0,0,0); /* temporary variable */ MPD_NEW_CONST(two,0,0,1,1,1,2); /* const 2 */ mpd_ssize_t klist[MPD_MAX_PREC_LOG2]; mpd_ssize_t adj, maxprec, initprec; uint8_t sign = mpd_sign(a); int i; assert(result != a); v = &vtmp; mpd_clear_flags(v); adj = v->digits + v->exp; v->exp = -v->digits; /* Initial approximation */ _mpd_qreciprocal_approx(z, v, status); mpd_maxcontext(&varcontext); mpd_maxcontext(&maxcontext); varcontext.round = maxcontext.round = MPD_ROUND_TRUNC; varcontext.emax = maxcontext.emax = MPD_MAX_EMAX + 100; varcontext.emin = maxcontext.emin = MPD_MIN_EMIN - 100; maxcontext.prec = MPD_MAX_PREC + 100; maxprec = ctx->prec; maxprec += 2; initprec = MPD_RDIGITS-3; i = recpr_schedule_prec(klist, maxprec, initprec); for (; i >= 0; i--) { /* Loop invariant: z->digits <= klist[i]+7 */ /* Let s := z**2, exact result */ _mpd_qmul_exact(&s, z, z, &maxcontext, status); varcontext.prec = 2*klist[i] + 5; if (v->digits > varcontext.prec) { /* Let t := v, truncated to n >= 2*k+5 fraction digits */ mpd_qshiftr(&t, v, v->digits-varcontext.prec, status); t.exp = -varcontext.prec; /* Let t := trunc(v)*s, truncated to n >= 2*k+1 fraction digits */ mpd_qmul(&t, &t, &s, &varcontext, status); } else { /* v->digits <= 2*k+5 */ /* Let t := v*s, truncated to n >= 2*k+1 fraction digits */ mpd_qmul(&t, v, &s, &varcontext, status); } /* Let s := 2*z, exact result */ _mpd_qmul_exact(&s, z, &two, &maxcontext, status); /* s.digits < t.digits <= 2*k+5, |adjexp(s)-adjexp(t)| <= 1, * so the subtraction generates at most 2*k+6 <= klist[i+1]+7 * digits. The loop invariant is preserved. */ _mpd_qsub_exact(z, &s, &t, &maxcontext, status); } if (!mpd_isspecial(z)) { z->exp -= adj; mpd_set_flags(z, sign); } mpd_del(&s); mpd_del(&t); mpd_qfinalize(z, ctx, status); } /* * Internal function for large numbers: * * q, r = divmod(coeff(a), coeff(b)) * * Strategy: Multiply the dividend by the reciprocal of the divisor. The * inexact result is fixed by a small loop, using at most one iteration. * * ACL2 proofs: * ------------ * 1) q is a natural number. (ndivmod-quotient-natp) * 2) r is a natural number. (ndivmod-remainder-natp) * 3) a = q * b + r (ndivmod-q*b+r==a) * 4) r < b (ndivmod-remainder-<-b) */ static void _mpd_base_ndivmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b, uint32_t *status) { mpd_context_t workctx; mpd_t *qq = q, *rr = r; mpd_t aa, bb; int k; _mpd_copy_shared(&aa, a); _mpd_copy_shared(&bb, b); mpd_set_positive(&aa); mpd_set_positive(&bb); aa.exp = 0; bb.exp = 0; if (q == a || q == b) { if ((qq = mpd_qnew()) == NULL) { *status |= MPD_Malloc_error; goto nanresult; } } if (r == a || r == b) { if ((rr = mpd_qnew()) == NULL) { *status |= MPD_Malloc_error; goto nanresult; } } mpd_maxcontext(&workctx); /* Let prec := adigits - bdigits + 4 */ workctx.prec = a->digits - b->digits + 1 + 3; if (a->digits > MPD_MAX_PREC || workctx.prec > MPD_MAX_PREC) { *status |= MPD_Division_impossible; goto nanresult; } /* Let x := _mpd_qreciprocal(b, prec) * Then x is bounded by: * 1) 1/b - 10**(-prec - bdigits) < x < 1/b + 10**(-prec - bdigits) * 2) 1/b - 10**(-adigits - 4) < x < 1/b + 10**(-adigits - 4) */ _mpd_qreciprocal(rr, &bb, &workctx, &workctx.status); /* Get an estimate for the quotient. Let q := a * x * Then q is bounded by: * 3) a/b - 10**-4 < q < a/b + 10**-4 */ _mpd_qmul(qq, &aa, rr, &workctx, &workctx.status); /* Truncate q to an integer: * 4) a/b - 2 < trunc(q) < a/b + 1 */ mpd_qtrunc(qq, qq, &workctx, &workctx.status); workctx.prec = aa.digits + 3; workctx.emax = MPD_MAX_EMAX + 3; workctx.emin = MPD_MIN_EMIN - 3; /* Multiply the estimate for q by b: * 5) a - 2 * b < trunc(q) * b < a + b */ _mpd_qmul(rr, &bb, qq, &workctx, &workctx.status); /* Get the estimate for r such that a = q * b + r. */ _mpd_qsub_exact(rr, &aa, rr, &workctx, &workctx.status); /* Fix the result. At this point -b < r < 2*b, so the correction loop takes at most one iteration. */ for (k = 0;; k++) { if (mpd_isspecial(qq) || mpd_isspecial(rr)) { *status |= (workctx.status&MPD_Errors); goto nanresult; } if (k > 2) { /* Allow two iterations despite the proof. */ mpd_err_warn("libmpdec: internal error in " /* GCOV_NOT_REACHED */ "_mpd_base_ndivmod: please report"); /* GCOV_NOT_REACHED */ *status |= MPD_Invalid_operation; /* GCOV_NOT_REACHED */ goto nanresult; /* GCOV_NOT_REACHED */ } /* r < 0 */ else if (_mpd_cmp(&zero, rr) == 1) { _mpd_qadd_exact(rr, rr, &bb, &workctx, &workctx.status); _mpd_qadd_exact(qq, qq, &minus_one, &workctx, &workctx.status); } /* 0 <= r < b */ else if (_mpd_cmp(rr, &bb) == -1) { break; } /* r >= b */ else { _mpd_qsub_exact(rr, rr, &bb, &workctx, &workctx.status); _mpd_qadd_exact(qq, qq, &one, &workctx, &workctx.status); } } if (qq != q) { if (!mpd_qcopy(q, qq, status)) { goto nanresult; /* GCOV_UNLIKELY */ } mpd_del(qq); } if (rr != r) { if (!mpd_qcopy(r, rr, status)) { goto nanresult; /* GCOV_UNLIKELY */ } mpd_del(rr); } *status |= (workctx.status&MPD_Errors); return; nanresult: if (qq && qq != q) mpd_del(qq); if (rr && rr != r) mpd_del(rr); mpd_setspecial(q, MPD_POS, MPD_NAN); mpd_setspecial(r, MPD_POS, MPD_NAN); } /* LIBMPDEC_ONLY */ /* * Schedule the optimal precision increase for the Newton iteration. * v := input operand * z_0 := initial approximation * initprec := natural number such that abs(sqrt(v) - z_0) < 10**-initprec * maxprec := target precision * * For convenience the output klist contains the elements in reverse order: * klist := [k_n-1, ..., k_0], where * 1) k_0 <= initprec and * 2) abs(sqrt(v) - result) < 10**(-2*k_n-1 + 2) <= 10**-maxprec. */ static inline int invroot_schedule_prec(mpd_ssize_t klist[MPD_MAX_PREC_LOG2], mpd_ssize_t maxprec, mpd_ssize_t initprec) { mpd_ssize_t k; int i; assert(maxprec >= 3 && initprec >= 3); if (maxprec <= initprec) return -1; i = 0; k = maxprec; do { k = (k+3) / 2; klist[i++] = k; } while (k > initprec); return i-1; } /* * Initial approximation for the inverse square root function. * Input: * v := rational number, with 1 <= v < 100 * vhat := floor(v * 10**6) * Output: * z := approximation to 1/sqrt(v), such that abs(z - 1/sqrt(v)) < 10**-3. */ static inline void _invroot_init_approx(mpd_t *z, mpd_uint_t vhat) { mpd_uint_t lo = 1000; mpd_uint_t hi = 10000; mpd_uint_t a, sq; assert(lo*lo <= vhat && vhat < (hi+1)*(hi+1)); for(;;) { a = (lo + hi) / 2; sq = a * a; if (vhat >= sq) { if (vhat < sq + 2*a + 1) { break; } lo = a + 1; } else { hi = a - 1; } } /* * After the binary search we have: * 1) a**2 <= floor(v * 10**6) < (a + 1)**2 * This implies: * 2) a**2 <= v * 10**6 < (a + 1)**2 * 3) a <= sqrt(v) * 10**3 < a + 1 * Since 10**3 <= a: * 4) 0 <= 10**prec/a - 1/sqrt(v) < 10**-prec * We have: * 5) 10**3/a - 10**-3 < floor(10**9/a) * 10**-6 <= 10**3/a * Merging 4) and 5): * 6) abs(floor(10**9/a) * 10**-6 - 1/sqrt(v)) < 10**-3 */ mpd_minalloc(z); mpd_clear_flags(z); z->data[0] = 1000000000UL / a; z->len = 1; z->exp = -6; mpd_setdigits(z); } /* * Set 'result' to 1/sqrt(a). * Relative error: abs(result - 1/sqrt(a)) < 10**-prec * 1/sqrt(a) */ static void _mpd_qinvroot(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { uint32_t workstatus = 0; mpd_context_t varcontext, maxcontext; mpd_t *z = result; /* current approximation */ mpd_t *v; /* a, normalized to a number between 1 and 100 */ MPD_NEW_SHARED(vtmp, a); /* by default v will share data with a */ MPD_NEW_STATIC(s,0,0,0,0); /* temporary variable */ MPD_NEW_STATIC(t,0,0,0,0); /* temporary variable */ MPD_NEW_CONST(one_half,0,-1,1,1,1,5); MPD_NEW_CONST(three,0,0,1,1,1,3); mpd_ssize_t klist[MPD_MAX_PREC_LOG2]; mpd_ssize_t ideal_exp, shift; mpd_ssize_t adj, tz; mpd_ssize_t maxprec, fracdigits; mpd_uint_t vhat, dummy; int i, n; ideal_exp = -(a->exp - (a->exp & 1)) / 2; v = &vtmp; if (result == a) { if ((v = mpd_qncopy(a)) == NULL) { mpd_seterror(result, MPD_Malloc_error, status); return; } } /* normalize a to 1 <= v < 100 */ if ((v->digits+v->exp) & 1) { fracdigits = v->digits - 1; v->exp = -fracdigits; n = (v->digits > 7) ? 7 : (int)v->digits; /* Let vhat := floor(v * 10**(2*initprec)) */ _mpd_get_msdigits(&dummy, &vhat, v, n); if (n < 7) { vhat *= mpd_pow10[7-n]; } } else { fracdigits = v->digits - 2; v->exp = -fracdigits; n = (v->digits > 8) ? 8 : (int)v->digits; /* Let vhat := floor(v * 10**(2*initprec)) */ _mpd_get_msdigits(&dummy, &vhat, v, n); if (n < 8) { vhat *= mpd_pow10[8-n]; } } adj = (a->exp-v->exp) / 2; /* initial approximation */ _invroot_init_approx(z, vhat); mpd_maxcontext(&maxcontext); mpd_maxcontext(&varcontext); varcontext.round = MPD_ROUND_TRUNC; maxprec = ctx->prec + 1; /* initprec == 3 */ i = invroot_schedule_prec(klist, maxprec, 3); for (; i >= 0; i--) { varcontext.prec = 2*klist[i]+2; mpd_qmul(&s, z, z, &maxcontext, &workstatus); if (v->digits > varcontext.prec) { shift = v->digits - varcontext.prec; mpd_qshiftr(&t, v, shift, &workstatus); t.exp += shift; mpd_qmul(&t, &t, &s, &varcontext, &workstatus); } else { mpd_qmul(&t, v, &s, &varcontext, &workstatus); } mpd_qsub(&t, &three, &t, &maxcontext, &workstatus); mpd_qmul(z, z, &t, &varcontext, &workstatus); mpd_qmul(z, z, &one_half, &maxcontext, &workstatus); } z->exp -= adj; tz = mpd_trail_zeros(result); shift = ideal_exp - result->exp; shift = (tz > shift) ? shift : tz; if (shift > 0) { mpd_qshiftr_inplace(result, shift); result->exp += shift; } mpd_del(&s); mpd_del(&t); if (v != &vtmp) mpd_del(v); *status |= (workstatus&MPD_Errors); *status |= (MPD_Rounded|MPD_Inexact); } void mpd_qinvroot(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t workctx; if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } if (mpd_isnegative(a)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } /* positive infinity */ _settriple(result, MPD_POS, 0, mpd_etiny(ctx)); *status |= MPD_Clamped; return; } if (mpd_iszero(a)) { mpd_setspecial(result, mpd_sign(a), MPD_INF); *status |= MPD_Division_by_zero; return; } if (mpd_isnegative(a)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } workctx = *ctx; workctx.prec += 2; workctx.round = MPD_ROUND_HALF_EVEN; _mpd_qinvroot(result, a, &workctx, status); mpd_qfinalize(result, ctx, status); } /* END LIBMPDEC_ONLY */ /* Algorithm from decimal.py */ void mpd_qsqrt(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, uint32_t *status) { mpd_context_t maxcontext; MPD_NEW_STATIC(c,0,0,0,0); MPD_NEW_STATIC(q,0,0,0,0); MPD_NEW_STATIC(r,0,0,0,0); MPD_NEW_CONST(two,0,0,1,1,1,2); mpd_ssize_t prec, ideal_exp; mpd_ssize_t l, shift; int exact = 0; ideal_exp = (a->exp - (a->exp & 1)) / 2; if (mpd_isspecial(a)) { if (mpd_qcheck_nan(result, a, ctx, status)) { return; } if (mpd_isnegative(a)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } mpd_setspecial(result, MPD_POS, MPD_INF); return; } if (mpd_iszero(a)) { _settriple(result, mpd_sign(a), 0, ideal_exp); mpd_qfinalize(result, ctx, status); return; } if (mpd_isnegative(a)) { mpd_seterror(result, MPD_Invalid_operation, status); return; } mpd_maxcontext(&maxcontext); prec = ctx->prec + 1; if (!mpd_qcopy(&c, a, status)) { goto malloc_error; } c.exp = 0; if (a->exp & 1) { if (!mpd_qshiftl(&c, &c, 1, status)) { goto malloc_error; } l = (a->digits >> 1) + 1; } else { l = (a->digits + 1) >> 1; } shift = prec - l; if (shift >= 0) { if (!mpd_qshiftl(&c, &c, 2*shift, status)) { goto malloc_error; } exact = 1; } else { exact = !mpd_qshiftr_inplace(&c, -2*shift); } ideal_exp -= shift; /* find result = floor(sqrt(c)) using Newton's method */ if (!mpd_qshiftl(result, &one, prec, status)) { goto malloc_error; } while (1) { _mpd_qdivmod(&q, &r, &c, result, &maxcontext, &maxcontext.status); if (mpd_isspecial(result) || mpd_isspecial(&q)) { mpd_seterror(result, maxcontext.status&MPD_Errors, status); goto out; } if (_mpd_cmp(result, &q) <= 0) { break; } _mpd_qadd_exact(result, result, &q, &maxcontext, &maxcontext.status); if (mpd_isspecial(result)) { mpd_seterror(result, maxcontext.status&MPD_Errors, status); goto out; } _mpd_qdivmod(result, &r, result, &two, &maxcontext, &maxcontext.status); } if (exact) { _mpd_qmul_exact(&r, result, result, &maxcontext, &maxcontext.status); if (mpd_isspecial(&r)) { mpd_seterror(result, maxcontext.status&MPD_Errors, status); goto out; } exact = (_mpd_cmp(&r, &c) == 0); } if (exact) { if (shift >= 0) { mpd_qshiftr_inplace(result, shift); } else { if (!mpd_qshiftl(result, result, -shift, status)) { goto malloc_error; } } ideal_exp += shift; } else { int lsd = (int)mpd_lsd(result->data[0]); if (lsd == 0 || lsd == 5) { result->data[0] += 1; } } result->exp = ideal_exp; out: mpd_del(&c); mpd_del(&q); mpd_del(&r); maxcontext = *ctx; maxcontext.round = MPD_ROUND_HALF_EVEN; mpd_qfinalize(result, &maxcontext, status); return; malloc_error: mpd_seterror(result, MPD_Malloc_error, status); goto out; } /******************************************************************************/ /* Base conversions */ /******************************************************************************/ /* Space needed to represent an integer mpd_t in base 'base'. */ size_t mpd_sizeinbase(const mpd_t *a, uint32_t base) { double x; size_t digits; assert(mpd_isinteger(a)); assert(base >= 2); if (mpd_iszero(a)) { return 1; } digits = a->digits+a->exp; assert(digits > 0); #ifdef CONFIG_64 /* ceil(2711437152599294 / log10(2)) + 4 == 2**53 */ if (digits > 2711437152599294ULL) { return SIZE_MAX; } #endif x = (double)digits / log10(base); return (x > SIZE_MAX-1) ? SIZE_MAX : (size_t)x + 1; } /* Space needed to import a base 'base' integer of length 'srclen'. */ static mpd_ssize_t _mpd_importsize(size_t srclen, uint32_t base) { double x; assert(srclen > 0); assert(base >= 2); #if SIZE_MAX == UINT64_MAX if (srclen > (1ULL<<53)) { return MPD_SSIZE_MAX; } #endif x = (double)srclen * (log10(base)/MPD_RDIGITS); return (x >= MPD_MAXIMPORT) ? MPD_SSIZE_MAX : (mpd_ssize_t)x + 1; } static uint8_t mpd_resize_u16(uint16_t **w, size_t nmemb) { uint8_t err = 0; *w = mpd_realloc(*w, nmemb, sizeof **w, &err); return !err; } static uint8_t mpd_resize_u32(uint32_t **w, size_t nmemb) { uint8_t err = 0; *w = mpd_realloc(*w, nmemb, sizeof **w, &err); return !err; } static size_t _baseconv_to_u16(uint16_t **w, size_t wlen, mpd_uint_t wbase, mpd_uint_t *u, mpd_ssize_t ulen) { size_t n = 0; assert(wlen > 0 && ulen > 0); assert(wbase <= (1U<<16)); do { if (n >= wlen) { if (!mpd_resize_u16(w, n+1)) { return SIZE_MAX; } wlen = n+1; } (*w)[n++] = (uint16_t)_mpd_shortdiv(u, u, ulen, wbase); /* ulen is at least 1. u[ulen-1] can only be zero if ulen == 1. */ ulen = _mpd_real_size(u, ulen); } while (u[ulen-1] != 0); return n; } static size_t _coeff_from_u16(mpd_t *w, mpd_ssize_t wlen, const mpd_uint_t *u, size_t ulen, uint32_t ubase, uint32_t *status) { mpd_ssize_t n = 0; mpd_uint_t carry; assert(wlen > 0 && ulen > 0); assert(ubase <= (1U<<16)); w->data[n++] = u[--ulen]; while (--ulen != SIZE_MAX) { carry = _mpd_shortmul_c(w->data, w->data, n, ubase); if (carry) { if (n >= wlen) { if (!mpd_qresize(w, n+1, status)) { return SIZE_MAX; } wlen = n+1; } w->data[n++] = carry; } carry = _mpd_shortadd(w->data, n, u[ulen]); if (carry) { if (n >= wlen) { if (!mpd_qresize(w, n+1, status)) { return SIZE_MAX; } wlen = n+1; } w->data[n++] = carry; } } return n; } /* target base wbase < source base ubase */ static size_t _baseconv_to_smaller(uint32_t **w, size_t wlen, uint32_t wbase, mpd_uint_t *u, mpd_ssize_t ulen, mpd_uint_t ubase) { size_t n = 0; assert(wlen > 0 && ulen > 0); assert(wbase < ubase); do { if (n >= wlen) { if (!mpd_resize_u32(w, n+1)) { return SIZE_MAX; } wlen = n+1; } (*w)[n++] = (uint32_t)_mpd_shortdiv_b(u, u, ulen, wbase, ubase); /* ulen is at least 1. u[ulen-1] can only be zero if ulen == 1. */ ulen = _mpd_real_size(u, ulen); } while (u[ulen-1] != 0); return n; } #ifdef CONFIG_32 /* target base 'wbase' == source base 'ubase' */ static size_t _copy_equal_base(uint32_t **w, size_t wlen, const uint32_t *u, size_t ulen) { if (wlen < ulen) { if (!mpd_resize_u32(w, ulen)) { return SIZE_MAX; } } memcpy(*w, u, ulen * (sizeof **w)); return ulen; } /* target base 'wbase' > source base 'ubase' */ static size_t _baseconv_to_larger(uint32_t **w, size_t wlen, mpd_uint_t wbase, const mpd_uint_t *u, size_t ulen, mpd_uint_t ubase) { size_t n = 0; mpd_uint_t carry; assert(wlen > 0 && ulen > 0); assert(ubase < wbase); (*w)[n++] = u[--ulen]; while (--ulen != SIZE_MAX) { carry = _mpd_shortmul_b(*w, *w, n, ubase, wbase); if (carry) { if (n >= wlen) { if (!mpd_resize_u32(w, n+1)) { return SIZE_MAX; } wlen = n+1; } (*w)[n++] = carry; } carry = _mpd_shortadd_b(*w, n, u[ulen], wbase); if (carry) { if (n >= wlen) { if (!mpd_resize_u32(w, n+1)) { return SIZE_MAX; } wlen = n+1; } (*w)[n++] = carry; } } return n; } /* target base wbase < source base ubase */ static size_t _coeff_from_larger_base(mpd_t *w, size_t wlen, mpd_uint_t wbase, mpd_uint_t *u, mpd_ssize_t ulen, mpd_uint_t ubase, uint32_t *status) { size_t n = 0; assert(wlen > 0 && ulen > 0); assert(wbase < ubase); do { if (n >= wlen) { if (!mpd_qresize(w, n+1, status)) { return SIZE_MAX; } wlen = n+1; } w->data[n++] = (uint32_t)_mpd_shortdiv_b(u, u, ulen, wbase, ubase); /* ulen is at least 1. u[ulen-1] can only be zero if ulen == 1. */ ulen = _mpd_real_size(u, ulen); } while (u[ulen-1] != 0); return n; } #endif /* target base 'wbase' > source base 'ubase' */ static size_t _coeff_from_smaller_base(mpd_t *w, mpd_ssize_t wlen, mpd_uint_t wbase, const uint32_t *u, size_t ulen, mpd_uint_t ubase, uint32_t *status) { mpd_ssize_t n = 0; mpd_uint_t carry; assert(wlen > 0 && ulen > 0); assert(wbase > ubase); w->data[n++] = u[--ulen]; while (--ulen != SIZE_MAX) { carry = _mpd_shortmul_b(w->data, w->data, n, ubase, wbase); if (carry) { if (n >= wlen) { if (!mpd_qresize(w, n+1, status)) { return SIZE_MAX; } wlen = n+1; } w->data[n++] = carry; } carry = _mpd_shortadd_b(w->data, n, u[ulen], wbase); if (carry) { if (n >= wlen) { if (!mpd_qresize(w, n+1, status)) { return SIZE_MAX; } wlen = n+1; } w->data[n++] = carry; } } return n; } /* * Convert an integer mpd_t to a multiprecision integer with base <= 2**16. * The least significant word of the result is (*rdata)[0]. * * If rdata is NULL, space is allocated by the function and rlen is irrelevant. * In case of an error any allocated storage is freed and rdata is set back to * NULL. * * If rdata is non-NULL, it MUST be allocated by one of libmpdec's allocation * functions and rlen MUST be correct. If necessary, the function will resize * rdata. In case of an error the caller must free rdata. * * Return value: In case of success, the exact length of rdata, SIZE_MAX * otherwise. */ size_t mpd_qexport_u16(uint16_t **rdata, size_t rlen, uint32_t rbase, const mpd_t *src, uint32_t *status) { MPD_NEW_STATIC(tsrc,0,0,0,0); int alloc = 0; /* rdata == NULL */ size_t n; assert(rbase <= (1U<<16)); if (mpd_isspecial(src) || !_mpd_isint(src)) { *status |= MPD_Invalid_operation; return SIZE_MAX; } if (*rdata == NULL) { rlen = mpd_sizeinbase(src, rbase); if (rlen == SIZE_MAX) { *status |= MPD_Invalid_operation; return SIZE_MAX; } *rdata = mpd_alloc(rlen, sizeof **rdata); if (*rdata == NULL) { goto malloc_error; } alloc = 1; } if (mpd_iszero(src)) { **rdata = 0; return 1; } if (src->exp >= 0) { if (!mpd_qshiftl(&tsrc, src, src->exp, status)) { goto malloc_error; } } else { if (mpd_qshiftr(&tsrc, src, -src->exp, status) == MPD_UINT_MAX) { goto malloc_error; } } n = _baseconv_to_u16(rdata, rlen, rbase, tsrc.data, tsrc.len); if (n == SIZE_MAX) { goto malloc_error; } out: mpd_del(&tsrc); return n; malloc_error: if (alloc) { mpd_free(*rdata); *rdata = NULL; } n = SIZE_MAX; *status |= MPD_Malloc_error; goto out; } /* * Convert an integer mpd_t to a multiprecision integer with base<=UINT32_MAX. * The least significant word of the result is (*rdata)[0]. * * If rdata is NULL, space is allocated by the function and rlen is irrelevant. * In case of an error any allocated storage is freed and rdata is set back to * NULL. * * If rdata is non-NULL, it MUST be allocated by one of libmpdec's allocation * functions and rlen MUST be correct. If necessary, the function will resize * rdata. In case of an error the caller must free rdata. * * Return value: In case of success, the exact length of rdata, SIZE_MAX * otherwise. */ size_t mpd_qexport_u32(uint32_t **rdata, size_t rlen, uint32_t rbase, const mpd_t *src, uint32_t *status) { MPD_NEW_STATIC(tsrc,0,0,0,0); int alloc = 0; /* rdata == NULL */ size_t n; if (mpd_isspecial(src) || !_mpd_isint(src)) { *status |= MPD_Invalid_operation; return SIZE_MAX; } if (*rdata == NULL) { rlen = mpd_sizeinbase(src, rbase); if (rlen == SIZE_MAX) { *status |= MPD_Invalid_operation; return SIZE_MAX; } *rdata = mpd_alloc(rlen, sizeof **rdata); if (*rdata == NULL) { goto malloc_error; } alloc = 1; } if (mpd_iszero(src)) { **rdata = 0; return 1; } if (src->exp >= 0) { if (!mpd_qshiftl(&tsrc, src, src->exp, status)) { goto malloc_error; } } else { if (mpd_qshiftr(&tsrc, src, -src->exp, status) == MPD_UINT_MAX) { goto malloc_error; } } #ifdef CONFIG_64 n = _baseconv_to_smaller(rdata, rlen, rbase, tsrc.data, tsrc.len, MPD_RADIX); #else if (rbase == MPD_RADIX) { n = _copy_equal_base(rdata, rlen, tsrc.data, tsrc.len); } else if (rbase < MPD_RADIX) { n = _baseconv_to_smaller(rdata, rlen, rbase, tsrc.data, tsrc.len, MPD_RADIX); } else { n = _baseconv_to_larger(rdata, rlen, rbase, tsrc.data, tsrc.len, MPD_RADIX); } #endif if (n == SIZE_MAX) { goto malloc_error; } out: mpd_del(&tsrc); return n; malloc_error: if (alloc) { mpd_free(*rdata); *rdata = NULL; } n = SIZE_MAX; *status |= MPD_Malloc_error; goto out; } /* * Converts a multiprecision integer with base <= UINT16_MAX+1 to an mpd_t. * The least significant word of the source is srcdata[0]. */ void mpd_qimport_u16(mpd_t *result, const uint16_t *srcdata, size_t srclen, uint8_t srcsign, uint32_t srcbase, const mpd_context_t *ctx, uint32_t *status) { mpd_uint_t *usrc; /* uint16_t src copied to an mpd_uint_t array */ mpd_ssize_t rlen; /* length of the result */ size_t n; assert(srclen > 0); assert(srcbase <= (1U<<16)); rlen = _mpd_importsize(srclen, srcbase); if (rlen == MPD_SSIZE_MAX) { mpd_seterror(result, MPD_Invalid_operation, status); return; } usrc = mpd_alloc((mpd_size_t)srclen, sizeof *usrc); if (usrc == NULL) { mpd_seterror(result, MPD_Malloc_error, status); return; } for (n = 0; n < srclen; n++) { usrc[n] = srcdata[n]; } if (!mpd_qresize(result, rlen, status)) { goto finish; } n = _coeff_from_u16(result, rlen, usrc, srclen, srcbase, status); if (n == SIZE_MAX) { goto finish; } mpd_set_flags(result, srcsign); result->exp = 0; result->len = n; mpd_setdigits(result); mpd_qresize(result, result->len, status); mpd_qfinalize(result, ctx, status); finish: mpd_free(usrc); } /* * Converts a multiprecision integer with base <= UINT32_MAX to an mpd_t. * The least significant word of the source is srcdata[0]. */ void mpd_qimport_u32(mpd_t *result, const uint32_t *srcdata, size_t srclen, uint8_t srcsign, uint32_t srcbase, const mpd_context_t *ctx, uint32_t *status) { mpd_ssize_t rlen; /* length of the result */ size_t n; assert(srclen > 0); rlen = _mpd_importsize(srclen, srcbase); if (rlen == MPD_SSIZE_MAX) { mpd_seterror(result, MPD_Invalid_operation, status); return; } if (!mpd_qresize(result, rlen, status)) { return; } #ifdef CONFIG_64 n = _coeff_from_smaller_base(result, rlen, MPD_RADIX, srcdata, srclen, srcbase, status); #else if (srcbase == MPD_RADIX) { if (!mpd_qresize(result, srclen, status)) { return; } memcpy(result->data, srcdata, srclen * (sizeof *srcdata)); n = srclen; } else if (srcbase < MPD_RADIX) { n = _coeff_from_smaller_base(result, rlen, MPD_RADIX, srcdata, srclen, srcbase, status); } else { mpd_uint_t *usrc = mpd_alloc((mpd_size_t)srclen, sizeof *usrc); if (usrc == NULL) { mpd_seterror(result, MPD_Malloc_error, status); return; } for (n = 0; n < srclen; n++) { usrc[n] = srcdata[n]; } n = _coeff_from_larger_base(result, rlen, MPD_RADIX, usrc, (mpd_ssize_t)srclen, srcbase, status); mpd_free(usrc); } #endif if (n == SIZE_MAX) { return; } mpd_set_flags(result, srcsign); result->exp = 0; result->len = n; mpd_setdigits(result); mpd_qresize(result, result->len, status); mpd_qfinalize(result, ctx, status); }