Merged revisions 70542 via svnmerge from
svn+ssh://pythondev@svn.python.org/python/trunk ........ r70542 | mark.dickinson | 2009-03-23 18:25:13 +0000 (Mon, 23 Mar 2009) | 14 lines Issue #5512: speed up the long division algorithm for Python longs. The basic algorithm remains the same; the most significant speedups come from the following three changes: (1) normalize by shifting instead of multiplying and dividing (2) the old algorithm usually did an unnecessary extra iteration of the outer loop; remove this. As a special case, this means that long divisions with a single-digit result run twice as fast as before. (3) make inner loop much tighter. Various benchmarks show speedups of between 50% and 150% for long integer divisions and modulo operations. ........
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Mark Dickinson
parent
798ee1a4c6
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@ -12,6 +12,10 @@ What's New in Python 3.1 alpha 2?
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Core and Builtins
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-----------------
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- Issue #5512: Rewrite PyLong long division algorithm (x_divrem) to
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improve its performance. Long divisions and remainder operations
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are now between 50% and 150% faster.
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- Issue #4258: Make it possible to use base 2**30 instead of base
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2**15 for the internal representation of integers, for performance
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reasons. Base 2**30 is enabled by default on 64-bit machines. Add
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@ -1363,6 +1363,26 @@ PyLong_AsUnsignedLongLongMask(register PyObject *op)
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return Py_NotImplemented; \
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}
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/* bits_in_digit(d) returns the unique integer k such that 2**(k-1) <= d <
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2**k if d is nonzero, else 0. */
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static const unsigned char BitLengthTable[32] = {
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0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4,
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5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5
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};
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static int
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bits_in_digit(digit d)
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{
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int d_bits = 0;
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while (d >= 32) {
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d_bits += 6;
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d >>= 6;
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}
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d_bits += (int)BitLengthTable[d];
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return d_bits;
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}
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/* x[0:m] and y[0:n] are digit vectors, LSD first, m >= n required. x[0:n]
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* is modified in place, by adding y to it. Carries are propagated as far as
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* x[m-1], and the remaining carry (0 or 1) is returned.
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@ -1415,25 +1435,41 @@ v_isub(digit *x, Py_ssize_t m, digit *y, Py_ssize_t n)
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return borrow;
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}
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/* Multiply by a single digit, ignoring the sign. */
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static PyLongObject *
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mul1(PyLongObject *a, digit n)
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/* Shift digit vector a[0:m] d bits left, with 0 <= d < PyLong_SHIFT. Put
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* result in z[0:m], and return the d bits shifted out of the top.
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*/
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static digit
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v_lshift(digit *z, digit *a, Py_ssize_t m, int d)
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{
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Py_ssize_t size_a = ABS(Py_SIZE(a));
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PyLongObject *z = _PyLong_New(size_a+1);
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twodigits carry = 0;
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Py_ssize_t i;
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digit carry = 0;
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if (z == NULL)
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return NULL;
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for (i = 0; i < size_a; ++i) {
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carry += (twodigits)a->ob_digit[i] * n;
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z->ob_digit[i] = (digit) (carry & PyLong_MASK);
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carry >>= PyLong_SHIFT;
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assert(0 <= d && d < PyLong_SHIFT);
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for (i=0; i < m; i++) {
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twodigits acc = (twodigits)a[i] << d | carry;
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z[i] = (digit)acc & PyLong_MASK;
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carry = (digit)(acc >> PyLong_SHIFT);
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}
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z->ob_digit[i] = (digit) carry;
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return long_normalize(z);
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return carry;
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}
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/* Shift digit vector a[0:m] d bits right, with 0 <= d < PyLong_SHIFT. Put
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* result in z[0:m], and return the d bits shifted out of the bottom.
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*/
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static digit
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v_rshift(digit *z, digit *a, Py_ssize_t m, int d)
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{
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Py_ssize_t i;
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digit carry = 0;
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digit mask = ((digit)1 << d) - 1U;
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assert(0 <= d && d < PyLong_SHIFT);
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for (i=m; i-- > 0;) {
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twodigits acc = (twodigits)carry << PyLong_SHIFT | a[i];
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carry = (digit)acc & mask;
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z[i] = (digit)(acc >> d);
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}
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return carry;
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}
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/* Divide long pin, w/ size digits, by non-zero digit n, storing quotient
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@ -2089,104 +2125,131 @@ long_divrem(PyLongObject *a, PyLongObject *b,
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return 0;
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}
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/* Unsigned long division with remainder -- the algorithm */
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/* Unsigned long division with remainder -- the algorithm. The arguments v1
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and w1 should satisfy 2 <= ABS(Py_SIZE(w1)) <= ABS(Py_SIZE(v1)). */
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static PyLongObject *
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x_divrem(PyLongObject *v1, PyLongObject *w1, PyLongObject **prem)
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{
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Py_ssize_t size_v = ABS(Py_SIZE(v1)), size_w = ABS(Py_SIZE(w1));
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digit d = (digit) ((twodigits)PyLong_BASE / (w1->ob_digit[size_w-1] + 1));
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PyLongObject *v = mul1(v1, d);
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PyLongObject *w = mul1(w1, d);
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PyLongObject *a;
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Py_ssize_t j, k;
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PyLongObject *v, *w, *a;
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Py_ssize_t i, k, size_v, size_w;
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int d;
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digit wm1, wm2, carry, q, r, vtop, *v0, *vk, *w0, *ak;
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twodigits vv;
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sdigit zhi;
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stwodigits z;
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if (v == NULL || w == NULL) {
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Py_XDECREF(v);
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Py_XDECREF(w);
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/* We follow Knuth [The Art of Computer Programming, Vol. 2 (3rd
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edn.), section 4.3.1, Algorithm D], except that we don't explicitly
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handle the special case when the initial estimate q for a quotient
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digit is >= PyLong_BASE: the max value for q is PyLong_BASE+1, and
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that won't overflow a digit. */
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/* allocate space; w will also be used to hold the final remainder */
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size_v = ABS(Py_SIZE(v1));
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size_w = ABS(Py_SIZE(w1));
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assert(size_v >= size_w && size_w >= 2); /* Assert checks by div() */
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v = _PyLong_New(size_v+1);
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if (v == NULL) {
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*prem = NULL;
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return NULL;
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}
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w = _PyLong_New(size_w);
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if (w == NULL) {
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Py_DECREF(v);
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*prem = NULL;
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return NULL;
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}
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assert(size_v >= size_w && size_w > 1); /* Assert checks by div() */
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assert(Py_REFCNT(v) == 1); /* Since v will be used as accumulator! */
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assert(size_w == ABS(Py_SIZE(w))); /* That's how d was calculated */
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/* normalize: shift w1 left so that its top digit is >= PyLong_BASE/2.
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shift v1 left by the same amount. Results go into w and v. */
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d = PyLong_SHIFT - bits_in_digit(w1->ob_digit[size_w-1]);
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carry = v_lshift(w->ob_digit, w1->ob_digit, size_w, d);
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assert(carry == 0);
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carry = v_lshift(v->ob_digit, v1->ob_digit, size_v, d);
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if (carry != 0 || v->ob_digit[size_v-1] >= w->ob_digit[size_w-1]) {
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v->ob_digit[size_v] = carry;
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size_v++;
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}
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size_v = ABS(Py_SIZE(v));
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/* Now v->ob_digit[size_v-1] < w->ob_digit[size_w-1], so quotient has
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at most (and usually exactly) k = size_v - size_w digits. */
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k = size_v - size_w;
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a = _PyLong_New(k + 1);
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for (j = size_v; a != NULL && k >= 0; --j, --k) {
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digit vj = (j >= size_v) ? 0 : v->ob_digit[j];
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twodigits q;
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stwodigits carry = 0;
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Py_ssize_t i;
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assert(k >= 0);
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a = _PyLong_New(k);
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if (a == NULL) {
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Py_DECREF(w);
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Py_DECREF(v);
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*prem = NULL;
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return NULL;
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}
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v0 = v->ob_digit;
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w0 = w->ob_digit;
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wm1 = w0[size_w-1];
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wm2 = w0[size_w-2];
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for (vk = v0+k, ak = a->ob_digit + k; vk-- > v0;) {
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/* inner loop: divide vk[0:size_w+1] by w0[0:size_w], giving
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single-digit quotient q, remainder in vk[0:size_w]. */
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SIGCHECK({
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Py_DECREF(a);
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a = NULL;
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break;
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})
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if (vj == w->ob_digit[size_w-1])
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q = PyLong_MASK;
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else
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q = (((twodigits)vj << PyLong_SHIFT) + v->ob_digit[j-1]) /
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w->ob_digit[size_w-1];
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while (w->ob_digit[size_w-2]*q >
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((
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((twodigits)vj << PyLong_SHIFT)
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+ v->ob_digit[j-1]
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- q*w->ob_digit[size_w-1]
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) << PyLong_SHIFT)
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+ v->ob_digit[j-2])
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--q;
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for (i = 0; i < size_w && i+k < size_v; ++i) {
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twodigits z = w->ob_digit[i] * q;
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digit zz = (digit) (z >> PyLong_SHIFT);
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carry += v->ob_digit[i+k] - z
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+ ((twodigits)zz << PyLong_SHIFT);
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v->ob_digit[i+k] = (digit)(carry & PyLong_MASK);
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carry = Py_ARITHMETIC_RIGHT_SHIFT(stwodigits,
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carry, PyLong_SHIFT);
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carry -= zz;
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}
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if (i+k < size_v) {
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carry += v->ob_digit[i+k];
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v->ob_digit[i+k] = 0;
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}
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if (carry == 0)
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a->ob_digit[k] = (digit) q;
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else {
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assert(carry == -1);
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a->ob_digit[k] = (digit) q-1;
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carry = 0;
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for (i = 0; i < size_w && i+k < size_v; ++i) {
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carry += v->ob_digit[i+k] + w->ob_digit[i];
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v->ob_digit[i+k] = (digit)(carry & PyLong_MASK);
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carry = Py_ARITHMETIC_RIGHT_SHIFT(
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stwodigits,
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carry, PyLong_SHIFT);
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}
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}
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} /* for j, k */
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if (a == NULL)
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*prem = NULL;
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else {
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a = long_normalize(a);
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*prem = divrem1(v, d, &d);
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/* d receives the (unused) remainder */
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if (*prem == NULL) {
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Py_DECREF(a);
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a = NULL;
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}
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}
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Py_DECREF(v);
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Py_DECREF(w);
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return a;
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Py_DECREF(v);
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*prem = NULL;
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return NULL;
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})
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/* estimate quotient digit q; may overestimate by 1 (rare) */
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vtop = vk[size_w];
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assert(vtop <= wm1);
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vv = ((twodigits)vtop << PyLong_SHIFT) | vk[size_w-1];
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q = (digit)(vv / wm1);
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r = (digit)(vv - (twodigits)wm1 * q); /* r = vv % wm1 */
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while ((twodigits)wm2 * q > (((twodigits)r << PyLong_SHIFT)
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--q;
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r += wm1;
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if (r >= PyLong_BASE)
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break;
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}
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assert(q <= PyLong_BASE);
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/* subtract q*w0[0:size_w] from vk[0:size_w+1] */
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zhi = 0;
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for (i = 0; i < size_w; ++i) {
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/* invariants: -PyLong_BASE <= -q <= zhi <= 0;
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-PyLong_BASE * q <= z < PyLong_BASE */
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z = (sdigit)vk[i] + zhi -
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(stwodigits)q * (stwodigits)w0[i];
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vk[i] = (digit)z & PyLong_MASK;
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zhi = (sdigit)Py_ARITHMETIC_RIGHT_SHIFT(stwodigits,
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z, PyLong_SHIFT);
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}
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/* add w back if q was too large (this branch taken rarely) */
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assert((sdigit)vtop + zhi == -1 || (sdigit)vtop + zhi == 0);
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if ((sdigit)vtop + zhi < 0) {
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carry = 0;
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for (i = 0; i < size_w; ++i) {
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carry += vk[i] + w0[i];
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vk[i] = carry & PyLong_MASK;
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carry >>= PyLong_SHIFT;
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}
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--q;
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}
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/* store quotient digit */
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assert(q < PyLong_BASE);
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*--ak = q;
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}
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/* unshift remainder; we reuse w to store the result */
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carry = v_rshift(w0, v0, size_w, d);
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assert(carry==0);
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Py_DECREF(v);
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*prem = long_normalize(w);
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return long_normalize(a);
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}
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/* Methods */
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@ -3793,11 +3856,6 @@ long_sizeof(PyLongObject *v)
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return PyLong_FromSsize_t(res);
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}
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static const unsigned char BitLengthTable[32] = {
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0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4,
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5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5
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};
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static PyObject *
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long_bit_length(PyLongObject *v)
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{
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