import unittest from test import support import sys import random import math # Used for lazy formatting of failure messages class Frm(object): def __init__(self, format, *args): self.format = format self.args = args def __str__(self): return self.format % self.args # decorator for skipping tests on non-IEEE 754 platforms requires_IEEE_754 = unittest.skipUnless( float.__getformat__("double").startswith("IEEE"), "test requires IEEE 754 doubles") # SHIFT should match the value in longintrepr.h for best testing. SHIFT = sys.int_info.bits_per_digit BASE = 2 ** SHIFT MASK = BASE - 1 KARATSUBA_CUTOFF = 70 # from longobject.c # Max number of base BASE digits to use in test cases. Doubling # this will more than double the runtime. MAXDIGITS = 15 # build some special values special = [0, 1, 2, BASE, BASE >> 1, 0x5555555555555555, 0xaaaaaaaaaaaaaaaa] # some solid strings of one bits p2 = 4 # 0 and 1 already added for i in range(2*SHIFT): special.append(p2 - 1) p2 = p2 << 1 del p2 # add complements & negations special += [~x for x in special] + [-x for x in special] DBL_MAX = sys.float_info.max DBL_MAX_EXP = sys.float_info.max_exp DBL_MIN_EXP = sys.float_info.min_exp DBL_MANT_DIG = sys.float_info.mant_dig DBL_MIN_OVERFLOW = 2**DBL_MAX_EXP - 2**(DBL_MAX_EXP - DBL_MANT_DIG - 1) # pure Python version of correctly-rounded true division def truediv(a, b): """Correctly-rounded true division for integers.""" negative = a^b < 0 a, b = abs(a), abs(b) # exceptions: division by zero, overflow if not b: raise ZeroDivisionError("division by zero") if a >= DBL_MIN_OVERFLOW * b: raise OverflowError("int/int too large to represent as a float") # find integer d satisfying 2**(d - 1) <= a/b < 2**d d = a.bit_length() - b.bit_length() if d >= 0 and a >= 2**d * b or d < 0 and a * 2**-d >= b: d += 1 # compute 2**-exp * a / b for suitable exp exp = max(d, DBL_MIN_EXP) - DBL_MANT_DIG a, b = a << max(-exp, 0), b << max(exp, 0) q, r = divmod(a, b) # round-half-to-even: fractional part is r/b, which is > 0.5 iff # 2*r > b, and == 0.5 iff 2*r == b. if 2*r > b or 2*r == b and q % 2 == 1: q += 1 result = float(q) * 2.**exp return -result if negative else result class LongTest(unittest.TestCase): # Get quasi-random long consisting of ndigits digits (in base BASE). # quasi == the most-significant digit will not be 0, and the number # is constructed to contain long strings of 0 and 1 bits. These are # more likely than random bits to provoke digit-boundary errors. # The sign of the number is also random. def getran(self, ndigits): self.assertTrue(ndigits > 0) nbits_hi = ndigits * SHIFT nbits_lo = nbits_hi - SHIFT + 1 answer = 0 nbits = 0 r = int(random.random() * (SHIFT * 2)) | 1 # force 1 bits to start while nbits < nbits_lo: bits = (r >> 1) + 1 bits = min(bits, nbits_hi - nbits) self.assertTrue(1 <= bits <= SHIFT) nbits = nbits + bits answer = answer << bits if r & 1: answer = answer | ((1 << bits) - 1) r = int(random.random() * (SHIFT * 2)) self.assertTrue(nbits_lo <= nbits <= nbits_hi) if random.random() < 0.5: answer = -answer return answer # Get random long consisting of ndigits random digits (relative to base # BASE). The sign bit is also random. def getran2(ndigits): answer = 0 for i in range(ndigits): answer = (answer << SHIFT) | random.randint(0, MASK) if random.random() < 0.5: answer = -answer return answer def check_division(self, x, y): eq = self.assertEqual q, r = divmod(x, y) q2, r2 = x//y, x%y pab, pba = x*y, y*x eq(pab, pba, Frm("multiplication does not commute for %r and %r", x, y)) eq(q, q2, Frm("divmod returns different quotient than / for %r and %r", x, y)) eq(r, r2, Frm("divmod returns different mod than %% for %r and %r", x, y)) eq(x, q*y + r, Frm("x != q*y + r after divmod on x=%r, y=%r", x, y)) if y > 0: self.assertTrue(0 <= r < y, Frm("bad mod from divmod on %r and %r", x, y)) else: self.assertTrue(y < r <= 0, Frm("bad mod from divmod on %r and %r", x, y)) def test_division(self): digits = list(range(1, MAXDIGITS+1)) + list(range(KARATSUBA_CUTOFF, KARATSUBA_CUTOFF + 14)) digits.append(KARATSUBA_CUTOFF * 3) for lenx in digits: x = self.getran(lenx) for leny in digits: y = self.getran(leny) or 1 self.check_division(x, y) # specific numbers chosen to exercise corner cases of the # current long division implementation # 30-bit cases involving a quotient digit estimate of BASE+1 self.check_division(1231948412290879395966702881, 1147341367131428698) self.check_division(815427756481275430342312021515587883, 707270836069027745) self.check_division(627976073697012820849443363563599041, 643588798496057020) self.check_division(1115141373653752303710932756325578065, 1038556335171453937726882627) # 30-bit cases that require the post-subtraction correction step self.check_division(922498905405436751940989320930368494, 949985870686786135626943396) self.check_division(768235853328091167204009652174031844, 1091555541180371554426545266) # 15-bit cases involving a quotient digit estimate of BASE+1 self.check_division(20172188947443, 615611397) self.check_division(1020908530270155025, 950795710) self.check_division(128589565723112408, 736393718) self.check_division(609919780285761575, 18613274546784) # 15-bit cases that require the post-subtraction correction step self.check_division(710031681576388032, 26769404391308) self.check_division(1933622614268221, 30212853348836) def test_karatsuba(self): digits = list(range(1, 5)) + list(range(KARATSUBA_CUTOFF, KARATSUBA_CUTOFF + 10)) digits.extend([KARATSUBA_CUTOFF * 10, KARATSUBA_CUTOFF * 100]) bits = [digit * SHIFT for digit in digits] # Test products of long strings of 1 bits -- (2**x-1)*(2**y-1) == # 2**(x+y) - 2**x - 2**y + 1, so the proper result is easy to check. for abits in bits: a = (1 << abits) - 1 for bbits in bits: if bbits < abits: continue b = (1 << bbits) - 1 x = a * b y = ((1 << (abits + bbits)) - (1 << abits) - (1 << bbits) + 1) self.assertEqual(x, y, Frm("bad result for a*b: a=%r, b=%r, x=%r, y=%r", a, b, x, y)) def check_bitop_identities_1(self, x): eq = self.assertEqual eq(x & 0, 0, Frm("x & 0 != 0 for x=%r", x)) eq(x | 0, x, Frm("x | 0 != x for x=%r", x)) eq(x ^ 0, x, Frm("x ^ 0 != x for x=%r", x)) eq(x & -1, x, Frm("x & -1 != x for x=%r", x)) eq(x | -1, -1, Frm("x | -1 != -1 for x=%r", x)) eq(x ^ -1, ~x, Frm("x ^ -1 != ~x for x=%r", x)) eq(x, ~~x, Frm("x != ~~x for x=%r", x)) eq(x & x, x, Frm("x & x != x for x=%r", x)) eq(x | x, x, Frm("x | x != x for x=%r", x)) eq(x ^ x, 0, Frm("x ^ x != 0 for x=%r", x)) eq(x & ~x, 0, Frm("x & ~x != 0 for x=%r", x)) eq(x | ~x, -1, Frm("x | ~x != -1 for x=%r", x)) eq(x ^ ~x, -1, Frm("x ^ ~x != -1 for x=%r", x)) eq(-x, 1 + ~x, Frm("not -x == 1 + ~x for x=%r", x)) eq(-x, ~(x-1), Frm("not -x == ~(x-1) forx =%r", x)) for n in range(2*SHIFT): p2 = 2 ** n eq(x << n >> n, x, Frm("x << n >> n != x for x=%r, n=%r", (x, n))) eq(x // p2, x >> n, Frm("x // p2 != x >> n for x=%r n=%r p2=%r", (x, n, p2))) eq(x * p2, x << n, Frm("x * p2 != x << n for x=%r n=%r p2=%r", (x, n, p2))) eq(x & -p2, x >> n << n, Frm("not x & -p2 == x >> n << n for x=%r n=%r p2=%r", (x, n, p2))) eq(x & -p2, x & ~(p2 - 1), Frm("not x & -p2 == x & ~(p2 - 1) for x=%r n=%r p2=%r", (x, n, p2))) def check_bitop_identities_2(self, x, y): eq = self.assertEqual eq(x & y, y & x, Frm("x & y != y & x for x=%r, y=%r", (x, y))) eq(x | y, y | x, Frm("x | y != y | x for x=%r, y=%r", (x, y))) eq(x ^ y, y ^ x, Frm("x ^ y != y ^ x for x=%r, y=%r", (x, y))) eq(x ^ y ^ x, y, Frm("x ^ y ^ x != y for x=%r, y=%r", (x, y))) eq(x & y, ~(~x | ~y), Frm("x & y != ~(~x | ~y) for x=%r, y=%r", (x, y))) eq(x | y, ~(~x & ~y), Frm("x | y != ~(~x & ~y) for x=%r, y=%r", (x, y))) eq(x ^ y, (x | y) & ~(x & y), Frm("x ^ y != (x | y) & ~(x & y) for x=%r, y=%r", (x, y))) eq(x ^ y, (x & ~y) | (~x & y), Frm("x ^ y == (x & ~y) | (~x & y) for x=%r, y=%r", (x, y))) eq(x ^ y, (x | y) & (~x | ~y), Frm("x ^ y == (x | y) & (~x | ~y) for x=%r, y=%r", (x, y))) def check_bitop_identities_3(self, x, y, z): eq = self.assertEqual eq((x & y) & z, x & (y & z), Frm("(x & y) & z != x & (y & z) for x=%r, y=%r, z=%r", (x, y, z))) eq((x | y) | z, x | (y | z), Frm("(x | y) | z != x | (y | z) for x=%r, y=%r, z=%r", (x, y, z))) eq((x ^ y) ^ z, x ^ (y ^ z), Frm("(x ^ y) ^ z != x ^ (y ^ z) for x=%r, y=%r, z=%r", (x, y, z))) eq(x & (y | z), (x & y) | (x & z), Frm("x & (y | z) != (x & y) | (x & z) for x=%r, y=%r, z=%r", (x, y, z))) eq(x | (y & z), (x | y) & (x | z), Frm("x | (y & z) != (x | y) & (x | z) for x=%r, y=%r, z=%r", (x, y, z))) def test_bitop_identities(self): for x in special: self.check_bitop_identities_1(x) digits = range(1, MAXDIGITS+1) for lenx in digits: x = self.getran(lenx) self.check_bitop_identities_1(x) for leny in digits: y = self.getran(leny) self.check_bitop_identities_2(x, y) self.check_bitop_identities_3(x, y, self.getran((lenx + leny)//2)) def slow_format(self, x, base): digits = [] sign = 0 if x < 0: sign, x = 1, -x while x: x, r = divmod(x, base) digits.append(int(r)) digits.reverse() digits = digits or [0] return '-'[:sign] + \ {2: '0b', 8: '0o', 10: '', 16: '0x'}[base] + \ "".join(map(lambda i: "0123456789abcdef"[i], digits)) def check_format_1(self, x): for base, mapper in (8, oct), (10, repr), (16, hex): got = mapper(x) expected = self.slow_format(x, base) msg = Frm("%s returned %r but expected %r for %r", mapper.__name__, got, expected, x) self.assertEqual(got, expected, msg) self.assertEqual(int(got, 0), x, Frm('int("%s", 0) != %r', got, x)) # str() has to be checked a little differently since there's no # trailing "L" got = str(x) expected = self.slow_format(x, 10) msg = Frm("%s returned %r but expected %r for %r", mapper.__name__, got, expected, x) self.assertEqual(got, expected, msg) def test_format(self): for x in special: self.check_format_1(x) for i in range(10): for lenx in range(1, MAXDIGITS+1): x = self.getran(lenx) self.check_format_1(x) def test_long(self): # Check conversions from string LL = [ ('1' + '0'*20, 10**20), ('1' + '0'*100, 10**100) ] for s, v in LL: for sign in "", "+", "-": for prefix in "", " ", "\t", " \t\t ": ss = prefix + sign + s vv = v if sign == "-" and v is not ValueError: vv = -v try: self.assertEqual(int(ss), vv) except ValueError: pass # trailing L should no longer be accepted... self.assertRaises(ValueError, int, '123L') self.assertRaises(ValueError, int, '123l') self.assertRaises(ValueError, int, '0L') self.assertRaises(ValueError, int, '-37L') self.assertRaises(ValueError, int, '0x32L', 16) self.assertRaises(ValueError, int, '1L', 21) # ... but it's just a normal digit if base >= 22 self.assertEqual(int('1L', 22), 43) def test_conversion(self): class JustLong: # test that __long__ no longer used in 3.x def __long__(self): return 42 self.assertRaises(TypeError, int, JustLong()) class LongTrunc: # __long__ should be ignored in 3.x def __long__(self): return 42 def __trunc__(self): return 1729 self.assertEqual(int(LongTrunc()), 1729) @unittest.skipUnless(float.__getformat__("double").startswith("IEEE"), "test requires IEEE 754 doubles") def test_float_conversion(self): exact_values = [0, 1, 2, 2**53-3, 2**53-2, 2**53-1, 2**53, 2**53+2, 2**54-4, 2**54-2, 2**54, 2**54+4] for x in exact_values: self.assertEqual(float(x), x) self.assertEqual(float(-x), -x) # test round-half-even for x, y in [(1, 0), (2, 2), (3, 4), (4, 4), (5, 4), (6, 6), (7, 8)]: for p in range(15): self.assertEqual(int(float(2**p*(2**53+x))), 2**p*(2**53+y)) for x, y in [(0, 0), (1, 0), (2, 0), (3, 4), (4, 4), (5, 4), (6, 8), (7, 8), (8, 8), (9, 8), (10, 8), (11, 12), (12, 12), (13, 12), (14, 16), (15, 16)]: for p in range(15): self.assertEqual(int(float(2**p*(2**54+x))), 2**p*(2**54+y)) # behaviour near extremes of floating-point range int_dbl_max = int(DBL_MAX) top_power = 2**DBL_MAX_EXP halfway = (int_dbl_max + top_power)//2 self.assertEqual(float(int_dbl_max), DBL_MAX) self.assertEqual(float(int_dbl_max+1), DBL_MAX) self.assertEqual(float(halfway-1), DBL_MAX) self.assertRaises(OverflowError, float, halfway) self.assertEqual(float(1-halfway), -DBL_MAX) self.assertRaises(OverflowError, float, -halfway) self.assertRaises(OverflowError, float, top_power-1) self.assertRaises(OverflowError, float, top_power) self.assertRaises(OverflowError, float, top_power+1) self.assertRaises(OverflowError, float, 2*top_power-1) self.assertRaises(OverflowError, float, 2*top_power) self.assertRaises(OverflowError, float, top_power*top_power) for p in range(100): x = 2**p * (2**53 + 1) + 1 y = 2**p * (2**53 + 2) self.assertEqual(int(float(x)), y) x = 2**p * (2**53 + 1) y = 2**p * 2**53 self.assertEqual(int(float(x)), y) def test_float_overflow(self): import math for x in -2.0, -1.0, 0.0, 1.0, 2.0: self.assertEqual(float(int(x)), x) shuge = '12345' * 120 huge = 1 << 30000 mhuge = -huge namespace = {'huge': huge, 'mhuge': mhuge, 'shuge': shuge, 'math': math} for test in ["float(huge)", "float(mhuge)", "complex(huge)", "complex(mhuge)", "complex(huge, 1)", "complex(mhuge, 1)", "complex(1, huge)", "complex(1, mhuge)", "1. + huge", "huge + 1.", "1. + mhuge", "mhuge + 1.", "1. - huge", "huge - 1.", "1. - mhuge", "mhuge - 1.", "1. * huge", "huge * 1.", "1. * mhuge", "mhuge * 1.", "1. // huge", "huge // 1.", "1. // mhuge", "mhuge // 1.", "1. / huge", "huge / 1.", "1. / mhuge", "mhuge / 1.", "1. ** huge", "huge ** 1.", "1. ** mhuge", "mhuge ** 1.", "math.sin(huge)", "math.sin(mhuge)", "math.sqrt(huge)", "math.sqrt(mhuge)", # should do better # math.floor() of an int returns an int now ##"math.floor(huge)", "math.floor(mhuge)", ]: self.assertRaises(OverflowError, eval, test, namespace) # XXX Perhaps float(shuge) can raise OverflowError on some box? # The comparison should not. self.assertNotEqual(float(shuge), int(shuge), "float(shuge) should not equal int(shuge)") def test_logs(self): import math LOG10E = math.log10(math.e) for exp in list(range(10)) + [100, 1000, 10000]: value = 10 ** exp log10 = math.log10(value) self.assertAlmostEqual(log10, exp) # log10(value) == exp, so log(value) == log10(value)/log10(e) == # exp/LOG10E expected = exp / LOG10E log = math.log(value) self.assertAlmostEqual(log, expected) for bad in -(1 << 10000), -2, 0: self.assertRaises(ValueError, math.log, bad) self.assertRaises(ValueError, math.log10, bad) def test_mixed_compares(self): eq = self.assertEqual import math # We're mostly concerned with that mixing floats and longs does the # right stuff, even when longs are too large to fit in a float. # The safest way to check the results is to use an entirely different # method, which we do here via a skeletal rational class (which # represents all Python ints, longs and floats exactly). class Rat: def __init__(self, value): if isinstance(value, int): self.n = value self.d = 1 elif isinstance(value, float): # Convert to exact rational equivalent. f, e = math.frexp(abs(value)) assert f == 0 or 0.5 <= f < 1.0 # |value| = f * 2**e exactly # Suck up CHUNK bits at a time; 28 is enough so that we suck # up all bits in 2 iterations for all known binary double- # precision formats, and small enough to fit in an int. CHUNK = 28 top = 0 # invariant: |value| = (top + f) * 2**e exactly while f: f = math.ldexp(f, CHUNK) digit = int(f) assert digit >> CHUNK == 0 top = (top << CHUNK) | digit f -= digit assert 0.0 <= f < 1.0 e -= CHUNK # Now |value| = top * 2**e exactly. if e >= 0: n = top << e d = 1 else: n = top d = 1 << -e if value < 0: n = -n self.n = n self.d = d assert float(n) / float(d) == value else: raise TypeError("can't deal with %r" % val) def _cmp__(self, other): if not isinstance(other, Rat): other = Rat(other) x, y = self.n * other.d, self.d * other.n return (x > y) - (x < y) def __eq__(self, other): return self._cmp__(other) == 0 def __ne__(self, other): return self._cmp__(other) != 0 def __ge__(self, other): return self._cmp__(other) >= 0 def __gt__(self, other): return self._cmp__(other) > 0 def __le__(self, other): return self._cmp__(other) <= 0 def __lt__(self, other): return self._cmp__(other) < 0 cases = [0, 0.001, 0.99, 1.0, 1.5, 1e20, 1e200] # 2**48 is an important boundary in the internals. 2**53 is an # important boundary for IEEE double precision. for t in 2.0**48, 2.0**50, 2.0**53: cases.extend([t - 1.0, t - 0.3, t, t + 0.3, t + 1.0, int(t-1), int(t), int(t+1)]) cases.extend([0, 1, 2, sys.maxsize, float(sys.maxsize)]) # 1 << 20000 should exceed all double formats. int(1e200) is to # check that we get equality with 1e200 above. t = int(1e200) cases.extend([0, 1, 2, 1 << 20000, t-1, t, t+1]) cases.extend([-x for x in cases]) for x in cases: Rx = Rat(x) for y in cases: Ry = Rat(y) Rcmp = (Rx > Ry) - (Rx < Ry) xycmp = (x > y) - (x < y) eq(Rcmp, xycmp, Frm("%r %r %d %d", x, y, Rcmp, xycmp)) eq(x == y, Rcmp == 0, Frm("%r == %r %d", x, y, Rcmp)) eq(x != y, Rcmp != 0, Frm("%r != %r %d", x, y, Rcmp)) eq(x < y, Rcmp < 0, Frm("%r < %r %d", x, y, Rcmp)) eq(x <= y, Rcmp <= 0, Frm("%r <= %r %d", x, y, Rcmp)) eq(x > y, Rcmp > 0, Frm("%r > %r %d", x, y, Rcmp)) eq(x >= y, Rcmp >= 0, Frm("%r >= %r %d", x, y, Rcmp)) def test__format__(self): self.assertEqual(format(123456789, 'd'), '123456789') self.assertEqual(format(123456789, 'd'), '123456789') # sign and aligning are interdependent self.assertEqual(format(1, "-"), '1') self.assertEqual(format(-1, "-"), '-1') self.assertEqual(format(1, "-3"), ' 1') self.assertEqual(format(-1, "-3"), ' -1') self.assertEqual(format(1, "+3"), ' +1') self.assertEqual(format(-1, "+3"), ' -1') self.assertEqual(format(1, " 3"), ' 1') self.assertEqual(format(-1, " 3"), ' -1') self.assertEqual(format(1, " "), ' 1') self.assertEqual(format(-1, " "), '-1') # hex self.assertEqual(format(3, "x"), "3") self.assertEqual(format(3, "X"), "3") self.assertEqual(format(1234, "x"), "4d2") self.assertEqual(format(-1234, "x"), "-4d2") self.assertEqual(format(1234, "8x"), " 4d2") self.assertEqual(format(-1234, "8x"), " -4d2") self.assertEqual(format(1234, "x"), "4d2") self.assertEqual(format(-1234, "x"), "-4d2") self.assertEqual(format(-3, "x"), "-3") self.assertEqual(format(-3, "X"), "-3") self.assertEqual(format(int('be', 16), "x"), "be") self.assertEqual(format(int('be', 16), "X"), "BE") self.assertEqual(format(-int('be', 16), "x"), "-be") self.assertEqual(format(-int('be', 16), "X"), "-BE") # octal self.assertEqual(format(3, "b"), "11") self.assertEqual(format(-3, "b"), "-11") self.assertEqual(format(1234, "b"), "10011010010") self.assertEqual(format(-1234, "b"), "-10011010010") self.assertEqual(format(1234, "-b"), "10011010010") self.assertEqual(format(-1234, "-b"), "-10011010010") self.assertEqual(format(1234, " b"), " 10011010010") self.assertEqual(format(-1234, " b"), "-10011010010") self.assertEqual(format(1234, "+b"), "+10011010010") self.assertEqual(format(-1234, "+b"), "-10011010010") # make sure these are errors self.assertRaises(ValueError, format, 3, "1.3") # precision disallowed self.assertRaises(ValueError, format, 3, "+c") # sign not allowed # with 'c' # ensure that only int and float type specifiers work for format_spec in ([chr(x) for x in range(ord('a'), ord('z')+1)] + [chr(x) for x in range(ord('A'), ord('Z')+1)]): if not format_spec in 'bcdoxXeEfFgGn%': self.assertRaises(ValueError, format, 0, format_spec) self.assertRaises(ValueError, format, 1, format_spec) self.assertRaises(ValueError, format, -1, format_spec) self.assertRaises(ValueError, format, 2**100, format_spec) self.assertRaises(ValueError, format, -(2**100), format_spec) # ensure that float type specifiers work; format converts # the int to a float for format_spec in 'eEfFgG%': for value in [0, 1, -1, 100, -100, 1234567890, -1234567890]: self.assertEqual(format(value, format_spec), format(float(value), format_spec)) def test_nan_inf(self): self.assertRaises(OverflowError, int, float('inf')) self.assertRaises(OverflowError, int, float('-inf')) self.assertRaises(ValueError, int, float('nan')) def test_true_division(self): huge = 1 << 40000 mhuge = -huge self.assertEqual(huge / huge, 1.0) self.assertEqual(mhuge / mhuge, 1.0) self.assertEqual(huge / mhuge, -1.0) self.assertEqual(mhuge / huge, -1.0) self.assertEqual(1 / huge, 0.0) self.assertEqual(1 / huge, 0.0) self.assertEqual(1 / mhuge, 0.0) self.assertEqual(1 / mhuge, 0.0) self.assertEqual((666 * huge + (huge >> 1)) / huge, 666.5) self.assertEqual((666 * mhuge + (mhuge >> 1)) / mhuge, 666.5) self.assertEqual((666 * huge + (huge >> 1)) / mhuge, -666.5) self.assertEqual((666 * mhuge + (mhuge >> 1)) / huge, -666.5) self.assertEqual(huge / (huge << 1), 0.5) self.assertEqual((1000000 * huge) / huge, 1000000) namespace = {'huge': huge, 'mhuge': mhuge} for overflow in ["float(huge)", "float(mhuge)", "huge / 1", "huge / 2", "huge / -1", "huge / -2", "mhuge / 100", "mhuge / 200"]: self.assertRaises(OverflowError, eval, overflow, namespace) for underflow in ["1 / huge", "2 / huge", "-1 / huge", "-2 / huge", "100 / mhuge", "200 / mhuge"]: result = eval(underflow, namespace) self.assertEqual(result, 0.0, "expected underflow to 0 from %r" % underflow) for zero in ["huge / 0", "mhuge / 0"]: self.assertRaises(ZeroDivisionError, eval, zero, namespace) def check_truediv(self, a, b, skip_small=True): """Verify that the result of a/b is correctly rounded, by comparing it with a pure Python implementation of correctly rounded division. b should be nonzero.""" # skip check for small a and b: in this case, the current # implementation converts the arguments to float directly and # then applies a float division. This can give doubly-rounded # results on x87-using machines (particularly 32-bit Linux). if skip_small and max(abs(a), abs(b)) < 2**DBL_MANT_DIG: return try: # use repr so that we can distinguish between -0.0 and 0.0 expected = repr(truediv(a, b)) except OverflowError: expected = 'overflow' except ZeroDivisionError: expected = 'zerodivision' try: got = repr(a / b) except OverflowError: got = 'overflow' except ZeroDivisionError: got = 'zerodivision' if expected != got: self.fail("Incorrectly rounded division {}/{}: expected {!r}, " "got {!r}.".format(a, b, expected, got)) @requires_IEEE_754 def test_correctly_rounded_true_division(self): # more stringent tests than those above, checking that the # result of true division of ints is always correctly rounded. # This test should probably be considered CPython-specific. # Exercise all the code paths not involving Gb-sized ints. # ... divisions involving zero self.check_truediv(123, 0) self.check_truediv(-456, 0) self.check_truediv(0, 3) self.check_truediv(0, -3) self.check_truediv(0, 0) # ... overflow or underflow by large margin self.check_truediv(671 * 12345 * 2**DBL_MAX_EXP, 12345) self.check_truediv(12345, 345678 * 2**(DBL_MANT_DIG - DBL_MIN_EXP)) # ... a much larger or smaller than b self.check_truediv(12345*2**100, 98765) self.check_truediv(12345*2**30, 98765*7**81) # ... a / b near a boundary: one of 1, 2**DBL_MANT_DIG, 2**DBL_MIN_EXP, # 2**DBL_MAX_EXP, 2**(DBL_MIN_EXP-DBL_MANT_DIG) bases = (0, DBL_MANT_DIG, DBL_MIN_EXP, DBL_MAX_EXP, DBL_MIN_EXP - DBL_MANT_DIG) for base in bases: for exp in range(base - 15, base + 15): self.check_truediv(75312*2**max(exp, 0), 69187*2**max(-exp, 0)) self.check_truediv(69187*2**max(exp, 0), 75312*2**max(-exp, 0)) # overflow corner case for m in [1, 2, 7, 17, 12345, 7**100, -1, -2, -5, -23, -67891, -41**50]: for n in range(-10, 10): self.check_truediv(m*DBL_MIN_OVERFLOW + n, m) self.check_truediv(m*DBL_MIN_OVERFLOW + n, -m) # check detection of inexactness in shifting stage for n in range(250): # (2**DBL_MANT_DIG+1)/(2**DBL_MANT_DIG) lies halfway # between two representable floats, and would usually be # rounded down under round-half-to-even. The tiniest of # additions to the numerator should cause it to be rounded # up instead. self.check_truediv((2**DBL_MANT_DIG + 1)*12345*2**200 + 2**n, 2**DBL_MANT_DIG*12345) # 1/2731 is one of the smallest division cases that's subject # to double rounding on IEEE 754 machines working internally with # 64-bit precision. On such machines, the next check would fail, # were it not explicitly skipped in check_truediv. self.check_truediv(1, 2731) # a particularly bad case for the old algorithm: gives an # error of close to 3.5 ulps. self.check_truediv(295147931372582273023, 295147932265116303360) for i in range(1000): self.check_truediv(10**(i+1), 10**i) self.check_truediv(10**i, 10**(i+1)) # test round-half-to-even behaviour, normal result for m in [1, 2, 4, 7, 8, 16, 17, 32, 12345, 7**100, -1, -2, -5, -23, -67891, -41**50]: for n in range(-10, 10): self.check_truediv(2**DBL_MANT_DIG*m + n, m) # test round-half-to-even, subnormal result for n in range(-20, 20): self.check_truediv(n, 2**1076) # largeish random divisions: a/b where |a| <= |b| <= # 2*|a|; |ans| is between 0.5 and 1.0, so error should # always be bounded by 2**-54 with equality possible only # if the least significant bit of q=ans*2**53 is zero. for M in [10**10, 10**100, 10**1000]: for i in range(1000): a = random.randrange(1, M) b = random.randrange(a, 2*a+1) self.check_truediv(a, b) self.check_truediv(-a, b) self.check_truediv(a, -b) self.check_truediv(-a, -b) # and some (genuinely) random tests for _ in range(10000): a_bits = random.randrange(1000) b_bits = random.randrange(1, 1000) x = random.randrange(2**a_bits) y = random.randrange(1, 2**b_bits) self.check_truediv(x, y) self.check_truediv(x, -y) self.check_truediv(-x, y) self.check_truediv(-x, -y) def test_small_ints(self): for i in range(-5, 257): self.assertTrue(i is i + 0) self.assertTrue(i is i * 1) self.assertTrue(i is i - 0) self.assertTrue(i is i // 1) self.assertTrue(i is i & -1) self.assertTrue(i is i | 0) self.assertTrue(i is i ^ 0) self.assertTrue(i is ~~i) self.assertTrue(i is i**1) self.assertTrue(i is int(str(i))) self.assertTrue(i is i<<2>>2, str(i)) # corner cases i = 1 << 70 self.assertTrue(i - i is 0) self.assertTrue(0 * i is 0) def test_bit_length(self): tiny = 1e-10 for x in range(-65000, 65000): k = x.bit_length() # Check equivalence with Python version self.assertEqual(k, len(bin(x).lstrip('-0b'))) # Behaviour as specified in the docs if x != 0: self.assertTrue(2**(k-1) <= abs(x) < 2**k) else: self.assertEqual(k, 0) # Alternative definition: x.bit_length() == 1 + floor(log_2(x)) if x != 0: # When x is an exact power of 2, numeric errors can # cause floor(log(x)/log(2)) to be one too small; for # small x this can be fixed by adding a small quantity # to the quotient before taking the floor. self.assertEqual(k, 1 + math.floor( math.log(abs(x))/math.log(2) + tiny)) self.assertEqual((0).bit_length(), 0) self.assertEqual((1).bit_length(), 1) self.assertEqual((-1).bit_length(), 1) self.assertEqual((2).bit_length(), 2) self.assertEqual((-2).bit_length(), 2) for i in [2, 3, 15, 16, 17, 31, 32, 33, 63, 64, 234]: a = 2**i self.assertEqual((a-1).bit_length(), i) self.assertEqual((1-a).bit_length(), i) self.assertEqual((a).bit_length(), i+1) self.assertEqual((-a).bit_length(), i+1) self.assertEqual((a+1).bit_length(), i+1) self.assertEqual((-a-1).bit_length(), i+1) def test_round(self): # check round-half-even algorithm. For round to nearest ten; # rounding map is invariant under adding multiples of 20 test_dict = {0:0, 1:0, 2:0, 3:0, 4:0, 5:0, 6:10, 7:10, 8:10, 9:10, 10:10, 11:10, 12:10, 13:10, 14:10, 15:20, 16:20, 17:20, 18:20, 19:20} for offset in range(-520, 520, 20): for k, v in test_dict.items(): got = round(k+offset, -1) expected = v+offset self.assertEqual(got, expected) self.assertTrue(type(got) is int) # larger second argument self.assertEqual(round(-150, -2), -200) self.assertEqual(round(-149, -2), -100) self.assertEqual(round(-51, -2), -100) self.assertEqual(round(-50, -2), 0) self.assertEqual(round(-49, -2), 0) self.assertEqual(round(-1, -2), 0) self.assertEqual(round(0, -2), 0) self.assertEqual(round(1, -2), 0) self.assertEqual(round(49, -2), 0) self.assertEqual(round(50, -2), 0) self.assertEqual(round(51, -2), 100) self.assertEqual(round(149, -2), 100) self.assertEqual(round(150, -2), 200) self.assertEqual(round(250, -2), 200) self.assertEqual(round(251, -2), 300) self.assertEqual(round(172500, -3), 172000) self.assertEqual(round(173500, -3), 174000) self.assertEqual(round(31415926535, -1), 31415926540) self.assertEqual(round(31415926535, -2), 31415926500) self.assertEqual(round(31415926535, -3), 31415927000) self.assertEqual(round(31415926535, -4), 31415930000) self.assertEqual(round(31415926535, -5), 31415900000) self.assertEqual(round(31415926535, -6), 31416000000) self.assertEqual(round(31415926535, -7), 31420000000) self.assertEqual(round(31415926535, -8), 31400000000) self.assertEqual(round(31415926535, -9), 31000000000) self.assertEqual(round(31415926535, -10), 30000000000) self.assertEqual(round(31415926535, -11), 0) self.assertEqual(round(31415926535, -12), 0) self.assertEqual(round(31415926535, -999), 0) # should get correct results even for huge inputs for k in range(10, 100): got = round(10**k + 324678, -3) expect = 10**k + 325000 self.assertEqual(got, expect) self.assertTrue(type(got) is int) # nonnegative second argument: round(x, n) should just return x for n in range(5): for i in range(100): x = random.randrange(-10000, 10000) got = round(x, n) self.assertEqual(got, x) self.assertTrue(type(got) is int) for huge_n in 2**31-1, 2**31, 2**63-1, 2**63, 2**100, 10**100: self.assertEqual(round(8979323, huge_n), 8979323) # omitted second argument for i in range(100): x = random.randrange(-10000, 10000) got = round(x) self.assertEqual(got, x) self.assertTrue(type(got) is int) # bad second argument bad_exponents = ('brian', 2.0, 0j, None) for e in bad_exponents: self.assertRaises(TypeError, round, 3, e) def test_main(): support.run_unittest(LongTest) if __name__ == "__main__": test_main()