from __future__ import division # When true division is the default, get rid of this and add it to # test_long.py instead. In the meantime, it's too obscure to try to # trick just part of test_long into using future division. import sys import random import math import unittest from test.test_support import run_unittest # decorator for skipping tests on non-IEEE 754 platforms requires_IEEE_754 = unittest.skipUnless( float.__getformat__("double").startswith("IEEE"), "test requires IEEE 754 doubles") 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 = math.ldexp(float(q), exp) return -result if negative else result class TrueDivisionTests(unittest.TestCase): def test(self): huge = 1L << 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(1L / huge, 0.0) self.assertEqual(1 / mhuge, 0.0) self.assertEqual(1L / 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 / 2L", "huge / -1", "huge / -2L", "mhuge / 100", "mhuge / 100L"]: # If the "eval" does not happen in this module, # true division is not enabled with self.assertRaises(OverflowError): eval(overflow, namespace) for underflow in ["1 / huge", "2L / huge", "-1 / huge", "-2L / huge", "100 / mhuge", "100L / mhuge"]: result = eval(underflow, namespace) self.assertEqual(result, 0.0, 'expected underflow to 0 ' 'from {!r}'.format(underflow)) for zero in ["huge / 0", "huge / 0L", "mhuge / 0", "mhuge / 0L"]: with 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.""" a, b = long(a), long(b) # 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' self.assertEqual(expected, got, "Incorrectly rounded division {}/{}: " "expected {}, got {}".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_main(): run_unittest(TrueDivisionTests) if __name__ == "__main__": test_main()