| 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() |
