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2217 lines (1928 loc) · 90.3 KB
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# Python test set -- math module
# XXXX Should not do tests around zero only
from test.support import run_unittest, verbose, requires_IEEE_754
from test import support
import unittest
import itertools
import decimal
import math
import os
import platform
import random
import struct
import sys
eps = 1E-05
NAN = float('nan')
INF = float('inf')
NINF = float('-inf')
FLOAT_MAX = sys.float_info.max
FLOAT_MIN = sys.float_info.min
# detect evidence of double-rounding: fsum is not always correctly
# rounded on machines that suffer from double rounding.
x, y = 1e16, 2.9999 # use temporary values to defeat peephole optimizer
HAVE_DOUBLE_ROUNDING = (x + y == 1e16 + 4)
# locate file with test values
if __name__ == '__main__':
file = sys.argv[0]
else:
file = __file__
test_dir = os.path.dirname(file) or os.curdir
math_testcases = os.path.join(test_dir, 'math_testcases.txt')
test_file = os.path.join(test_dir, 'cmath_testcases.txt')
def to_ulps(x):
"""Convert a non-NaN float x to an integer, in such a way that
adjacent floats are converted to adjacent integers. Then
abs(ulps(x) - ulps(y)) gives the difference in ulps between two
floats.
The results from this function will only make sense on platforms
where native doubles are represented in IEEE 754 binary64 format.
Note: 0.0 and -0.0 are converted to 0 and -1, respectively.
"""
n = struct.unpack('<q', struct.pack('<d', x))[0]
if n < 0:
n = ~(n+2**63)
return n
# Here's a pure Python version of the math.factorial algorithm, for
# documentation and comparison purposes.
#
# Formula:
#
# factorial(n) = factorial_odd_part(n) << (n - count_set_bits(n))
#
# where
#
# factorial_odd_part(n) = product_{i >= 0} product_{0 < j <= n >> i; j odd} j
#
# The outer product above is an infinite product, but once i >= n.bit_length,
# (n >> i) < 1 and the corresponding term of the product is empty. So only the
# finitely many terms for 0 <= i < n.bit_length() contribute anything.
#
# We iterate downwards from i == n.bit_length() - 1 to i == 0. The inner
# product in the formula above starts at 1 for i == n.bit_length(); for each i
# < n.bit_length() we get the inner product for i from that for i + 1 by
# multiplying by all j in {n >> i+1 < j <= n >> i; j odd}. In Python terms,
# this set is range((n >> i+1) + 1 | 1, (n >> i) + 1 | 1, 2).
def count_set_bits(n):
"""Number of '1' bits in binary expansion of a nonnnegative integer."""
return 1 + count_set_bits(n & n - 1) if n else 0
def partial_product(start, stop):
"""Product of integers in range(start, stop, 2), computed recursively.
start and stop should both be odd, with start <= stop.
"""
numfactors = (stop - start) >> 1
if not numfactors:
return 1
elif numfactors == 1:
return start
else:
mid = (start + numfactors) | 1
return partial_product(start, mid) * partial_product(mid, stop)
def py_factorial(n):
"""Factorial of nonnegative integer n, via "Binary Split Factorial Formula"
described at http://www.luschny.de/math/factorial/binarysplitfact.html
"""
inner = outer = 1
for i in reversed(range(n.bit_length())):
inner *= partial_product((n >> i + 1) + 1 | 1, (n >> i) + 1 | 1)
outer *= inner
return outer << (n - count_set_bits(n))
def ulp_abs_check(expected, got, ulp_tol, abs_tol):
"""Given finite floats `expected` and `got`, check that they're
approximately equal to within the given number of ulps or the
given absolute tolerance, whichever is bigger.
Returns None on success and an error message on failure.
"""
ulp_error = abs(to_ulps(expected) - to_ulps(got))
abs_error = abs(expected - got)
# Succeed if either abs_error <= abs_tol or ulp_error <= ulp_tol.
if abs_error <= abs_tol or ulp_error <= ulp_tol:
return None
else:
fmt = ("error = {:.3g} ({:d} ulps); "
"permitted error = {:.3g} or {:d} ulps")
return fmt.format(abs_error, ulp_error, abs_tol, ulp_tol)
def parse_mtestfile(fname):
"""Parse a file with test values
-- starts a comment
blank lines, or lines containing only a comment, are ignored
other lines are expected to have the form
id fn arg -> expected [flag]*
"""
with open(fname) as fp:
for line in fp:
# strip comments, and skip blank lines
if '--' in line:
line = line[:line.index('--')]
if not line.strip():
continue
lhs, rhs = line.split('->')
id, fn, arg = lhs.split()
rhs_pieces = rhs.split()
exp = rhs_pieces[0]
flags = rhs_pieces[1:]
yield (id, fn, float(arg), float(exp), flags)
def parse_testfile(fname):
"""Parse a file with test values
Empty lines or lines starting with -- are ignored
yields id, fn, arg_real, arg_imag, exp_real, exp_imag
"""
with open(fname) as fp:
for line in fp:
# skip comment lines and blank lines
if line.startswith('--') or not line.strip():
continue
lhs, rhs = line.split('->')
id, fn, arg_real, arg_imag = lhs.split()
rhs_pieces = rhs.split()
exp_real, exp_imag = rhs_pieces[0], rhs_pieces[1]
flags = rhs_pieces[2:]
yield (id, fn,
float(arg_real), float(arg_imag),
float(exp_real), float(exp_imag),
flags)
def result_check(expected, got, ulp_tol=5, abs_tol=0.0):
# Common logic of MathTests.(ftest, test_testcases, test_mtestcases)
"""Compare arguments expected and got, as floats, if either
is a float, using a tolerance expressed in multiples of
ulp(expected) or absolutely (if given and greater).
As a convenience, when neither argument is a float, and for
non-finite floats, exact equality is demanded. Also, nan==nan
as far as this function is concerned.
Returns None on success and an error message on failure.
"""
# Check exactly equal (applies also to strings representing exceptions)
if got == expected:
return None
failure = "not equal"
# Turn mixed float and int comparison (e.g. floor()) to all-float
if isinstance(expected, float) and isinstance(got, int):
got = float(got)
elif isinstance(got, float) and isinstance(expected, int):
expected = float(expected)
if isinstance(expected, float) and isinstance(got, float):
if math.isnan(expected) and math.isnan(got):
# Pass, since both nan
failure = None
elif math.isinf(expected) or math.isinf(got):
# We already know they're not equal, drop through to failure
pass
else:
# Both are finite floats (now). Are they close enough?
failure = ulp_abs_check(expected, got, ulp_tol, abs_tol)
# arguments are not equal, and if numeric, are too far apart
if failure is not None:
fail_fmt = "expected {!r}, got {!r}"
fail_msg = fail_fmt.format(expected, got)
fail_msg += ' ({})'.format(failure)
return fail_msg
else:
return None
class FloatLike:
def __init__(self, value):
self.value = value
def __float__(self):
return self.value
class IntSubclass(int):
pass
# Class providing an __index__ method.
class MyIndexable(object):
def __init__(self, value):
self.value = value
def __index__(self):
return self.value
class MathTests(unittest.TestCase):
def ftest(self, name, got, expected, ulp_tol=5, abs_tol=0.0):
"""Compare arguments expected and got, as floats, if either
is a float, using a tolerance expressed in multiples of
ulp(expected) or absolutely, whichever is greater.
As a convenience, when neither argument is a float, and for
non-finite floats, exact equality is demanded. Also, nan==nan
in this function.
"""
failure = result_check(expected, got, ulp_tol, abs_tol)
if failure is not None:
self.fail("{}: {}".format(name, failure))
def testConstants(self):
# Ref: Abramowitz & Stegun (Dover, 1965)
self.ftest('pi', math.pi, 3.141592653589793238462643)
self.ftest('e', math.e, 2.718281828459045235360287)
self.assertEqual(math.tau, 2*math.pi)
def testAcos(self):
self.assertRaises(TypeError, math.acos)
self.ftest('acos(-1)', math.acos(-1), math.pi)
self.ftest('acos(0)', math.acos(0), math.pi/2)
self.ftest('acos(1)', math.acos(1), 0)
self.assertRaises(ValueError, math.acos, INF)
self.assertRaises(ValueError, math.acos, NINF)
self.assertRaises(ValueError, math.acos, 1 + eps)
self.assertRaises(ValueError, math.acos, -1 - eps)
self.assertTrue(math.isnan(math.acos(NAN)))
def testAcosh(self):
self.assertRaises(TypeError, math.acosh)
self.ftest('acosh(1)', math.acosh(1), 0)
self.ftest('acosh(2)', math.acosh(2), 1.3169578969248168)
self.assertRaises(ValueError, math.acosh, 0)
self.assertRaises(ValueError, math.acosh, -1)
self.assertEqual(math.acosh(INF), INF)
self.assertRaises(ValueError, math.acosh, NINF)
self.assertTrue(math.isnan(math.acosh(NAN)))
def testAsin(self):
self.assertRaises(TypeError, math.asin)
self.ftest('asin(-1)', math.asin(-1), -math.pi/2)
self.ftest('asin(0)', math.asin(0), 0)
self.ftest('asin(1)', math.asin(1), math.pi/2)
self.assertRaises(ValueError, math.asin, INF)
self.assertRaises(ValueError, math.asin, NINF)
self.assertRaises(ValueError, math.asin, 1 + eps)
self.assertRaises(ValueError, math.asin, -1 - eps)
self.assertTrue(math.isnan(math.asin(NAN)))
def testAsinh(self):
self.assertRaises(TypeError, math.asinh)
self.ftest('asinh(0)', math.asinh(0), 0)
self.ftest('asinh(1)', math.asinh(1), 0.88137358701954305)
self.ftest('asinh(-1)', math.asinh(-1), -0.88137358701954305)
self.assertEqual(math.asinh(INF), INF)
self.assertEqual(math.asinh(NINF), NINF)
self.assertTrue(math.isnan(math.asinh(NAN)))
def testAtan(self):
self.assertRaises(TypeError, math.atan)
self.ftest('atan(-1)', math.atan(-1), -math.pi/4)
self.ftest('atan(0)', math.atan(0), 0)
self.ftest('atan(1)', math.atan(1), math.pi/4)
self.ftest('atan(inf)', math.atan(INF), math.pi/2)
self.ftest('atan(-inf)', math.atan(NINF), -math.pi/2)
self.assertTrue(math.isnan(math.atan(NAN)))
def testAtanh(self):
self.assertRaises(TypeError, math.atan)
self.ftest('atanh(0)', math.atanh(0), 0)
self.ftest('atanh(0.5)', math.atanh(0.5), 0.54930614433405489)
self.ftest('atanh(-0.5)', math.atanh(-0.5), -0.54930614433405489)
self.assertRaises(ValueError, math.atanh, 1)
self.assertRaises(ValueError, math.atanh, -1)
self.assertRaises(ValueError, math.atanh, INF)
self.assertRaises(ValueError, math.atanh, NINF)
self.assertTrue(math.isnan(math.atanh(NAN)))
def testAtan2(self):
self.assertRaises(TypeError, math.atan2)
self.ftest('atan2(-1, 0)', math.atan2(-1, 0), -math.pi/2)
self.ftest('atan2(-1, 1)', math.atan2(-1, 1), -math.pi/4)
self.ftest('atan2(0, 1)', math.atan2(0, 1), 0)
self.ftest('atan2(1, 1)', math.atan2(1, 1), math.pi/4)
self.ftest('atan2(1, 0)', math.atan2(1, 0), math.pi/2)
# math.atan2(0, x)
self.ftest('atan2(0., -inf)', math.atan2(0., NINF), math.pi)
self.ftest('atan2(0., -2.3)', math.atan2(0., -2.3), math.pi)
self.ftest('atan2(0., -0.)', math.atan2(0., -0.), math.pi)
self.assertEqual(math.atan2(0., 0.), 0.)
self.assertEqual(math.atan2(0., 2.3), 0.)
self.assertEqual(math.atan2(0., INF), 0.)
self.assertTrue(math.isnan(math.atan2(0., NAN)))
# math.atan2(-0, x)
self.ftest('atan2(-0., -inf)', math.atan2(-0., NINF), -math.pi)
self.ftest('atan2(-0., -2.3)', math.atan2(-0., -2.3), -math.pi)
self.ftest('atan2(-0., -0.)', math.atan2(-0., -0.), -math.pi)
self.assertEqual(math.atan2(-0., 0.), -0.)
self.assertEqual(math.atan2(-0., 2.3), -0.)
self.assertEqual(math.atan2(-0., INF), -0.)
self.assertTrue(math.isnan(math.atan2(-0., NAN)))
# math.atan2(INF, x)
self.ftest('atan2(inf, -inf)', math.atan2(INF, NINF), math.pi*3/4)
self.ftest('atan2(inf, -2.3)', math.atan2(INF, -2.3), math.pi/2)
self.ftest('atan2(inf, -0.)', math.atan2(INF, -0.0), math.pi/2)
self.ftest('atan2(inf, 0.)', math.atan2(INF, 0.0), math.pi/2)
self.ftest('atan2(inf, 2.3)', math.atan2(INF, 2.3), math.pi/2)
self.ftest('atan2(inf, inf)', math.atan2(INF, INF), math.pi/4)
self.assertTrue(math.isnan(math.atan2(INF, NAN)))
# math.atan2(NINF, x)
self.ftest('atan2(-inf, -inf)', math.atan2(NINF, NINF), -math.pi*3/4)
self.ftest('atan2(-inf, -2.3)', math.atan2(NINF, -2.3), -math.pi/2)
self.ftest('atan2(-inf, -0.)', math.atan2(NINF, -0.0), -math.pi/2)
self.ftest('atan2(-inf, 0.)', math.atan2(NINF, 0.0), -math.pi/2)
self.ftest('atan2(-inf, 2.3)', math.atan2(NINF, 2.3), -math.pi/2)
self.ftest('atan2(-inf, inf)', math.atan2(NINF, INF), -math.pi/4)
self.assertTrue(math.isnan(math.atan2(NINF, NAN)))
# math.atan2(+finite, x)
self.ftest('atan2(2.3, -inf)', math.atan2(2.3, NINF), math.pi)
self.ftest('atan2(2.3, -0.)', math.atan2(2.3, -0.), math.pi/2)
self.ftest('atan2(2.3, 0.)', math.atan2(2.3, 0.), math.pi/2)
self.assertEqual(math.atan2(2.3, INF), 0.)
self.assertTrue(math.isnan(math.atan2(2.3, NAN)))
# math.atan2(-finite, x)
self.ftest('atan2(-2.3, -inf)', math.atan2(-2.3, NINF), -math.pi)
self.ftest('atan2(-2.3, -0.)', math.atan2(-2.3, -0.), -math.pi/2)
self.ftest('atan2(-2.3, 0.)', math.atan2(-2.3, 0.), -math.pi/2)
self.assertEqual(math.atan2(-2.3, INF), -0.)
self.assertTrue(math.isnan(math.atan2(-2.3, NAN)))
# math.atan2(NAN, x)
self.assertTrue(math.isnan(math.atan2(NAN, NINF)))
self.assertTrue(math.isnan(math.atan2(NAN, -2.3)))
self.assertTrue(math.isnan(math.atan2(NAN, -0.)))
self.assertTrue(math.isnan(math.atan2(NAN, 0.)))
self.assertTrue(math.isnan(math.atan2(NAN, 2.3)))
self.assertTrue(math.isnan(math.atan2(NAN, INF)))
self.assertTrue(math.isnan(math.atan2(NAN, NAN)))
def testCeil(self):
self.assertRaises(TypeError, math.ceil)
self.assertEqual(int, type(math.ceil(0.5)))
self.assertEqual(math.ceil(0.5), 1)
self.assertEqual(math.ceil(1.0), 1)
self.assertEqual(math.ceil(1.5), 2)
self.assertEqual(math.ceil(-0.5), 0)
self.assertEqual(math.ceil(-1.0), -1)
self.assertEqual(math.ceil(-1.5), -1)
self.assertEqual(math.ceil(0.0), 0)
self.assertEqual(math.ceil(-0.0), 0)
#self.assertEqual(math.ceil(INF), INF)
#self.assertEqual(math.ceil(NINF), NINF)
#self.assertTrue(math.isnan(math.ceil(NAN)))
class TestCeil:
def __ceil__(self):
return 42
class FloatCeil(float):
def __ceil__(self):
return 42
class TestNoCeil:
pass
self.assertEqual(math.ceil(TestCeil()), 42)
self.assertEqual(math.ceil(FloatCeil()), 42)
self.assertEqual(math.ceil(FloatLike(42.5)), 43)
self.assertRaises(TypeError, math.ceil, TestNoCeil())
t = TestNoCeil()
t.__ceil__ = lambda *args: args
self.assertRaises(TypeError, math.ceil, t)
self.assertRaises(TypeError, math.ceil, t, 0)
@requires_IEEE_754
def testCopysign(self):
self.assertEqual(math.copysign(1, 42), 1.0)
self.assertEqual(math.copysign(0., 42), 0.0)
self.assertEqual(math.copysign(1., -42), -1.0)
self.assertEqual(math.copysign(3, 0.), 3.0)
self.assertEqual(math.copysign(4., -0.), -4.0)
self.assertRaises(TypeError, math.copysign)
# copysign should let us distinguish signs of zeros
self.assertEqual(math.copysign(1., 0.), 1.)
self.assertEqual(math.copysign(1., -0.), -1.)
self.assertEqual(math.copysign(INF, 0.), INF)
self.assertEqual(math.copysign(INF, -0.), NINF)
self.assertEqual(math.copysign(NINF, 0.), INF)
self.assertEqual(math.copysign(NINF, -0.), NINF)
# and of infinities
self.assertEqual(math.copysign(1., INF), 1.)
self.assertEqual(math.copysign(1., NINF), -1.)
self.assertEqual(math.copysign(INF, INF), INF)
self.assertEqual(math.copysign(INF, NINF), NINF)
self.assertEqual(math.copysign(NINF, INF), INF)
self.assertEqual(math.copysign(NINF, NINF), NINF)
self.assertTrue(math.isnan(math.copysign(NAN, 1.)))
self.assertTrue(math.isnan(math.copysign(NAN, INF)))
self.assertTrue(math.isnan(math.copysign(NAN, NINF)))
self.assertTrue(math.isnan(math.copysign(NAN, NAN)))
# copysign(INF, NAN) may be INF or it may be NINF, since
# we don't know whether the sign bit of NAN is set on any
# given platform.
self.assertTrue(math.isinf(math.copysign(INF, NAN)))
# similarly, copysign(2., NAN) could be 2. or -2.
self.assertEqual(abs(math.copysign(2., NAN)), 2.)
def testCos(self):
self.assertRaises(TypeError, math.cos)
self.ftest('cos(-pi/2)', math.cos(-math.pi/2), 0, abs_tol=math.ulp(1))
self.ftest('cos(0)', math.cos(0), 1)
self.ftest('cos(pi/2)', math.cos(math.pi/2), 0, abs_tol=math.ulp(1))
self.ftest('cos(pi)', math.cos(math.pi), -1)
try:
self.assertTrue(math.isnan(math.cos(INF)))
self.assertTrue(math.isnan(math.cos(NINF)))
except ValueError:
self.assertRaises(ValueError, math.cos, INF)
self.assertRaises(ValueError, math.cos, NINF)
self.assertTrue(math.isnan(math.cos(NAN)))
@unittest.skipIf(sys.platform == 'win32' and platform.machine() in ('ARM', 'ARM64'),
"Windows UCRT is off by 2 ULP this test requires accuracy within 1 ULP")
def testCosh(self):
self.assertRaises(TypeError, math.cosh)
self.ftest('cosh(0)', math.cosh(0), 1)
self.ftest('cosh(2)-2*cosh(1)**2', math.cosh(2)-2*math.cosh(1)**2, -1) # Thanks to Lambert
self.assertEqual(math.cosh(INF), INF)
self.assertEqual(math.cosh(NINF), INF)
self.assertTrue(math.isnan(math.cosh(NAN)))
def testDegrees(self):
self.assertRaises(TypeError, math.degrees)
self.ftest('degrees(pi)', math.degrees(math.pi), 180.0)
self.ftest('degrees(pi/2)', math.degrees(math.pi/2), 90.0)
self.ftest('degrees(-pi/4)', math.degrees(-math.pi/4), -45.0)
self.ftest('degrees(0)', math.degrees(0), 0)
def testExp(self):
self.assertRaises(TypeError, math.exp)
self.ftest('exp(-1)', math.exp(-1), 1/math.e)
self.ftest('exp(0)', math.exp(0), 1)
self.ftest('exp(1)', math.exp(1), math.e)
self.assertEqual(math.exp(INF), INF)
self.assertEqual(math.exp(NINF), 0.)
self.assertTrue(math.isnan(math.exp(NAN)))
self.assertRaises(OverflowError, math.exp, 1000000)
def testFabs(self):
self.assertRaises(TypeError, math.fabs)
self.ftest('fabs(-1)', math.fabs(-1), 1)
self.ftest('fabs(0)', math.fabs(0), 0)
self.ftest('fabs(1)', math.fabs(1), 1)
def testFactorial(self):
self.assertEqual(math.factorial(0), 1)
total = 1
for i in range(1, 1000):
total *= i
self.assertEqual(math.factorial(i), total)
self.assertEqual(math.factorial(i), py_factorial(i))
self.assertRaises(ValueError, math.factorial, -1)
self.assertRaises(ValueError, math.factorial, -10**100)
def testFactorialNonIntegers(self):
self.assertRaises(TypeError, math.factorial, 5.0)
self.assertRaises(TypeError, math.factorial, 5.2)
self.assertRaises(TypeError, math.factorial, -1.0)
self.assertRaises(TypeError, math.factorial, -1e100)
self.assertRaises(TypeError, math.factorial, decimal.Decimal('5'))
self.assertRaises(TypeError, math.factorial, decimal.Decimal('5.2'))
self.assertRaises(TypeError, math.factorial, "5")
# Other implementations may place different upper bounds.
@support.cpython_only
def testFactorialHugeInputs(self):
# Currently raises OverflowError for inputs that are too large
# to fit into a C long.
self.assertRaises(OverflowError, math.factorial, 10**100)
self.assertRaises(TypeError, math.factorial, 1e100)
def testFloor(self):
self.assertRaises(TypeError, math.floor)
self.assertEqual(int, type(math.floor(0.5)))
self.assertEqual(math.floor(0.5), 0)
self.assertEqual(math.floor(1.0), 1)
self.assertEqual(math.floor(1.5), 1)
self.assertEqual(math.floor(-0.5), -1)
self.assertEqual(math.floor(-1.0), -1)
self.assertEqual(math.floor(-1.5), -2)
#self.assertEqual(math.ceil(INF), INF)
#self.assertEqual(math.ceil(NINF), NINF)
#self.assertTrue(math.isnan(math.floor(NAN)))
class TestFloor:
def __floor__(self):
return 42
class FloatFloor(float):
def __floor__(self):
return 42
class TestNoFloor:
pass
self.assertEqual(math.floor(TestFloor()), 42)
self.assertEqual(math.floor(FloatFloor()), 42)
self.assertEqual(math.floor(FloatLike(41.9)), 41)
self.assertRaises(TypeError, math.floor, TestNoFloor())
t = TestNoFloor()
t.__floor__ = lambda *args: args
self.assertRaises(TypeError, math.floor, t)
self.assertRaises(TypeError, math.floor, t, 0)
def testFmod(self):
self.assertRaises(TypeError, math.fmod)
self.ftest('fmod(10, 1)', math.fmod(10, 1), 0.0)
self.ftest('fmod(10, 0.5)', math.fmod(10, 0.5), 0.0)
self.ftest('fmod(10, 1.5)', math.fmod(10, 1.5), 1.0)
self.ftest('fmod(-10, 1)', math.fmod(-10, 1), -0.0)
self.ftest('fmod(-10, 0.5)', math.fmod(-10, 0.5), -0.0)
self.ftest('fmod(-10, 1.5)', math.fmod(-10, 1.5), -1.0)
self.assertTrue(math.isnan(math.fmod(NAN, 1.)))
self.assertTrue(math.isnan(math.fmod(1., NAN)))
self.assertTrue(math.isnan(math.fmod(NAN, NAN)))
self.assertRaises(ValueError, math.fmod, 1., 0.)
self.assertRaises(ValueError, math.fmod, INF, 1.)
self.assertRaises(ValueError, math.fmod, NINF, 1.)
self.assertRaises(ValueError, math.fmod, INF, 0.)
self.assertEqual(math.fmod(3.0, INF), 3.0)
self.assertEqual(math.fmod(-3.0, INF), -3.0)
self.assertEqual(math.fmod(3.0, NINF), 3.0)
self.assertEqual(math.fmod(-3.0, NINF), -3.0)
self.assertEqual(math.fmod(0.0, 3.0), 0.0)
self.assertEqual(math.fmod(0.0, NINF), 0.0)
def testFrexp(self):
self.assertRaises(TypeError, math.frexp)
def testfrexp(name, result, expected):
(mant, exp), (emant, eexp) = result, expected
if abs(mant-emant) > eps or exp != eexp:
self.fail('%s returned %r, expected %r'%\
(name, result, expected))
testfrexp('frexp(-1)', math.frexp(-1), (-0.5, 1))
testfrexp('frexp(0)', math.frexp(0), (0, 0))
testfrexp('frexp(1)', math.frexp(1), (0.5, 1))
testfrexp('frexp(2)', math.frexp(2), (0.5, 2))
self.assertEqual(math.frexp(INF)[0], INF)
self.assertEqual(math.frexp(NINF)[0], NINF)
self.assertTrue(math.isnan(math.frexp(NAN)[0]))
@requires_IEEE_754
@unittest.skipIf(HAVE_DOUBLE_ROUNDING,
"fsum is not exact on machines with double rounding")
def testFsum(self):
# math.fsum relies on exact rounding for correct operation.
# There's a known problem with IA32 floating-point that causes
# inexact rounding in some situations, and will cause the
# math.fsum tests below to fail; see issue #2937. On non IEEE
# 754 platforms, and on IEEE 754 platforms that exhibit the
# problem described in issue #2937, we simply skip the whole
# test.
# Python version of math.fsum, for comparison. Uses a
# different algorithm based on frexp, ldexp and integer
# arithmetic.
from sys import float_info
mant_dig = float_info.mant_dig
etiny = float_info.min_exp - mant_dig
def msum(iterable):
"""Full precision summation. Compute sum(iterable) without any
intermediate accumulation of error. Based on the 'lsum' function
at https://github.com/ActiveState/code/tree/master/recipes/Python/393090_Binary_floating_point_summatiaccurate_full/recipe-393090.py
"""
tmant, texp = 0, 0
for x in iterable:
mant, exp = math.frexp(x)
mant, exp = int(math.ldexp(mant, mant_dig)), exp - mant_dig
if texp > exp:
tmant <<= texp-exp
texp = exp
else:
mant <<= exp-texp
tmant += mant
# Round tmant * 2**texp to a float. The original recipe
# used float(str(tmant)) * 2.0**texp for this, but that's
# a little unsafe because str -> float conversion can't be
# relied upon to do correct rounding on all platforms.
tail = max(len(bin(abs(tmant)))-2 - mant_dig, etiny - texp)
if tail > 0:
h = 1 << (tail-1)
tmant = tmant // (2*h) + bool(tmant & h and tmant & 3*h-1)
texp += tail
return math.ldexp(tmant, texp)
test_values = [
([], 0.0),
([0.0], 0.0),
([1e100, 1.0, -1e100, 1e-100, 1e50, -1.0, -1e50], 1e-100),
([2.0**53, -0.5, -2.0**-54], 2.0**53-1.0),
([2.0**53, 1.0, 2.0**-100], 2.0**53+2.0),
([2.0**53+10.0, 1.0, 2.0**-100], 2.0**53+12.0),
([2.0**53-4.0, 0.5, 2.0**-54], 2.0**53-3.0),
([1./n for n in range(1, 1001)],
float.fromhex('0x1.df11f45f4e61ap+2')),
([(-1.)**n/n for n in range(1, 1001)],
float.fromhex('-0x1.62a2af1bd3624p-1')),
([1e16, 1., 1e-16], 10000000000000002.0),
([1e16-2., 1.-2.**-53, -(1e16-2.), -(1.-2.**-53)], 0.0),
# exercise code for resizing partials array
([2.**n - 2.**(n+50) + 2.**(n+52) for n in range(-1074, 972, 2)] +
[-2.**1022],
float.fromhex('0x1.5555555555555p+970')),
]
# Telescoping sum, with exact differences (due to Sterbenz)
terms = [1.7**i for i in range(1001)]
test_values.append((
[terms[i+1] - terms[i] for i in range(1000)] + [-terms[1000]],
-terms[0]
))
for i, (vals, expected) in enumerate(test_values):
try:
actual = math.fsum(vals)
except OverflowError:
self.fail("test %d failed: got OverflowError, expected %r "
"for math.fsum(%.100r)" % (i, expected, vals))
except ValueError:
self.fail("test %d failed: got ValueError, expected %r "
"for math.fsum(%.100r)" % (i, expected, vals))
self.assertEqual(actual, expected)
from random import random, gauss, shuffle
for j in range(1000):
vals = [7, 1e100, -7, -1e100, -9e-20, 8e-20] * 10
s = 0
for i in range(200):
v = gauss(0, random()) ** 7 - s
s += v
vals.append(v)
shuffle(vals)
s = msum(vals)
self.assertEqual(msum(vals), math.fsum(vals))
def testGcd(self):
gcd = math.gcd
self.assertEqual(gcd(0, 0), 0)
self.assertEqual(gcd(1, 0), 1)
self.assertEqual(gcd(-1, 0), 1)
self.assertEqual(gcd(0, 1), 1)
self.assertEqual(gcd(0, -1), 1)
self.assertEqual(gcd(7, 1), 1)
self.assertEqual(gcd(7, -1), 1)
self.assertEqual(gcd(-23, 15), 1)
self.assertEqual(gcd(120, 84), 12)
self.assertEqual(gcd(84, -120), 12)
self.assertEqual(gcd(1216342683557601535506311712,
436522681849110124616458784), 32)
x = 434610456570399902378880679233098819019853229470286994367836600566
y = 1064502245825115327754847244914921553977
for c in (652560,
576559230871654959816130551884856912003141446781646602790216406874):
a = x * c
b = y * c
self.assertEqual(gcd(a, b), c)
self.assertEqual(gcd(b, a), c)
self.assertEqual(gcd(-a, b), c)
self.assertEqual(gcd(b, -a), c)
self.assertEqual(gcd(a, -b), c)
self.assertEqual(gcd(-b, a), c)
self.assertEqual(gcd(-a, -b), c)
self.assertEqual(gcd(-b, -a), c)
self.assertEqual(gcd(), 0)
self.assertEqual(gcd(120), 120)
self.assertEqual(gcd(-120), 120)
self.assertEqual(gcd(120, 84, 102), 6)
self.assertEqual(gcd(120, 1, 84), 1)
self.assertRaises(TypeError, gcd, 120.0)
self.assertRaises(TypeError, gcd, 120.0, 84)
self.assertRaises(TypeError, gcd, 120, 84.0)
self.assertRaises(TypeError, gcd, 120, 1, 84.0)
self.assertEqual(gcd(MyIndexable(120), MyIndexable(84)), 12)
def testHypot(self):
from decimal import Decimal
from fractions import Fraction
hypot = math.hypot
# Test different numbers of arguments (from zero to five)
# against a straightforward pure python implementation
args = math.e, math.pi, math.sqrt(2.0), math.gamma(3.5), math.sin(2.1)
for i in range(len(args)+1):
self.assertAlmostEqual(
hypot(*args[:i]),
math.sqrt(sum(s**2 for s in args[:i]))
)
# Test allowable types (those with __float__)
self.assertEqual(hypot(12.0, 5.0), 13.0)
self.assertEqual(hypot(12, 5), 13)
self.assertEqual(hypot(Decimal(12), Decimal(5)), 13)
self.assertEqual(hypot(Fraction(12, 32), Fraction(5, 32)), Fraction(13, 32))
self.assertEqual(hypot(bool(1), bool(0), bool(1), bool(1)), math.sqrt(3))
# Test corner cases
self.assertEqual(hypot(0.0, 0.0), 0.0) # Max input is zero
self.assertEqual(hypot(-10.5), 10.5) # Negative input
self.assertEqual(hypot(), 0.0) # Negative input
self.assertEqual(1.0,
math.copysign(1.0, hypot(-0.0)) # Convert negative zero to positive zero
)
self.assertEqual( # Handling of moving max to the end
hypot(1.5, 1.5, 0.5),
hypot(1.5, 0.5, 1.5),
)
# Test handling of bad arguments
with self.assertRaises(TypeError): # Reject keyword args
hypot(x=1)
with self.assertRaises(TypeError): # Reject values without __float__
hypot(1.1, 'string', 2.2)
int_too_big_for_float = 10 ** (sys.float_info.max_10_exp + 5)
with self.assertRaises((ValueError, OverflowError)):
hypot(1, int_too_big_for_float)
# Any infinity gives positive infinity.
self.assertEqual(hypot(INF), INF)
self.assertEqual(hypot(0, INF), INF)
self.assertEqual(hypot(10, INF), INF)
self.assertEqual(hypot(-10, INF), INF)
self.assertEqual(hypot(NAN, INF), INF)
self.assertEqual(hypot(INF, NAN), INF)
self.assertEqual(hypot(NINF, NAN), INF)
self.assertEqual(hypot(NAN, NINF), INF)
self.assertEqual(hypot(-INF, INF), INF)
self.assertEqual(hypot(-INF, -INF), INF)
self.assertEqual(hypot(10, -INF), INF)
# If no infinity, any NaN gives a NaN.
self.assertTrue(math.isnan(hypot(NAN)))
self.assertTrue(math.isnan(hypot(0, NAN)))
self.assertTrue(math.isnan(hypot(NAN, 10)))
self.assertTrue(math.isnan(hypot(10, NAN)))
self.assertTrue(math.isnan(hypot(NAN, NAN)))
self.assertTrue(math.isnan(hypot(NAN)))
# Verify scaling for extremely large values
fourthmax = FLOAT_MAX / 4.0
for n in range(32):
self.assertTrue(math.isclose(hypot(*([fourthmax]*n)),
fourthmax * math.sqrt(n)))
# Verify scaling for extremely small values
for exp in range(32):
scale = FLOAT_MIN / 2.0 ** exp
self.assertEqual(math.hypot(4*scale, 3*scale), 5*scale)
@requires_IEEE_754
@unittest.skipIf(HAVE_DOUBLE_ROUNDING,
"hypot() loses accuracy on machines with double rounding")
def testHypotAccuracy(self):
# Verify improved accuracy in cases that were known to be inaccurate.
#
# The new algorithm's accuracy depends on IEEE 754 arithmetic
# guarantees, on having the usual ROUND HALF EVEN rounding mode, on
# the system not having double rounding due to extended precision,
# and on the compiler maintaining the specified order of operations.
#
# This test is known to succeed on most of our builds. If it fails
# some build, we either need to add another skipIf if the cause is
# identifiable; otherwise, we can remove this test entirely.
hypot = math.hypot
Decimal = decimal.Decimal
high_precision = decimal.Context(prec=500)
for hx, hy in [
# Cases with a 1 ulp error in Python 3.7 compiled with Clang
('0x1.10e89518dca48p+29', '0x1.1970f7565b7efp+30'),
('0x1.10106eb4b44a2p+29', '0x1.ef0596cdc97f8p+29'),
('0x1.459c058e20bb7p+30', '0x1.993ca009b9178p+29'),
('0x1.378371ae67c0cp+30', '0x1.fbe6619854b4cp+29'),
('0x1.f4cd0574fb97ap+29', '0x1.50fe31669340ep+30'),
('0x1.494b2cdd3d446p+29', '0x1.212a5367b4c7cp+29'),
('0x1.f84e649f1e46dp+29', '0x1.1fa56bef8eec4p+30'),
('0x1.2e817edd3d6fap+30', '0x1.eb0814f1e9602p+29'),
('0x1.0d3a6e3d04245p+29', '0x1.32a62fea52352p+30'),
('0x1.888e19611bfc5p+29', '0x1.52b8e70b24353p+29'),
# Cases with 2 ulp error in Python 3.8
('0x1.538816d48a13fp+29', '0x1.7967c5ca43e16p+29'),
('0x1.57b47b7234530p+29', '0x1.74e2c7040e772p+29'),
('0x1.821b685e9b168p+30', '0x1.677dc1c1e3dc6p+29'),
('0x1.9e8247f67097bp+29', '0x1.24bd2dc4f4baep+29'),
('0x1.b73b59e0cb5f9p+29', '0x1.da899ab784a97p+28'),
('0x1.94a8d2842a7cfp+30', '0x1.326a51d4d8d8ap+30'),
('0x1.e930b9cd99035p+29', '0x1.5a1030e18dff9p+30'),
('0x1.1592bbb0e4690p+29', '0x1.a9c337b33fb9ap+29'),
('0x1.1243a50751fd4p+29', '0x1.a5a10175622d9p+29'),
('0x1.57a8596e74722p+30', '0x1.42d1af9d04da9p+30'),
# Cases with 1 ulp error in version fff3c28052e6b0
('0x1.ee7dbd9565899p+29', '0x1.7ab4d6fc6e4b4p+29'),
('0x1.5c6bfbec5c4dcp+30', '0x1.02511184b4970p+30'),
('0x1.59dcebba995cap+30', '0x1.50ca7e7c38854p+29'),
('0x1.768cdd94cf5aap+29', '0x1.9cfdc5571d38ep+29'),
('0x1.dcf137d60262ep+29', '0x1.1101621990b3ep+30'),
('0x1.3a2d006e288b0p+30', '0x1.e9a240914326cp+29'),
('0x1.62a32f7f53c61p+29', '0x1.47eb6cd72684fp+29'),
('0x1.d3bcb60748ef2p+29', '0x1.3f13c4056312cp+30'),
('0x1.282bdb82f17f3p+30', '0x1.640ba4c4eed3ap+30'),
('0x1.89d8c423ea0c6p+29', '0x1.d35dcfe902bc3p+29'),
]:
x = float.fromhex(hx)
y = float.fromhex(hy)
with self.subTest(hx=hx, hy=hy, x=x, y=y):
with decimal.localcontext(high_precision):
z = float((Decimal(x)**2 + Decimal(y)**2).sqrt())
self.assertEqual(hypot(x, y), z)
def testDist(self):
from decimal import Decimal as D
from fractions import Fraction as F
dist = math.dist
sqrt = math.sqrt
# Simple exact cases
self.assertEqual(dist((1.0, 2.0, 3.0), (4.0, 2.0, -1.0)), 5.0)
self.assertEqual(dist((1, 2, 3), (4, 2, -1)), 5.0)
# Test different numbers of arguments (from zero to nine)
# against a straightforward pure python implementation
for i in range(9):
for j in range(5):
p = tuple(random.uniform(-5, 5) for k in range(i))
q = tuple(random.uniform(-5, 5) for k in range(i))
self.assertAlmostEqual(
dist(p, q),
sqrt(sum((px - qx) ** 2.0 for px, qx in zip(p, q)))
)
# Test non-tuple inputs
self.assertEqual(dist([1.0, 2.0, 3.0], [4.0, 2.0, -1.0]), 5.0)
self.assertEqual(dist(iter([1.0, 2.0, 3.0]), iter([4.0, 2.0, -1.0])), 5.0)
# Test allowable types (those with __float__)
self.assertEqual(dist((14.0, 1.0), (2.0, -4.0)), 13.0)
self.assertEqual(dist((14, 1), (2, -4)), 13)
self.assertEqual(dist((D(14), D(1)), (D(2), D(-4))), D(13))
self.assertEqual(dist((F(14, 32), F(1, 32)), (F(2, 32), F(-4, 32))),
F(13, 32))
self.assertEqual(dist((True, True, False, True, False),
(True, False, True, True, False)),
sqrt(2.0))
# Test corner cases
self.assertEqual(dist((13.25, 12.5, -3.25),
(13.25, 12.5, -3.25)),
0.0) # Distance with self is zero
self.assertEqual(dist((), ()), 0.0) # Zero-dimensional case
self.assertEqual(1.0, # Convert negative zero to positive zero
math.copysign(1.0, dist((-0.0,), (0.0,)))
)
self.assertEqual(1.0, # Convert negative zero to positive zero
math.copysign(1.0, dist((0.0,), (-0.0,)))
)
self.assertEqual( # Handling of moving max to the end
dist((1.5, 1.5, 0.5), (0, 0, 0)),
dist((1.5, 0.5, 1.5), (0, 0, 0))
)
# Verify tuple subclasses are allowed
class T(tuple):
pass
self.assertEqual(dist(T((1, 2, 3)), ((4, 2, -1))), 5.0)
# Test handling of bad arguments
with self.assertRaises(TypeError): # Reject keyword args
dist(p=(1, 2, 3), q=(4, 5, 6))
with self.assertRaises(TypeError): # Too few args
dist((1, 2, 3))
with self.assertRaises(TypeError): # Too many args
dist((1, 2, 3), (4, 5, 6), (7, 8, 9))
with self.assertRaises(TypeError): # Scalars not allowed
dist(1, 2)
with self.assertRaises(TypeError): # Reject values without __float__
dist((1.1, 'string', 2.2), (1, 2, 3))
with self.assertRaises(ValueError): # Check dimension agree
dist((1, 2, 3, 4), (5, 6, 7))
with self.assertRaises(ValueError): # Check dimension agree
dist((1, 2, 3), (4, 5, 6, 7))
with self.assertRaises(TypeError): # Rejects invalid types
dist("abc", "xyz")
int_too_big_for_float = 10 ** (sys.float_info.max_10_exp + 5)
with self.assertRaises((ValueError, OverflowError)):
dist((1, int_too_big_for_float), (2, 3))
with self.assertRaises((ValueError, OverflowError)):
dist((2, 3), (1, int_too_big_for_float))
# Verify that the one dimensional case is equivalent to abs()
for i in range(20):
p, q = random.random(), random.random()
self.assertEqual(dist((p,), (q,)), abs(p - q))
# Test special values
values = [NINF, -10.5, -0.0, 0.0, 10.5, INF, NAN]
for p in itertools.product(values, repeat=3):
for q in itertools.product(values, repeat=3):
diffs = [px - qx for px, qx in zip(p, q)]
if any(map(math.isinf, diffs)):
# Any infinite difference gives positive infinity.
self.assertEqual(dist(p, q), INF)
elif any(map(math.isnan, diffs)):
# If no infinity, any NaN gives a NaN.
self.assertTrue(math.isnan(dist(p, q)))
# Verify scaling for extremely large values
fourthmax = FLOAT_MAX / 4.0
for n in range(32):
p = (fourthmax,) * n
q = (0.0,) * n
self.assertTrue(math.isclose(dist(p, q), fourthmax * math.sqrt(n)))
self.assertTrue(math.isclose(dist(q, p), fourthmax * math.sqrt(n)))
# Verify scaling for extremely small values
for exp in range(32):
scale = FLOAT_MIN / 2.0 ** exp
p = (4*scale, 3*scale)
q = (0.0, 0.0)
self.assertEqual(math.dist(p, q), 5*scale)
self.assertEqual(math.dist(q, p), 5*scale)
def testIsqrt(self):
# Test a variety of inputs, large and small.
test_values = (
list(range(1000))
+ list(range(10**6 - 1000, 10**6 + 1000))
+ [2**e + i for e in range(60, 200) for i in range(-40, 40)]
+ [3**9999, 10**5001]
)
for value in test_values:
with self.subTest(value=value):
s = math.isqrt(value)
self.assertIs(type(s), int)
self.assertLessEqual(s*s, value)
self.assertLess(value, (s+1)*(s+1))
# Negative values
with self.assertRaises(ValueError):
math.isqrt(-1)