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cpython/Lib/test/test_math.py

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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
import sysconfig
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
def ulp(x):
"""Return the value of the least significant bit of a
float x, such that the first float bigger than x is x+ulp(x).
Then, given an expected result x and a tolerance of n ulps,
the result y should be such that abs(y-x) <= n * ulp(x).
The results from this function will only make sense on platforms
where native doubles are represented in IEEE 754 binary64 format.
"""
x = abs(float(x))
if math.isnan(x) or math.isinf(x):
return x
# Find next float up from x.
n = struct.unpack('<q', struct.pack('<d', x))[0]
x_next = struct.unpack('<d', struct.pack('<q', n + 1))[0]
if math.isinf(x_next):
# Corner case: x was the largest finite float. Then it's
# not an exact power of two, so we can take the difference
# between x and the previous float.
x_prev = struct.unpack('<d', struct.pack('<q', n - 1))[0]
return x - x_prev
else:
return x_next - x
# 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 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.ftest('ceil(0.5)', math.ceil(0.5), 1)
self.ftest('ceil(1.0)', math.ceil(1.0), 1)
self.ftest('ceil(1.5)', math.ceil(1.5), 2)
self.ftest('ceil(-0.5)', math.ceil(-0.5), 0)
self.ftest('ceil(-1.0)', math.ceil(-1.0), -1)
self.ftest('ceil(-1.5)', math.ceil(-1.5), -1)
#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 TestNoCeil:
pass
self.ftest('ceil(TestCeil())', math.ceil(TestCeil()), 42)
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=ulp(1))
self.ftest('cos(0)', math.cos(0), 1)
self.ftest('cos(pi/2)', math.cos(math.pi/2), 0, abs_tol=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)))
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)
self.assertEqual(math.factorial(0.0), 1)
total = 1
for i in range(1, 1000):
total *= i
self.assertEqual(math.factorial(i), total)
self.assertEqual(math.factorial(float(i)), total)
self.assertEqual(math.factorial(i), py_factorial(i))
self.assertRaises(ValueError, math.factorial, -1)
self.assertRaises(ValueError, math.factorial, -1.0)
self.assertRaises(ValueError, math.factorial, -10**100)
self.assertRaises(ValueError, math.factorial, -1e100)
self.assertRaises(ValueError, math.factorial, math.pi)
def testFactorialNonIntegers(self):
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 ValueError for inputs that are too large
# to fit into a C long.
self.assertRaises(OverflowError, math.factorial, 10**100)
self.assertRaises(OverflowError, math.factorial, 1e100)
def testFloor(self):
self.assertRaises(TypeError, math.floor)
self.assertEqual(int, type(math.floor(0.5)))
self.ftest('floor(0.5)', math.floor(0.5), 0)
self.ftest('floor(1.0)', math.floor(1.0), 1)
self.ftest('floor(1.5)', math.floor(1.5), 1)
self.ftest('floor(-0.5)', math.floor(-0.5), -1)
self.ftest('floor(-1.0)', math.floor(-1.0), -1)
self.ftest('floor(-1.5)', math.floor(-1.5), -2)
# pow() relies on floor() to check for integers
# This fails on some platforms - so check it here
self.ftest('floor(1.23e167)', math.floor(1.23e167), 1.23e167)
self.ftest('floor(-1.23e167)', math.floor(-1.23e167), -1.23e167)
#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 TestNoFloor:
pass
self.ftest('floor(TestFloor())', math.floor(TestFloor()), 42)
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 http://code.activestate.com/recipes/393090/
"""
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')),
([1.7**(i+1)-1.7**i for i in range(1000)] + [-1.7**1000], -1.0),
([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')),
]
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)
c = 652560
x = 434610456570399902378880679233098819019853229470286994367836600566
y = 1064502245825115327754847244914921553977
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)
c = 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.assertRaises(TypeError, gcd, 120.0, 84)
self.assertRaises(TypeError, gcd, 120, 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.assertEqual(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)
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 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): # Lists not allowed
dist([1, 2, 3], [4, 5, 6])
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.assertEqual(dist(p, q), fourthmax * math.sqrt(n))
self.assertEqual(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 testLdexp(self):
self.assertRaises(TypeError, math.ldexp)
self.ftest('ldexp(0,1)', math.ldexp(0,1), 0)
self.ftest('ldexp(1,1)', math.ldexp(1,1), 2)
self.ftest('ldexp(1,-1)', math.ldexp(1,-1), 0.5)
self.ftest('ldexp(-1,1)', math.ldexp(-1,1), -2)
self.assertRaises(OverflowError, math.ldexp, 1., 1000000)
self.assertRaises(OverflowError, math.ldexp, -1., 1000000)
self.assertEqual(math.ldexp(1., -1000000), 0.)
self.assertEqual(math.ldexp(-1., -1000000), -0.)
self.assertEqual(math.ldexp(INF, 30), INF)
self.assertEqual(math.ldexp(NINF, -213), NINF)
self.assertTrue(math.isnan(math.ldexp(NAN, 0)))
# large second argument
for n in [10**5, 10**10, 10**20, 10**40]:
self.assertEqual(math.ldexp(INF, -n), INF)
self.assertEqual(math.ldexp(NINF, -n), NINF)
self.assertEqual(math.ldexp(1., -n), 0.)
self.assertEqual(math.ldexp(-1., -n), -0.)
self.assertEqual(math.ldexp(0., -n), 0.)
self.assertEqual(math.ldexp(-0., -n), -0.)
self.assertTrue(math.isnan(math.ldexp(NAN, -n)))
self.assertRaises(OverflowError, math.ldexp, 1., n)
self.assertRaises(OverflowError, math.ldexp, -1., n)
self.assertEqual(math.ldexp(0., n), 0.)
self.assertEqual(math.ldexp(-0., n), -0.)
self.assertEqual(math.ldexp(INF, n), INF)
self.assertEqual(math.ldexp(NINF, n), NINF)
self.assertTrue(math.isnan(math.ldexp(NAN, n)))
def testLog(self):
self.assertRaises(TypeError, math.log)
self.ftest('log(1/e)', math.log(1/math.e), -1)
self.ftest('log(1)', math.log(1), 0)
self.ftest('log(e)', math.log(math.e), 1)
self.ftest('log(32,2)', math.log(32,2), 5)
self.ftest('log(10**40, 10)', math.log(10**40, 10), 40)
self.ftest('log(10**40, 10**20)', math.log(10**40, 10**20), 2)
self.ftest('log(10**1000)', math.log(10**1000),
2302.5850929940457)
self.assertRaises(ValueError, math.log, -1.5)
self.assertRaises(ValueError, math.log, -10**1000)
self.assertRaises(ValueError, math.log, NINF)
self.assertEqual(math.log(INF), INF)
self.assertTrue(math.isnan(math.log(NAN)))
def testLog1p(self):
self.assertRaises(TypeError, math.log1p)
for n in [2, 2**90, 2**300]:
self.assertAlmostEqual(math.log1p(n), math.log1p(float(n)))
self.assertRaises(ValueError, math.log1p, -1)
self.assertEqual(math.log1p(INF), INF)
@requires_IEEE_754
def testLog2(self):
self.assertRaises(TypeError, math.log2)
# Check some integer values
self.assertEqual(math.log2(1), 0.0)
self.assertEqual(math.log2(2), 1.0)
self.assertEqual(math.log2(4), 2.0)
# Large integer values
self.assertEqual(math.log2(2**1023), 1023.0)
self.assertEqual(math.log2(2**1024), 1024.0)
self.assertEqual(math.log2(2**2000), 2000.0)
self.assertRaises(ValueError, math.log2, -1.5)
self.assertRaises(ValueError, math.log2, NINF)
self.assertTrue(math.isnan(math.log2(NAN)))
@requires_IEEE_754
# log2() is not accurate enough on Mac OS X Tiger (10.4)
@support.requires_mac_ver(10, 5)
def testLog2Exact(self):
# Check that we get exact equality for log2 of powers of 2.
actual = [math.log2(math.ldexp(1.0, n)) for n in range(-1074, 1024)]
expected = [float(n) for n in range(-1074, 1024)]
self.assertEqual(actual, expected)
def testLog10(self):
self.assertRaises(TypeError, math.log10)
self.ftest('log10(0.1)', math.log10(0.1), -1)
self.ftest('log10(1)', math.log10(1), 0)
self.ftest('log10(10)', math.log10(10), 1)
self.ftest('log10(10**1000)', math.log10(10**1000), 1000.0)
self.assertRaises(ValueError, math.log10, -1.5)
self.assertRaises(ValueError, math.log10, -10**1000)
self.assertRaises(ValueError, math.log10, NINF)
self.assertEqual(math.log(INF), INF)
self.assertTrue(math.isnan(math.log10(NAN)))
def testModf(self):
self.assertRaises(TypeError, math.modf)
def testmodf(name, result, expected):
(v1, v2), (e1, e2) = result, expected
if abs(v1-e1) > eps or abs(v2-e2):
self.fail('%s returned %r, expected %r'%\
(name, result, expected))
testmodf('modf(1.5)', math.modf(1.5), (0.5, 1.0))
testmodf('modf(-1.5)', math.modf(-1.5), (-0.5, -1.0))
self.assertEqual(math.modf(INF), (0.0, INF))
self.assertEqual(math.modf(NINF), (-0.0, NINF))
modf_nan = math.modf(NAN)
self.assertTrue(math.isnan(modf_nan[0]))
self.assertTrue(math.isnan(modf_nan[1]))
def testPow(self):
self.assertRaises(TypeError, math.pow)
self.ftest('pow(0,1)', math.pow(0,1), 0)
self.ftest('pow(1,0)', math.pow(1,0), 1)
self.ftest('pow(2,1)', math.pow(2,1), 2)
self.ftest('pow(2,-1)', math.pow(2,-1), 0.5)
self.assertEqual(math.pow(INF, 1), INF)
self.assertEqual(math.pow(NINF, 1), NINF)
self.assertEqual((math.pow(1, INF)), 1.)
self.assertEqual((math.pow(1, NINF)), 1.)
self.assertTrue(math.isnan(math.pow(NAN, 1)))
self.assertTrue(math.isnan(math.pow(2, NAN)))
self.assertTrue(math.isnan(math.pow(0, NAN)))
self.assertEqual(math.pow(1, NAN), 1)
# pow(0., x)
self.assertEqual(math.pow(0., INF), 0.)
self.assertEqual(math.pow(0., 3.), 0.)
self.assertEqual(math.pow(0., 2.3), 0.)
self.assertEqual(math.pow(0., 2.), 0.)
self.assertEqual(math.pow(0., 0.), 1.)
self.assertEqual(math.pow(0., -0.), 1.)
self.assertRaises(ValueError, math.pow, 0., -2.)
self.assertRaises(ValueError, math.pow, 0., -2.3)
self.assertRaises(ValueError, math.pow, 0., -3.)
self.assertRaises(ValueError, math.pow, 0., NINF)
self.assertTrue(math.isnan(math.pow(0., NAN)))
# pow(INF, x)
self.assertEqual(math.pow(INF, INF), INF)
self.assertEqual(math.pow(INF, 3.), INF)
self.assertEqual(math.pow(INF, 2.3), INF)
self.assertEqual(math.pow(INF, 2.), INF)
self.assertEqual(math.pow(INF, 0.), 1.)
self.assertEqual(math.pow(INF, -0.), 1.)
self.assertEqual(math.pow(INF, -2.), 0.)
self.assertEqual(math.pow(INF, -2.3), 0.)
self.assertEqual(math.pow(INF, -3.), 0.)
self.assertEqual(math.pow(INF, NINF), 0.)
self.assertTrue(math.isnan(math.pow(INF, NAN)))
# pow(-0., x)
self.assertEqual(math.pow(-0., INF), 0.)
self.assertEqual(math.pow(-0., 3.), -0.)
self.assertEqual(math.pow(-0., 2.3), 0.)
self.assertEqual(math.pow(-0., 2.), 0.)
self.assertEqual(math.pow(-0., 0.), 1.)
self.assertEqual(math.pow(-0., -0.), 1.)
self.assertRaises(ValueError, math.pow, -0., -2.)
self.assertRaises(ValueError, math.pow, -0., -2.3)
self.assertRaises(ValueError, math.pow, -0., -3.)
self.assertRaises(ValueError, math.pow, -0., NINF)
self.assertTrue(math.isnan(math.pow(-0., NAN)))
# pow(NINF, x)
self.assertEqual(math.pow(NINF, INF), INF)
self.assertEqual(math.pow(NINF, 3.), NINF)
self.assertEqual(math.pow(NINF, 2.3), INF)
self.assertEqual(math.pow(NINF, 2.), INF)
self.assertEqual(math.pow(NINF, 0.), 1.)
self.assertEqual(math.pow(NINF, -0.), 1.)
self.assertEqual(math.pow(NINF, -2.), 0.)
self.assertEqual(math.pow(NINF, -2.3), 0.)
self.assertEqual(math.pow(NINF, -3.), -0.)
self.assertEqual(math.pow(NINF, NINF), 0.)
self.assertTrue(math.isnan(math.pow(NINF, NAN)))
# pow(-1, x)
self.assertEqual(math.pow(-1., INF), 1.)
self.assertEqual(math.pow(-1., 3.), -1.)
self.assertRaises(ValueError, math.pow, -1., 2.3)
self.assertEqual(math.pow(-1., 2.), 1.)
self.assertEqual(math.pow(-1., 0.), 1.)
self.assertEqual(math.pow(-1., -0.), 1.)
self.assertEqual(math.pow(-1., -2.), 1.)
self.assertRaises(ValueError, math.pow, -1., -2.3)
self.assertEqual(math.pow(-1., -3.), -1.)
self.assertEqual(math.pow(-1., NINF), 1.)
self.assertTrue(math.isnan(math.pow(-1., NAN)))
# pow(1, x)
self.assertEqual(math.pow(1., INF), 1.)
self.assertEqual(math.pow(1., 3.), 1.)
self.assertEqual(math.pow(1., 2.3), 1.)
self.assertEqual(math.pow(1., 2.), 1.)
self.assertEqual(math.pow(1., 0.), 1.)
self.assertEqual(math.pow(1., -0.), 1.)
self.assertEqual(math.pow(1., -2.), 1.)
self.assertEqual(math.pow(1., -2.3), 1.)
self.assertEqual(math.pow(1., -3.), 1.)
self.assertEqual(math.pow(1., NINF), 1.)
self.assertEqual(math.pow(1., NAN), 1.)
# pow(x, 0) should be 1 for any x
self.assertEqual(math.pow(2.3, 0.), 1.)
self.assertEqual(math.pow(-2.3, 0.), 1.)
self.assertEqual(math.pow(NAN, 0.), 1.)
self.assertEqual(math.pow(2.3, -0.), 1.)
self.assertEqual(math.pow(-2.3, -0.), 1.)
self.assertEqual(math.pow(NAN, -0.), 1.)
# pow(x, y) is invalid if x is negative and y is not integral
self.assertRaises(ValueError, math.pow, -1., 2.3)
self.assertRaises(ValueError, math.pow, -15., -3.1)
# pow(x, NINF)
self.assertEqual(math.pow(1.9, NINF), 0.)
self.assertEqual(math.pow(1.1, NINF), 0.)
self.assertEqual(math.pow(0.9, NINF), INF)
self.assertEqual(math.pow(0.1, NINF), INF)
self.assertEqual(math.pow(-0.1, NINF), INF)
self.assertEqual(math.pow(-0.9, NINF), INF)
self.assertEqual(math.pow(-1.1, NINF), 0.)
self.assertEqual(math.pow(-1.9, NINF), 0.)
# pow(x, INF)
self.assertEqual(math.pow(1.9, INF), INF)
self.assertEqual(math.pow(1.1, INF), INF)
self.assertEqual(math.pow(0.9, INF), 0.)
self.assertEqual(math.pow(0.1, INF), 0.)
self.assertEqual(math.pow(-0.1, INF), 0.)
self.assertEqual(math.pow(-0.9, INF), 0.)
self.assertEqual(math.pow(-1.1, INF), INF)
self.assertEqual(math.pow(-1.9, INF), INF)
# pow(x, y) should work for x negative, y an integer
self.ftest('(-2.)**3.', math.pow(-2.0, 3.0), -8.0)
self.ftest('(-2.)**2.', math.pow(-2.0, 2.0), 4.0)
self.ftest('(-2.)**1.', math.pow(-2.0, 1.0), -2.0)
self.ftest('(-2.)**0.', math.pow(-2.0, 0.0), 1.0)
self.ftest('(-2.)**-0.', math.pow(-2.0, -0.0), 1.0)
self.ftest('(-2.)**-1.', math.pow(-2.0, -1.0), -0.5)
self.ftest('(-2.)**-2.', math.pow(-2.0, -2.0), 0.25)
self.ftest('(-2.)**-3.', math.pow(-2.0, -3.0), -0.125)
self.assertRaises(ValueError, math.pow, -2.0, -0.5)
self.assertRaises(ValueError, math.pow, -2.0, 0.5)
# the following tests have been commented out since they don't
# really belong here: the implementation of ** for floats is
# independent of the implementation of math.pow
#self.assertEqual(1**NAN, 1)
#self.assertEqual(1**INF, 1)
#self.assertEqual(1**NINF, 1)
#self.assertEqual(1**0, 1)
#self.assertEqual(1.**NAN, 1)
#self.assertEqual(1.**INF, 1)
#self.assertEqual(1.**NINF, 1)
#self.assertEqual(1.**0, 1)
def testRadians(self):
self.assertRaises(TypeError, math.radians)
self.ftest('radians(180)', math.radians(180), math.pi)
self.ftest('radians(90)', math.radians(90), math.pi/2)
self.ftest('radians(-45)', math.radians(-45), -math.pi/4)
self.ftest('radians(0)', math.radians(0), 0)
@requires_IEEE_754
def testRemainder(self):
from fractions import Fraction
def validate_spec(x, y, r):
"""
Check that r matches remainder(x, y) according to the IEEE 754
specification. Assumes that x, y and r are finite and y is nonzero.
"""
fx, fy, fr = Fraction(x), Fraction(y), Fraction(r)
# r should not exceed y/2 in absolute value
self.assertLessEqual(abs(fr), abs(fy/2))
# x - r should be an exact integer multiple of y
n = (fx - fr) / fy
self.assertEqual(n, int(n))
if abs(fr) == abs(fy/2):
# If |r| == |y/2|, n should be even.
self.assertEqual(n/2, int(n/2))
# triples (x, y, remainder(x, y)) in hexadecimal form.
testcases = [
# Remainders modulo 1, showing the ties-to-even behaviour.
'-4.0 1 -0.0',
'-3.8 1 0.8',
'-3.0 1 -0.0',
'-2.8 1 -0.8',
'-2.0 1 -0.0',
'-1.8 1 0.8',
'-1.0 1 -0.0',
'-0.8 1 -0.8',
'-0.0 1 -0.0',
' 0.0 1 0.0',
' 0.8 1 0.8',
' 1.0 1 0.0',
' 1.8 1 -0.8',
' 2.0 1 0.0',
' 2.8 1 0.8',
' 3.0 1 0.0',
' 3.8 1 -0.8',
' 4.0 1 0.0',
# Reductions modulo 2*pi
'0x0.0p+0 0x1.921fb54442d18p+2 0x0.0p+0',
'0x1.921fb54442d18p+0 0x1.921fb54442d18p+2 0x1.921fb54442d18p+0',
'0x1.921fb54442d17p+1 0x1.921fb54442d18p+2 0x1.921fb54442d17p+1',
'0x1.921fb54442d18p+1 0x1.921fb54442d18p+2 0x1.921fb54442d18p+1',
'0x1.921fb54442d19p+1 0x1.921fb54442d18p+2 -0x1.921fb54442d17p+1',
'0x1.921fb54442d17p+2 0x1.921fb54442d18p+2 -0x0.0000000000001p+2',
'0x1.921fb54442d18p+2 0x1.921fb54442d18p+2 0x0p0',
'0x1.921fb54442d19p+2 0x1.921fb54442d18p+2 0x0.0000000000001p+2',
'0x1.2d97c7f3321d1p+3 0x1.921fb54442d18p+2 0x1.921fb54442d14p+1',
'0x1.2d97c7f3321d2p+3 0x1.921fb54442d18p+2 -0x1.921fb54442d18p+1',
'0x1.2d97c7f3321d3p+3 0x1.921fb54442d18p+2 -0x1.921fb54442d14p+1',
'0x1.921fb54442d17p+3 0x1.921fb54442d18p+2 -0x0.0000000000001p+3',
'0x1.921fb54442d18p+3 0x1.921fb54442d18p+2 0x0p0',
'0x1.921fb54442d19p+3 0x1.921fb54442d18p+2 0x0.0000000000001p+3',
'0x1.f6a7a2955385dp+3 0x1.921fb54442d18p+2 0x1.921fb54442d14p+1',
'0x1.f6a7a2955385ep+3 0x1.921fb54442d18p+2 0x1.921fb54442d18p+1',
'0x1.f6a7a2955385fp+3 0x1.921fb54442d18p+2 -0x1.921fb54442d14p+1',
'0x1.1475cc9eedf00p+5 0x1.921fb54442d18p+2 0x1.921fb54442d10p+1',
'0x1.1475cc9eedf01p+5 0x1.921fb54442d18p+2 -0x1.921fb54442d10p+1',
# Symmetry with respect to signs.
' 1 0.c 0.4',
'-1 0.c -0.4',
' 1 -0.c 0.4',
'-1 -0.c -0.4',
' 1.4 0.c -0.4',
'-1.4 0.c 0.4',
' 1.4 -0.c -0.4',
'-1.4 -0.c 0.4',
# Huge modulus, to check that the underlying algorithm doesn't
# rely on 2.0 * modulus being representable.
'0x1.dp+1023 0x1.4p+1023 0x0.9p+1023',
'0x1.ep+1023 0x1.4p+1023 -0x0.ap+1023',
'0x1.fp+1023 0x1.4p+1023 -0x0.9p+1023',
]
for case in testcases:
with self.subTest(case=case):
x_hex, y_hex, expected_hex = case.split()
x = float.fromhex(x_hex)
y = float.fromhex(y_hex)
expected = float.fromhex(expected_hex)
validate_spec(x, y, expected)
actual = math.remainder(x, y)
# Cheap way of checking that the floats are
# as identical as we need them to be.
self.assertEqual(actual.hex(), expected.hex())
# Test tiny subnormal modulus: there's potential for
# getting the implementation wrong here (for example,
# by assuming that modulus/2 is exactly representable).
tiny = float.fromhex('1p-1074') # min +ve subnormal
for n in range(-25, 25):
if n == 0:
continue
y = n * tiny
for m in range(100):
x = m * tiny
actual = math.remainder(x, y)
validate_spec(x, y, actual)
actual = math.remainder(-x, y)
validate_spec(-x, y, actual)
# Special values.
# NaNs should propagate as usual.
for value in [NAN, 0.0, -0.0, 2.0, -2.3, NINF, INF]:
self.assertIsNaN(math.remainder(NAN, value))
self.assertIsNaN(math.remainder(value, NAN))
# remainder(x, inf) is x, for non-nan non-infinite x.
for value in [-2.3, -0.0, 0.0, 2.3]:
self.assertEqual(math.remainder(value, INF), value)
self.assertEqual(math.remainder(value, NINF), value)
# remainder(x, 0) and remainder(infinity, x) for non-NaN x are invalid
# operations according to IEEE 754-2008 7.2(f), and should raise.
for value in [NINF, -2.3, -0.0, 0.0, 2.3, INF]:
with self.assertRaises(ValueError):
math.remainder(INF, value)
with self.assertRaises(ValueError):
math.remainder(NINF, value)
with self.assertRaises(ValueError):
math.remainder(value, 0.0)
with self.assertRaises(ValueError):
math.remainder(value, -0.0)
def testSin(self):
self.assertRaises(TypeError, math.sin)
self.ftest('sin(0)', math.sin(0), 0)
self.ftest('sin(pi/2)', math.sin(math.pi/2), 1)
self.ftest('sin(-pi/2)', math.sin(-math.pi/2), -1)
try:
self.assertTrue(math.isnan(math.sin(INF)))
self.assertTrue(math.isnan(math.sin(NINF)))
except ValueError:
self.assertRaises(ValueError, math.sin, INF)
self.assertRaises(ValueError, math.sin, NINF)
self.assertTrue(math.isnan(math.sin(NAN)))
def testSinh(self):
self.assertRaises(TypeError, math.sinh)
self.ftest('sinh(0)', math.sinh(0), 0)
self.ftest('sinh(1)**2-cosh(1)**2', math.sinh(1)**2-math.cosh(1)**2, -1)
self.ftest('sinh(1)+sinh(-1)', math.sinh(1)+math.sinh(-1), 0)
self.assertEqual(math.sinh(INF), INF)
self.assertEqual(math.sinh(NINF), NINF)
self.assertTrue(math.isnan(math.sinh(NAN)))
def testSqrt(self):
self.assertRaises(TypeError, math.sqrt)
self.ftest('sqrt(0)', math.sqrt(0), 0)
self.ftest('sqrt(1)', math.sqrt(1), 1)
self.ftest('sqrt(4)', math.sqrt(4), 2)
self.assertEqual(math.sqrt(INF), INF)
self.assertRaises(ValueError, math.sqrt, -1)
self.assertRaises(ValueError, math.sqrt, NINF)
self.assertTrue(math.isnan(math.sqrt(NAN)))
def testTan(self):
self.assertRaises(TypeError, math.tan)
self.ftest('tan(0)', math.tan(0), 0)
self.ftest('tan(pi/4)', math.tan(math.pi/4), 1)
self.ftest('tan(-pi/4)', math.tan(-math.pi/4), -1)
try:
self.assertTrue(math.isnan(math.tan(INF)))
self.assertTrue(math.isnan(math.tan(NINF)))
except:
self.assertRaises(ValueError, math.tan, INF)
self.assertRaises(ValueError, math.tan, NINF)
self.assertTrue(math.isnan(math.tan(NAN)))
def testTanh(self):
self.assertRaises(TypeError, math.tanh)
self.ftest('tanh(0)', math.tanh(0), 0)
self.ftest('tanh(1)+tanh(-1)', math.tanh(1)+math.tanh(-1), 0,
abs_tol=ulp(1))
self.ftest('tanh(inf)', math.tanh(INF), 1)
self.ftest('tanh(-inf)', math.tanh(NINF), -1)
self.assertTrue(math.isnan(math.tanh(NAN)))
@requires_IEEE_754
def testTanhSign(self):
# check that tanh(-0.) == -0. on IEEE 754 systems
self.assertEqual(math.tanh(-0.), -0.)
self.assertEqual(math.copysign(1., math.tanh(-0.)),
math.copysign(1., -0.))
def test_trunc(self):
self.assertEqual(math.trunc(1), 1)
self.assertEqual(math.trunc(-1), -1)
self.assertEqual(type(math.trunc(1)), int)
self.assertEqual(type(math.trunc(1.5)), int)
self.assertEqual(math.trunc(1.5), 1)
self.assertEqual(math.trunc(-1.5), -1)
self.assertEqual(math.trunc(1.999999), 1)
self.assertEqual(math.trunc(-1.999999), -1)
self.assertEqual(math.trunc(-0.999999), -0)
self.assertEqual(math.trunc(-100.999), -100)
class TestTrunc(object):
def __trunc__(self):
return 23
class TestNoTrunc(object):
pass
self.assertEqual(math.trunc(TestTrunc()), 23)
self.assertRaises(TypeError, math.trunc)
self.assertRaises(TypeError, math.trunc, 1, 2)
self.assertRaises(TypeError, math.trunc, TestNoTrunc())
def testIsfinite(self):
self.assertTrue(math.isfinite(0.0))
self.assertTrue(math.isfinite(-0.0))
self.assertTrue(math.isfinite(1.0))
self.assertTrue(math.isfinite(-1.0))
self.assertFalse(math.isfinite(float("nan")))
self.assertFalse(math.isfinite(float("inf")))
self.assertFalse(math.isfinite(float("-inf")))
def testIsnan(self):
self.assertTrue(math.isnan(float("nan")))
self.assertTrue(math.isnan(float("-nan")))
self.assertTrue(math.isnan(float("inf") * 0.))
self.assertFalse(math.isnan(float("inf")))
self.assertFalse(math.isnan(0.))
self.assertFalse(math.isnan(1.))
def testIsinf(self):
self.assertTrue(math.isinf(float("inf")))
self.assertTrue(math.isinf(float("-inf")))
self.assertTrue(math.isinf(1E400))
self.assertTrue(math.isinf(-1E400))
self.assertFalse(math.isinf(float("nan")))
self.assertFalse(math.isinf(0.))
self.assertFalse(math.isinf(1.))
@requires_IEEE_754
def test_nan_constant(self):
self.assertTrue(math.isnan(math.nan))
@requires_IEEE_754
def test_inf_constant(self):
self.assertTrue(math.isinf(math.inf))
self.assertGreater(math.inf, 0.0)
self.assertEqual(math.inf, float("inf"))
self.assertEqual(-math.inf, float("-inf"))
# RED_FLAG 16-Oct-2000 Tim
# While 2.0 is more consistent about exceptions than previous releases, it
# still fails this part of the test on some platforms. For now, we only
# *run* test_exceptions() in verbose mode, so that this isn't normally
# tested.
@unittest.skipUnless(verbose, 'requires verbose mode')
def test_exceptions(self):
try:
x = math.exp(-1000000000)
except:
# mathmodule.c is failing to weed out underflows from libm, or
# we've got an fp format with huge dynamic range
self.fail("underflowing exp() should not have raised "
"an exception")
if x != 0:
self.fail("underflowing exp() should have returned 0")
# If this fails, probably using a strict IEEE-754 conforming libm, and x
# is +Inf afterwards. But Python wants overflows detected by default.
try:
x = math.exp(1000000000)
except OverflowError:
pass
else:
self.fail("overflowing exp() didn't trigger OverflowError")
# If this fails, it could be a puzzle. One odd possibility is that
# mathmodule.c's macros are getting confused while comparing
# Inf (HUGE_VAL) to a NaN, and artificially setting errno to ERANGE
# as a result (and so raising OverflowError instead).
try:
x = math.sqrt(-1.0)
except ValueError:
pass
else:
self.fail("sqrt(-1) didn't raise ValueError")
@requires_IEEE_754
def test_testfile(self):
# Some tests need to be skipped on ancient OS X versions.
# See issue #27953.
SKIP_ON_TIGER = {'tan0064'}
osx_version = None
if sys.platform == 'darwin':
version_txt = platform.mac_ver()[0]
try:
osx_version = tuple(map(int, version_txt.split('.')))
except ValueError:
pass
fail_fmt = "{}: {}({!r}): {}"
failures = []
for id, fn, ar, ai, er, ei, flags in parse_testfile(test_file):
# Skip if either the input or result is complex
if ai != 0.0 or ei != 0.0:
continue
if fn in ['rect', 'polar']:
# no real versions of rect, polar
continue
# Skip certain tests on OS X 10.4.
if osx_version is not None and osx_version < (10, 5):
if id in SKIP_ON_TIGER:
continue
func = getattr(math, fn)
if 'invalid' in flags or 'divide-by-zero' in flags:
er = 'ValueError'
elif 'overflow' in flags:
er = 'OverflowError'
try:
result = func(ar)
except ValueError:
result = 'ValueError'
except OverflowError:
result = 'OverflowError'
# Default tolerances
ulp_tol, abs_tol = 5, 0.0
failure = result_check(er, result, ulp_tol, abs_tol)
if failure is None:
continue
msg = fail_fmt.format(id, fn, ar, failure)
failures.append(msg)
if failures:
self.fail('Failures in test_testfile:\n ' +
'\n '.join(failures))
@requires_IEEE_754
def test_mtestfile(self):
fail_fmt = "{}: {}({!r}): {}"
failures = []
for id, fn, arg, expected, flags in parse_mtestfile(math_testcases):
func = getattr(math, fn)
if 'invalid' in flags or 'divide-by-zero' in flags:
expected = 'ValueError'
elif 'overflow' in flags:
expected = 'OverflowError'
try:
got = func(arg)
except ValueError:
got = 'ValueError'
except OverflowError:
got = 'OverflowError'
# Default tolerances
ulp_tol, abs_tol = 5, 0.0
# Exceptions to the defaults
if fn == 'gamma':
# Experimental results on one platform gave
# an accuracy of <= 10 ulps across the entire float
# domain. We weaken that to require 20 ulp accuracy.
ulp_tol = 20
elif fn == 'lgamma':
# we use a weaker accuracy test for lgamma;
# lgamma only achieves an absolute error of
# a few multiples of the machine accuracy, in
# general.
abs_tol = 1e-15
elif fn == 'erfc' and arg >= 0.0:
# erfc has less-than-ideal accuracy for large
# arguments (x ~ 25 or so), mainly due to the
# error involved in computing exp(-x*x).
#
# Observed between CPython and mpmath at 25 dp:
# x < 0 : err <= 2 ulp
# 0 <= x < 1 : err <= 10 ulp
# 1 <= x < 10 : err <= 100 ulp
# 10 <= x < 20 : err <= 300 ulp
# 20 <= x : < 600 ulp
#
if arg < 1.0:
ulp_tol = 10
elif arg < 10.0:
ulp_tol = 100
else:
ulp_tol = 1000
failure = result_check(expected, got, ulp_tol, abs_tol)
if failure is None:
continue
msg = fail_fmt.format(id, fn, arg, failure)
failures.append(msg)
if failures:
self.fail('Failures in test_mtestfile:\n ' +
'\n '.join(failures))
# Custom assertions.
def assertIsNaN(self, value):
if not math.isnan(value):
self.fail("Expected a NaN, got {!r}.".format(value))
class IsCloseTests(unittest.TestCase):
isclose = math.isclose # subclasses should override this
def assertIsClose(self, a, b, *args, **kwargs):
self.assertTrue(self.isclose(a, b, *args, **kwargs),
msg="%s and %s should be close!" % (a, b))
def assertIsNotClose(self, a, b, *args, **kwargs):
self.assertFalse(self.isclose(a, b, *args, **kwargs),
msg="%s and %s should not be close!" % (a, b))
def assertAllClose(self, examples, *args, **kwargs):
for a, b in examples:
self.assertIsClose(a, b, *args, **kwargs)
def assertAllNotClose(self, examples, *args, **kwargs):
for a, b in examples:
self.assertIsNotClose(a, b, *args, **kwargs)
def test_negative_tolerances(self):
# ValueError should be raised if either tolerance is less than zero
with self.assertRaises(ValueError):
self.assertIsClose(1, 1, rel_tol=-1e-100)
with self.assertRaises(ValueError):
self.assertIsClose(1, 1, rel_tol=1e-100, abs_tol=-1e10)
def test_identical(self):
# identical values must test as close
identical_examples = [(2.0, 2.0),
(0.1e200, 0.1e200),
(1.123e-300, 1.123e-300),
(12345, 12345.0),
(0.0, -0.0),
(345678, 345678)]
self.assertAllClose(identical_examples, rel_tol=0.0, abs_tol=0.0)
def test_eight_decimal_places(self):
# examples that are close to 1e-8, but not 1e-9
eight_decimal_places_examples = [(1e8, 1e8 + 1),
(-1e-8, -1.000000009e-8),
(1.12345678, 1.12345679)]
self.assertAllClose(eight_decimal_places_examples, rel_tol=1e-8)
self.assertAllNotClose(eight_decimal_places_examples, rel_tol=1e-9)
def test_near_zero(self):
# values close to zero
near_zero_examples = [(1e-9, 0.0),
(-1e-9, 0.0),
(-1e-150, 0.0)]
# these should not be close to any rel_tol
self.assertAllNotClose(near_zero_examples, rel_tol=0.9)
# these should be close to abs_tol=1e-8
self.assertAllClose(near_zero_examples, abs_tol=1e-8)
def test_identical_infinite(self):
# these are close regardless of tolerance -- i.e. they are equal
self.assertIsClose(INF, INF)
self.assertIsClose(INF, INF, abs_tol=0.0)
self.assertIsClose(NINF, NINF)
self.assertIsClose(NINF, NINF, abs_tol=0.0)
def test_inf_ninf_nan(self):
# these should never be close (following IEEE 754 rules for equality)
not_close_examples = [(NAN, NAN),
(NAN, 1e-100),
(1e-100, NAN),
(INF, NAN),
(NAN, INF),
(INF, NINF),
(INF, 1.0),
(1.0, INF),
(INF, 1e308),
(1e308, INF)]
# use largest reasonable tolerance
self.assertAllNotClose(not_close_examples, abs_tol=0.999999999999999)
def test_zero_tolerance(self):
# test with zero tolerance
zero_tolerance_close_examples = [(1.0, 1.0),
(-3.4, -3.4),
(-1e-300, -1e-300)]
self.assertAllClose(zero_tolerance_close_examples, rel_tol=0.0)
zero_tolerance_not_close_examples = [(1.0, 1.000000000000001),
(0.99999999999999, 1.0),
(1.0e200, .999999999999999e200)]
self.assertAllNotClose(zero_tolerance_not_close_examples, rel_tol=0.0)
def test_asymmetry(self):
# test the asymmetry example from PEP 485
self.assertAllClose([(9, 10), (10, 9)], rel_tol=0.1)
def test_integers(self):
# test with integer values
integer_examples = [(100000001, 100000000),
(123456789, 123456788)]
self.assertAllClose(integer_examples, rel_tol=1e-8)
self.assertAllNotClose(integer_examples, rel_tol=1e-9)
def test_decimals(self):
# test with Decimal values
from decimal import Decimal
decimal_examples = [(Decimal('1.00000001'), Decimal('1.0')),
(Decimal('1.00000001e-20'), Decimal('1.0e-20')),
(Decimal('1.00000001e-100'), Decimal('1.0e-100')),
(Decimal('1.00000001e20'), Decimal('1.0e20'))]
self.assertAllClose(decimal_examples, rel_tol=1e-8)
self.assertAllNotClose(decimal_examples, rel_tol=1e-9)
def test_fractions(self):
# test with Fraction values
from fractions import Fraction
fraction_examples = [
(Fraction(1, 100000000) + 1, Fraction(1)),
(Fraction(100000001), Fraction(100000000)),
(Fraction(10**8 + 1, 10**28), Fraction(1, 10**20))]
self.assertAllClose(fraction_examples, rel_tol=1e-8)
self.assertAllNotClose(fraction_examples, rel_tol=1e-9)
def test_main():
from doctest import DocFileSuite
suite = unittest.TestSuite()
suite.addTest(unittest.makeSuite(MathTests))
suite.addTest(unittest.makeSuite(IsCloseTests))
suite.addTest(DocFileSuite("ieee754.txt"))
run_unittest(suite)
if __name__ == '__main__':
test_main()
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