Lib/test/test_math.py

# 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 math
import os
import platform
import sys
import struct
import sysconfig

eps = 1E-05
NAN = float('nan')
INF = float('inf')
NINF = float('-inf')

# 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 C doubles are represented in IEEE 754 binary64 format.

    """
    n = struct.unpack('<q', struct.pack('<d', x))[0]
    if n < 0:
        n = ~(n+2**63)
    return n

def ulps_check(expected, got, ulps=20):
    """Given non-NaN floats `expected` and `got`,
    check that they're equal to within the given number of ulps.

    Returns None on success and an error message on failure."""

    ulps_error = to_ulps(got) - to_ulps(expected)
    if abs(ulps_error) <= ulps:
        return None
    return "error = {} ulps; permitted error = {} ulps".format(ulps_error,
                                                               ulps)

# 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 acc_check(expected, got, rel_err=2e-15, abs_err = 5e-323):
    """Determine whether non-NaN floats a and b are equal to within a
    (small) rounding error.  The default values for rel_err and
    abs_err are chosen to be suitable for platforms where a float is
    represented by an IEEE 754 double.  They allow an error of between
    9 and 19 ulps."""

    # need to special case infinities, since inf - inf gives nan
    if math.isinf(expected) and got == expected:
        return None

    error = got - expected

    permitted_error = max(abs_err, rel_err * abs(expected))
    if abs(error) < permitted_error:
        return None
    return "error = {}; permitted error = {}".format(error,
                                                     permitted_error)

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
                  )

# 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, value, expected):
        if abs(value-expected) > eps:
            # Use %r instead of %f so the error message
            # displays full precision. Otherwise discrepancies
            # in the last few bits will lead to very confusing
            # error messages
            self.fail('%s returned %r, expected %r' %
                      (name, value, expected))

    def testConstants(self):
        self.ftest('pi', math.pi, 3.1415926)
        self.ftest('e', math.e, 2.7182818)

    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.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.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)
        self.ftest('cos(0)', math.cos(0), 1)
        self.ftest('cos(pi/2)', math.cos(math.pi/2), 0)
        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)

    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)))

    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)

    # 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):
        self.assertRaises(TypeError, math.hypot)
        self.ftest('hypot(0,0)', math.hypot(0,0), 0)
        self.ftest('hypot(3,4)', math.hypot(3,4), 5)
        self.assertEqual(math.hypot(NAN, INF), INF)
        self.assertEqual(math.hypot(INF, NAN), INF)
        self.assertEqual(math.hypot(NAN, NINF), INF)
        self.assertEqual(math.hypot(NINF, NAN), INF)
        self.assertTrue(math.isnan(math.hypot(1.0, NAN)))
        self.assertTrue(math.isnan(math.hypot(NAN, -2.0)))

    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)
        n= 2**90
        self.assertAlmostEqual(math.log1p(n), math.log1p(float(n)))

    @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)

    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, 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)
        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
    @unittest.skipIf(sysconfig.get_config_var('TANH_PRESERVES_ZERO_SIGN') == 0,
                     "system tanh() function doesn't copy the sign")
    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("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):
        for id, fn, ar, ai, er, ei, flags in parse_testfile(test_file):
            # Skip if either the input or result is complex, or if
            # flags is nonempty
            if ai != 0. or ei != 0. or flags:
                continue
            if fn in ['rect', 'polar']:
                # no real versions of rect, polar
                continue
            func = getattr(math, fn)
            try:
                result = func(ar)
            except ValueError as exc:
                message = (("Unexpected ValueError: %s\n        " +
                           "in test %s:%s(%r)\n") % (exc.args[0], id, fn, ar))
                self.fail(message)
            except OverflowError:
                message = ("Unexpected OverflowError in " +
                           "test %s:%s(%r)\n" % (id, fn, ar))
                self.fail(message)
            self.ftest("%s:%s(%r)" % (id, fn, ar), result, er)

    @requires_IEEE_754
    def test_mtestfile(self):
        fail_fmt = "{}:{}({!r}): expected {!r}, got {!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'

            accuracy_failure = None
            if isinstance(got, float) and isinstance(expected, float):
                if math.isnan(expected) and math.isnan(got):
                    continue
                if not math.isnan(expected) and not math.isnan(got):
                    if 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.
                        accuracy_failure = acc_check(expected, got,
                                                  rel_err = 5e-15,
                                                  abs_err = 5e-15)
                    elif fn == 'erfc':
                        # erfc has less-than-ideal accuracy for large
                        # arguments (x ~ 25 or so), mainly due to the
                        # error involved in computing exp(-x*x).
                        #
                        # XXX Would be better to weaken this test only
                        # for large x, instead of for all x.
                        accuracy_failure = ulps_check(expected, got, 2000)

                    else:
                        accuracy_failure = ulps_check(expected, got, 20)
                    if accuracy_failure is None:
                        continue

            if isinstance(got, str) and isinstance(expected, str):
                if got == expected:
                    continue

            fail_msg = fail_fmt.format(id, fn, arg, expected, got)
            if accuracy_failure is not None:
                fail_msg += ' ({})'.format(accuracy_failure)
            failures.append(fail_msg)

        if failures:
            self.fail('Failures in test_mtestfile:\n  ' +
                      '\n  '.join(failures))


class IsCloseTests(unittest.TestCase):
    isclose = math.isclose # sublcasses 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'))]
        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

        # could use some more examples here!
        fraction_examples = [(Fraction(1, 100000000) + 1, Fraction(1))]
        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()