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//usr/local/lib/python3.9/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 itertools import decimal import math import os import platform import random import struct import sys eps = 1E-05 NAN = float('nan') INF = float('inf') NINF = float('-inf') FLOAT_MAX = sys.float_info.max FLOAT_MIN = sys.float_info.min # detect evidence of double-rounding: fsum is not always correctly # rounded on machines that suffer from double rounding. x, y = 1e16, 2.9999 # use temporary values to defeat peephole optimizer HAVE_DOUBLE_ROUNDING = (x + y == 1e16 + 4) # locate file with test values if __name__ == '__main__': file = sys.argv[0] else: file = __file__ test_dir = os.path.dirname(file) or os.curdir math_testcases = os.path.join(test_dir, 'math_testcases.txt') test_file = os.path.join(test_dir, 'cmath_testcases.txt') def to_ulps(x): """Convert a non-NaN float x to an integer, in such a way that adjacent floats are converted to adjacent integers. Then abs(ulps(x) - ulps(y)) gives the difference in ulps between two floats. The results from this function will only make sense on platforms where native doubles are represented in IEEE 754 binary64 format. Note: 0.0 and -0.0 are converted to 0 and -1, respectively. """ n = struct.unpack('<q', struct.pack('<d', x))[0] if n < 0: n = ~(n+2**63) return n # Here's a pure Python version of the math.factorial algorithm, for # documentation and comparison purposes. # # Formula: # # factorial(n) = factorial_odd_part(n) << (n - count_set_bits(n)) # # where # # factorial_odd_part(n) = product_{i >= 0} product_{0 < j <= n >> i; j odd} j # # The outer product above is an infinite product, but once i >= n.bit_length, # (n >> i) < 1 and the corresponding term of the product is empty. So only the # finitely many terms for 0 <= i < n.bit_length() contribute anything. # # We iterate downwards from i == n.bit_length() - 1 to i == 0. The inner # product in the formula above starts at 1 for i == n.bit_length(); for each i # < n.bit_length() we get the inner product for i from that for i + 1 by # multiplying by all j in {n >> i+1 < j <= n >> i; j odd}. In Python terms, # this set is range((n >> i+1) + 1 | 1, (n >> i) + 1 | 1, 2). def count_set_bits(n): """Number of '1' bits in binary expansion of a nonnnegative integer.""" return 1 + count_set_bits(n & n - 1) if n else 0 def partial_product(start, stop): """Product of integers in range(start, stop, 2), computed recursively. start and stop should both be odd, with start <= stop. """ numfactors = (stop - start) >> 1 if not numfactors: return 1 elif numfactors == 1: return start else: mid = (start + numfactors) | 1 return partial_product(start, mid) * partial_product(mid, stop) def py_factorial(n): """Factorial of nonnegative integer n, via "Binary Split Factorial Formula" described at http://www.luschny.de/math/factorial/binarysplitfact.html """ inner = outer = 1 for i in reversed(range(n.bit_length())): inner *= partial_product((n >> i + 1) + 1 | 1, (n >> i) + 1 | 1) outer *= inner return outer << (n - count_set_bits(n)) def ulp_abs_check(expected, got, ulp_tol, abs_tol): """Given finite floats `expected` and `got`, check that they're approximately equal to within the given number of ulps or the given absolute tolerance, whichever is bigger. Returns None on success and an error message on failure. """ ulp_error = abs(to_ulps(expected) - to_ulps(got)) abs_error = abs(expected - got) # Succeed if either abs_error <= abs_tol or ulp_error <= ulp_tol. if abs_error <= abs_tol or ulp_error <= ulp_tol: return None else: fmt = ("error = {:.3g} ({:d} ulps); " "permitted error = {:.3g} or {:d} ulps") return fmt.format(abs_error, ulp_error, abs_tol, ulp_tol) def parse_mtestfile(fname): """Parse a file with test values -- starts a comment blank lines, or lines containing only a comment, are ignored other lines are expected to have the form id fn arg -> expected [flag]* """ with open(fname) as fp: for line in fp: # strip comments, and skip blank lines if '--' in line: line = line[:line.index('--')] if not line.strip(): continue lhs, rhs = line.split('->') id, fn, arg = lhs.split() rhs_pieces = rhs.split() exp = rhs_pieces[0] flags = rhs_pieces[1:] yield (id, fn, float(arg), float(exp), flags) def parse_testfile(fname): """Parse a file with test values Empty lines or lines starting with -- are ignored yields id, fn, arg_real, arg_imag, exp_real, exp_imag """ with open(fname) as fp: for line in fp: # skip comment lines and blank lines if line.startswith('--') or not line.strip(): continue lhs, rhs = line.split('->') id, fn, arg_real, arg_imag = lhs.split() rhs_pieces = rhs.split() exp_real, exp_imag = rhs_pieces[0], rhs_pieces[1] flags = rhs_pieces[2:] yield (id, fn, float(arg_real), float(arg_imag), float(exp_real), float(exp_imag), flags) def result_check(expected, got, ulp_tol=5, abs_tol=0.0): # Common logic of MathTests.(ftest, test_testcases, test_mtestcases) """Compare arguments expected and got, as floats, if either is a float, using a tolerance expressed in multiples of ulp(expected) or absolutely (if given and greater). As a convenience, when neither argument is a float, and for non-finite floats, exact equality is demanded. Also, nan==nan as far as this function is concerned. Returns None on success and an error message on failure. """ # Check exactly equal (applies also to strings representing exceptions) if got == expected: return None failure = "not equal" # Turn mixed float and int comparison (e.g. floor()) to all-float if isinstance(expected, float) and isinstance(got, int): got = float(got) elif isinstance(got, float) and isinstance(expected, int): expected = float(expected) if isinstance(expected, float) and isinstance(got, float): if math.isnan(expected) and math.isnan(got): # Pass, since both nan failure = None elif math.isinf(expected) or math.isinf(got): # We already know they're not equal, drop through to failure pass else: # Both are finite floats (now). Are they close enough? failure = ulp_abs_check(expected, got, ulp_tol, abs_tol) # arguments are not equal, and if numeric, are too far apart if failure is not None: fail_fmt = "expected {!r}, got {!r}" fail_msg = fail_fmt.format(expected, got) fail_msg += ' ({})'.format(failure) return fail_msg else: return None class FloatLike: def __init__(self, value): self.value = value def __float__(self): return self.value class IntSubclass(int): pass # Class providing an __index__ method. class MyIndexable(object): def __init__(self, value): self.value = value def __index__(self): return self.value class MathTests(unittest.TestCase): def ftest(self, name, got, expected, ulp_tol=5, abs_tol=0.0): """Compare arguments expected and got, as floats, if either is a float, using a tolerance expressed in multiples of ulp(expected) or absolutely, whichever is greater. As a convenience, when neither argument is a float, and for non-finite floats, exact equality is demanded. Also, nan==nan in this function. """ failure = result_check(expected, got, ulp_tol, abs_tol) if failure is not None: self.fail("{}: {}".format(name, failure)) def testConstants(self): # Ref: Abramowitz & Stegun (Dover, 1965) self.ftest('pi', math.pi, 3.141592653589793238462643) self.ftest('e', math.e, 2.718281828459045235360287) self.assertEqual(math.tau, 2*math.pi) def testAcos(self): self.assertRaises(TypeError, math.acos) self.ftest('acos(-1)', math.acos(-1), math.pi) self.ftest('acos(0)', math.acos(0), math.pi/2) self.ftest('acos(1)', math.acos(1), 0) self.assertRaises(ValueError, math.acos, INF) self.assertRaises(ValueError, math.acos, NINF) self.assertRaises(ValueError, math.acos, 1 + eps) self.assertRaises(ValueError, math.acos, -1 - eps) self.assertTrue(math.isnan(math.acos(NAN))) def testAcosh(self): self.assertRaises(TypeError, math.acosh) self.ftest('acosh(1)', math.acosh(1), 0) self.ftest('acosh(2)', math.acosh(2), 1.3169578969248168) self.assertRaises(ValueError, math.acosh, 0) self.assertRaises(ValueError, math.acosh, -1) self.assertEqual(math.acosh(INF), INF) self.assertRaises(ValueError, math.acosh, NINF) self.assertTrue(math.isnan(math.acosh(NAN))) def testAsin(self): self.assertRaises(TypeError, math.asin) self.ftest('asin(-1)', math.asin(-1), -math.pi/2) self.ftest('asin(0)', math.asin(0), 0) self.ftest('asin(1)', math.asin(1), math.pi/2) self.assertRaises(ValueError, math.asin, INF) self.assertRaises(ValueError, math.asin, NINF) self.assertRaises(ValueError, math.asin, 1 + eps) self.assertRaises(ValueError, math.asin, -1 - eps) self.assertTrue(math.isnan(math.asin(NAN))) def testAsinh(self): self.assertRaises(TypeError, math.asinh) self.ftest('asinh(0)', math.asinh(0), 0) self.ftest('asinh(1)', math.asinh(1), 0.88137358701954305) self.ftest('asinh(-1)', math.asinh(-1), -0.88137358701954305) self.assertEqual(math.asinh(INF), INF) self.assertEqual(math.asinh(NINF), NINF) self.assertTrue(math.isnan(math.asinh(NAN))) def testAtan(self): self.assertRaises(TypeError, math.atan) self.ftest('atan(-1)', math.atan(-1), -math.pi/4) self.ftest('atan(0)', math.atan(0), 0) self.ftest('atan(1)', math.atan(1), math.pi/4) self.ftest('atan(inf)', math.atan(INF), math.pi/2) self.ftest('atan(-inf)', math.atan(NINF), -math.pi/2) self.assertTrue(math.isnan(math.atan(NAN))) def testAtanh(self): self.assertRaises(TypeError, math.atan) self.ftest('atanh(0)', math.atanh(0), 0) self.ftest('atanh(0.5)', math.atanh(0.5), 0.54930614433405489) self.ftest('atanh(-0.5)', math.atanh(-0.5), -0.54930614433405489) self.assertRaises(ValueError, math.atanh, 1) self.assertRaises(ValueError, math.atanh, -1) self.assertRaises(ValueError, math.atanh, INF) self.assertRaises(ValueError, math.atanh, NINF) self.assertTrue(math.isnan(math.atanh(NAN))) def testAtan2(self): self.assertRaises(TypeError, math.atan2) self.ftest('atan2(-1, 0)', math.atan2(-1, 0), -math.pi/2) self.ftest('atan2(-1, 1)', math.atan2(-1, 1), -math.pi/4) self.ftest('atan2(0, 1)', math.atan2(0, 1), 0) self.ftest('atan2(1, 1)', math.atan2(1, 1), math.pi/4) self.ftest('atan2(1, 0)', math.atan2(1, 0), math.pi/2) # math.atan2(0, x) self.ftest('atan2(0., -inf)', math.atan2(0., NINF), math.pi) self.ftest('atan2(0., -2.3)', math.atan2(0., -2.3), math.pi) self.ftest('atan2(0., -0.)', math.atan2(0., -0.), math.pi) self.assertEqual(math.atan2(0., 0.), 0.) self.assertEqual(math.atan2(0., 2.3), 0.) self.assertEqual(math.atan2(0., INF), 0.) self.assertTrue(math.isnan(math.atan2(0., NAN))) # math.atan2(-0, x) self.ftest('atan2(-0., -inf)', math.atan2(-0., NINF), -math.pi) self.ftest('atan2(-0., -2.3)', math.atan2(-0., -2.3), -math.pi) self.ftest('atan2(-0., -0.)', math.atan2(-0., -0.), -math.pi) self.assertEqual(math.atan2(-0., 0.), -0.) self.assertEqual(math.atan2(-0., 2.3), -0.) self.assertEqual(math.atan2(-0., INF), -0.) self.assertTrue(math.isnan(math.atan2(-0., NAN))) # math.atan2(INF, x) self.ftest('atan2(inf, -inf)', math.atan2(INF, NINF), math.pi*3/4) self.ftest('atan2(inf, -2.3)', math.atan2(INF, -2.3), math.pi/2) self.ftest('atan2(inf, -0.)', math.atan2(INF, -0.0), math.pi/2) self.ftest('atan2(inf, 0.)', math.atan2(INF, 0.0), math.pi/2) self.ftest('atan2(inf, 2.3)', math.atan2(INF, 2.3), math.pi/2) self.ftest('atan2(inf, inf)', math.atan2(INF, INF), math.pi/4) self.assertTrue(math.isnan(math.atan2(INF, NAN))) # math.atan2(NINF, x) self.ftest('atan2(-inf, -inf)', math.atan2(NINF, NINF), -math.pi*3/4) self.ftest('atan2(-inf, -2.3)', math.atan2(NINF, -2.3), -math.pi/2) self.ftest('atan2(-inf, -0.)', math.atan2(NINF, -0.0), -math.pi/2) self.ftest('atan2(-inf, 0.)', math.atan2(NINF, 0.0), -math.pi/2) self.ftest('atan2(-inf, 2.3)', math.atan2(NINF, 2.3), -math.pi/2) self.ftest('atan2(-inf, inf)', math.atan2(NINF, INF), -math.pi/4) self.assertTrue(math.isnan(math.atan2(NINF, NAN))) # math.atan2(+finite, x) self.ftest('atan2(2.3, -inf)', math.atan2(2.3, NINF), math.pi) self.ftest('atan2(2.3, -0.)', math.atan2(2.3, -0.), math.pi/2) self.ftest('atan2(2.3, 0.)', math.atan2(2.3, 0.), math.pi/2) self.assertEqual(math.atan2(2.3, INF), 0.) self.assertTrue(math.isnan(math.atan2(2.3, NAN))) # math.atan2(-finite, x) self.ftest('atan2(-2.3, -inf)', math.atan2(-2.3, NINF), -math.pi) self.ftest('atan2(-2.3, -0.)', math.atan2(-2.3, -0.), -math.pi/2) self.ftest('atan2(-2.3, 0.)', math.atan2(-2.3, 0.), -math.pi/2) self.assertEqual(math.atan2(-2.3, INF), -0.) self.assertTrue(math.isnan(math.atan2(-2.3, NAN))) # math.atan2(NAN, x) self.assertTrue(math.isnan(math.atan2(NAN, NINF))) self.assertTrue(math.isnan(math.atan2(NAN, -2.3))) self.assertTrue(math.isnan(math.atan2(NAN, -0.))) self.assertTrue(math.isnan(math.atan2(NAN, 0.))) self.assertTrue(math.isnan(math.atan2(NAN, 2.3))) self.assertTrue(math.isnan(math.atan2(NAN, INF))) self.assertTrue(math.isnan(math.atan2(NAN, NAN))) def testCeil(self): self.assertRaises(TypeError, math.ceil) self.assertEqual(int, type(math.ceil(0.5))) self.assertEqual(math.ceil(0.5), 1) self.assertEqual(math.ceil(1.0), 1) self.assertEqual(math.ceil(1.5), 2) self.assertEqual(math.ceil(-0.5), 0) self.assertEqual(math.ceil(-1.0), -1) self.assertEqual(math.ceil(-1.5), -1) self.assertEqual(math.ceil(0.0), 0) self.assertEqual(math.ceil(-0.0), 0) #self.assertEqual(math.ceil(INF), INF) #self.assertEqual(math.ceil(NINF), NINF) #self.assertTrue(math.isnan(math.ceil(NAN))) class TestCeil: def __ceil__(self): return 42 class FloatCeil(float): def __ceil__(self): return 42 class TestNoCeil: pass self.assertEqual(math.ceil(TestCeil()), 42) self.assertEqual(math.ceil(FloatCeil()), 42) self.assertEqual(math.ceil(FloatLike(42.5)), 43) self.assertRaises(TypeError, math.ceil, TestNoCeil()) t = TestNoCeil() t.__ceil__ = lambda *args: args self.assertRaises(TypeError, math.ceil, t) self.assertRaises(TypeError, math.ceil, t, 0) @requires_IEEE_754 def testCopysign(self): self.assertEqual(math.copysign(1, 42), 1.0) self.assertEqual(math.copysign(0., 42), 0.0) self.assertEqual(math.copysign(1., -42), -1.0) self.assertEqual(math.copysign(3, 0.), 3.0) self.assertEqual(math.copysign(4., -0.), -4.0) self.assertRaises(TypeError, math.copysign) # copysign should let us distinguish signs of zeros self.assertEqual(math.copysign(1., 0.), 1.) self.assertEqual(math.copysign(1., -0.), -1.) self.assertEqual(math.copysign(INF, 0.), INF) self.assertEqual(math.copysign(INF, -0.), NINF) self.assertEqual(math.copysign(NINF, 0.), INF) self.assertEqual(math.copysign(NINF, -0.), NINF) # and of infinities self.assertEqual(math.copysign(1., INF), 1.) self.assertEqual(math.copysign(1., NINF), -1.) self.assertEqual(math.copysign(INF, INF), INF) self.assertEqual(math.copysign(INF, NINF), NINF) self.assertEqual(math.copysign(NINF, INF), INF) self.assertEqual(math.copysign(NINF, NINF), NINF) self.assertTrue(math.isnan(math.copysign(NAN, 1.))) self.assertTrue(math.isnan(math.copysign(NAN, INF))) self.assertTrue(math.isnan(math.copysign(NAN, NINF))) self.assertTrue(math.isnan(math.copysign(NAN, NAN))) # copysign(INF, NAN) may be INF or it may be NINF, since # we don't know whether the sign bit of NAN is set on any # given platform. self.assertTrue(math.isinf(math.copysign(INF, NAN))) # similarly, copysign(2., NAN) could be 2. or -2. self.assertEqual(abs(math.copysign(2., NAN)), 2.) def testCos(self): self.assertRaises(TypeError, math.cos) self.ftest('cos(-pi/2)', math.cos(-math.pi/2), 0, abs_tol=math.ulp(1)) self.ftest('cos(0)', math.cos(0), 1) self.ftest('cos(pi/2)', math.cos(math.pi/2), 0, abs_tol=math.ulp(1)) self.ftest('cos(pi)', math.cos(math.pi), -1) try: self.assertTrue(math.isnan(math.cos(INF))) self.assertTrue(math.isnan(math.cos(NINF))) except ValueError: self.assertRaises(ValueError, math.cos, INF) self.assertRaises(ValueError, math.cos, NINF) self.assertTrue(math.isnan(math.cos(NAN))) @unittest.skipIf(sys.platform == 'win32' and platform.machine() in ('ARM', 'ARM64'), "Windows UCRT is off by 2 ULP this test requires accuracy within 1 ULP") def testCosh(self): self.assertRaises(TypeError, math.cosh) self.ftest('cosh(0)', math.cosh(0), 1) self.ftest('cosh(2)-2*cosh(1)**2', math.cosh(2)-2*math.cosh(1)**2, -1) # Thanks to Lambert self.assertEqual(math.cosh(INF), INF) self.assertEqual(math.cosh(NINF), INF) self.assertTrue(math.isnan(math.cosh(NAN))) def testDegrees(self): self.assertRaises(TypeError, math.degrees) self.ftest('degrees(pi)', math.degrees(math.pi), 180.0) self.ftest('degrees(pi/2)', math.degrees(math.pi/2), 90.0) self.ftest('degrees(-pi/4)', math.degrees(-math.pi/4), -45.0) self.ftest('degrees(0)', math.degrees(0), 0) def testExp(self): self.assertRaises(TypeError, math.exp) self.ftest('exp(-1)', math.exp(-1), 1/math.e) self.ftest('exp(0)', math.exp(0), 1) self.ftest('exp(1)', math.exp(1), math.e) self.assertEqual(math.exp(INF), INF) self.assertEqual(math.exp(NINF), 0.) self.assertTrue(math.isnan(math.exp(NAN))) self.assertRaises(OverflowError, math.exp, 1000000) def testFabs(self): self.assertRaises(TypeError, math.fabs) self.ftest('fabs(-1)', math.fabs(-1), 1) self.ftest('fabs(0)', math.fabs(0), 0) self.ftest('fabs(1)', math.fabs(1), 1) def testFactorial(self): self.assertEqual(math.factorial(0), 1) total = 1 for i in range(1, 1000): total *= i self.assertEqual(math.factorial(i), total) self.assertEqual(math.factorial(i), py_factorial(i)) self.assertRaises(ValueError, math.factorial, -1) self.assertRaises(ValueError, math.factorial, -10**100) def testFactorialNonIntegers(self): with self.assertWarns(DeprecationWarning): self.assertEqual(math.factorial(5.0), 120) with self.assertWarns(DeprecationWarning): self.assertRaises(ValueError, math.factorial, 5.2) with self.assertWarns(DeprecationWarning): self.assertRaises(ValueError, math.factorial, -1.0) with self.assertWarns(DeprecationWarning): self.assertRaises(ValueError, math.factorial, -1e100) self.assertRaises(TypeError, math.factorial, decimal.Decimal('5')) self.assertRaises(TypeError, math.factorial, decimal.Decimal('5.2')) self.assertRaises(TypeError, math.factorial, "5") # Other implementations may place different upper bounds. @support.cpython_only def testFactorialHugeInputs(self): # Currently raises OverflowError for inputs that are too large # to fit into a C long. self.assertRaises(OverflowError, math.factorial, 10**100) with self.assertWarns(DeprecationWarning): self.assertRaises(OverflowError, math.factorial, 1e100) def testFloor(self): self.assertRaises(TypeError, math.floor) self.assertEqual(int, type(math.floor(0.5))) self.assertEqual(math.floor(0.5), 0) self.assertEqual(math.floor(1.0), 1) self.assertEqual(math.floor(1.5), 1) self.assertEqual(math.floor(-0.5), -1) self.assertEqual(math.floor(-1.0), -1) self.assertEqual(math.floor(-1.5), -2) #self.assertEqual(math.ceil(INF), INF) #self.assertEqual(math.ceil(NINF), NINF) #self.assertTrue(math.isnan(math.floor(NAN))) class TestFloor: def __floor__(self): return 42 class FloatFloor(float): def __floor__(self): return 42 class TestNoFloor: pass self.assertEqual(math.floor(TestFloor()), 42) self.assertEqual(math.floor(FloatFloor()), 42) self.assertEqual(math.floor(FloatLike(41.9)), 41) self.assertRaises(TypeError, math.floor, TestNoFloor()) t = TestNoFloor() t.__floor__ = lambda *args: args self.assertRaises(TypeError, math.floor, t) self.assertRaises(TypeError, math.floor, t, 0) def testFmod(self): self.assertRaises(TypeError, math.fmod) self.ftest('fmod(10, 1)', math.fmod(10, 1), 0.0) self.ftest('fmod(10, 0.5)', math.fmod(10, 0.5), 0.0) self.ftest('fmod(10, 1.5)', math.fmod(10, 1.5), 1.0) self.ftest('fmod(-10, 1)', math.fmod(-10, 1), -0.0) self.ftest('fmod(-10, 0.5)', math.fmod(-10, 0.5), -0.0) self.ftest('fmod(-10, 1.5)', math.fmod(-10, 1.5), -1.0) self.assertTrue(math.isnan(math.fmod(NAN, 1.))) self.assertTrue(math.isnan(math.fmod(1., NAN))) self.assertTrue(math.isnan(math.fmod(NAN, NAN))) self.assertRaises(ValueError, math.fmod, 1., 0.) self.assertRaises(ValueError, math.fmod, INF, 1.) self.assertRaises(ValueError, math.fmod, NINF, 1.) self.assertRaises(ValueError, math.fmod, INF, 0.) self.assertEqual(math.fmod(3.0, INF), 3.0) self.assertEqual(math.fmod(-3.0, INF), -3.0) self.assertEqual(math.fmod(3.0, NINF), 3.0) self.assertEqual(math.fmod(-3.0, NINF), -3.0) self.assertEqual(math.fmod(0.0, 3.0), 0.0) self.assertEqual(math.fmod(0.0, NINF), 0.0) def testFrexp(self): self.assertRaises(TypeError, math.frexp) def testfrexp(name, result, expected): (mant, exp), (emant, eexp) = result, expected if abs(mant-emant) > eps or exp != eexp: self.fail('%s returned %r, expected %r'%\ (name, result, expected)) testfrexp('frexp(-1)', math.frexp(-1), (-0.5, 1)) testfrexp('frexp(0)', math.frexp(0), (0, 0)) testfrexp('frexp(1)', math.frexp(1), (0.5, 1)) testfrexp('frexp(2)', math.frexp(2), (0.5, 2)) self.assertEqual(math.frexp(INF)[0], INF) self.assertEqual(math.frexp(NINF)[0], NINF) self.assertTrue(math.isnan(math.frexp(NAN)[0])) @requires_IEEE_754 @unittest.skipIf(HAVE_DOUBLE_ROUNDING, "fsum is not exact on machines with double rounding") def testFsum(self): # math.fsum relies on exact rounding for correct operation. # There's a known problem with IA32 floating-point that causes # inexact rounding in some situations, and will cause the # math.fsum tests below to fail; see issue #2937. On non IEEE # 754 platforms, and on IEEE 754 platforms that exhibit the # problem described in issue #2937, we simply skip the whole # test. # Python version of math.fsum, for comparison. Uses a # different algorithm based on frexp, ldexp and integer # arithmetic. from sys import float_info mant_dig = float_info.mant_dig etiny = float_info.min_exp - mant_dig def msum(iterable): """Full precision summation. Compute sum(iterable) without any intermediate accumulation of error. Based on the 'lsum' function at 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')), ([1e16, 1., 1e-16], 10000000000000002.0), ([1e16-2., 1.-2.**-53, -(1e16-2.), -(1.-2.**-53)], 0.0), # exercise code for resizing partials array ([2.**n - 2.**(n+50) + 2.**(n+52) for n in range(-1074, 972, 2)] + [-2.**1022], float.fromhex('0x1.5555555555555p+970')), ] # Telescoping sum, with exact differences (due to Sterbenz) terms = [1.7**i for i in range(1001)] test_values.append(( [terms[i+1] - terms[i] for i in range(1000)] + [-terms[1000]], -terms[0] )) for i, (vals, expected) in enumerate(test_values): try: actual = math.fsum(vals) except OverflowError: self.fail("test %d failed: got OverflowError, expected %r " "for math.fsum(%.100r)" % (i, expected, vals)) except ValueError: self.fail("test %d failed: got ValueError, expected %r " "for math.fsum(%.100r)" % (i, expected, vals)) self.assertEqual(actual, expected) from random import random, gauss, shuffle for j in range(1000): vals = [7, 1e100, -7, -1e100, -9e-20, 8e-20] * 10 s = 0 for i in range(200): v = gauss(0, random()) ** 7 - s s += v vals.append(v) shuffle(vals) s = msum(vals) self.assertEqual(msum(vals), math.fsum(vals)) def testGcd(self): gcd = math.gcd self.assertEqual(gcd(0, 0), 0) self.assertEqual(gcd(1, 0), 1) self.assertEqual(gcd(-1, 0), 1) self.assertEqual(gcd(0, 1), 1) self.assertEqual(gcd(0, -1), 1) self.assertEqual(gcd(7, 1), 1) self.assertEqual(gcd(7, -1), 1) self.assertEqual(gcd(-23, 15), 1) self.assertEqual(gcd(120, 84), 12) self.assertEqual(gcd(84, -120), 12) self.assertEqual(gcd(1216342683557601535506311712, 436522681849110124616458784), 32) x = 434610456570399902378880679233098819019853229470286994367836600566 y = 1064502245825115327754847244914921553977 for c in (652560, 576559230871654959816130551884856912003141446781646602790216406874): a = x * c b = y * c self.assertEqual(gcd(a, b), c) self.assertEqual(gcd(b, a), c) self.assertEqual(gcd(-a, b), c) self.assertEqual(gcd(b, -a), c) self.assertEqual(gcd(a, -b), c) self.assertEqual(gcd(-b, a), c) self.assertEqual(gcd(-a, -b), c) self.assertEqual(gcd(-b, -a), c) self.assertEqual(gcd(), 0) self.assertEqual(gcd(120), 120) self.assertEqual(gcd(-120), 120) self.assertEqual(gcd(120, 84, 102), 6) self.assertEqual(gcd(120, 1, 84), 1) self.assertRaises(TypeError, gcd, 120.0) self.assertRaises(TypeError, gcd, 120.0, 84) self.assertRaises(TypeError, gcd, 120, 84.0) self.assertRaises(TypeError, gcd, 120, 1, 84.0) self.assertEqual(gcd(MyIndexable(120), MyIndexable(84)), 12) def testHypot(self): from decimal import Decimal from fractions import Fraction hypot = math.hypot # Test different numbers of arguments (from zero to five) # against a straightforward pure python implementation args = math.e, math.pi, math.sqrt(2.0), math.gamma(3.5), math.sin(2.1) for i in range(len(args)+1): self.assertAlmostEqual( hypot(*args[:i]), math.sqrt(sum(s**2 for s in args[:i])) ) # Test allowable types (those with __float__) self.assertEqual(hypot(12.0, 5.0), 13.0) self.assertEqual(hypot(12, 5), 13) self.assertEqual(hypot(Decimal(12), Decimal(5)), 13) self.assertEqual(hypot(Fraction(12, 32), Fraction(5, 32)), Fraction(13, 32)) self.assertEqual(hypot(bool(1), bool(0), bool(1), bool(1)), math.sqrt(3)) # Test corner cases self.assertEqual(hypot(0.0, 0.0), 0.0) # Max input is zero self.assertEqual(hypot(-10.5), 10.5) # Negative input self.assertEqual(hypot(), 0.0) # Negative input self.assertEqual(1.0, math.copysign(1.0, hypot(-0.0)) # Convert negative zero to positive zero ) self.assertEqual( # Handling of moving max to the end hypot(1.5, 1.5, 0.5), hypot(1.5, 0.5, 1.5), ) # Test handling of bad arguments with self.assertRaises(TypeError): # Reject keyword args hypot(x=1) with self.assertRaises(TypeError): # Reject values without __float__ hypot(1.1, 'string', 2.2) int_too_big_for_float = 10 ** (sys.float_info.max_10_exp + 5) with self.assertRaises((ValueError, OverflowError)): hypot(1, int_too_big_for_float) # Any infinity gives positive infinity. self.assertEqual(hypot(INF), INF) self.assertEqual(hypot(0, INF), INF) self.assertEqual(hypot(10, INF), INF) self.assertEqual(hypot(-10, INF), INF) self.assertEqual(hypot(NAN, INF), INF) self.assertEqual(hypot(INF, NAN), INF) self.assertEqual(hypot(NINF, NAN), INF) self.assertEqual(hypot(NAN, NINF), INF) self.assertEqual(hypot(-INF, INF), INF) self.assertEqual(hypot(-INF, -INF), INF) self.assertEqual(hypot(10, -INF), INF) # If no infinity, any NaN gives a NaN. self.assertTrue(math.isnan(hypot(NAN))) self.assertTrue(math.isnan(hypot(0, NAN))) self.assertTrue(math.isnan(hypot(NAN, 10))) self.assertTrue(math.isnan(hypot(10, NAN))) self.assertTrue(math.isnan(hypot(NAN, NAN))) self.assertTrue(math.isnan(hypot(NAN))) # Verify scaling for extremely large values fourthmax = FLOAT_MAX / 4.0 for n in range(32): self.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 non-tuple inputs self.assertEqual(dist([1.0, 2.0, 3.0], [4.0, 2.0, -1.0]), 5.0) self.assertEqual(dist(iter([1.0, 2.0, 3.0]), iter([4.0, 2.0, -1.0])), 5.0) # Test allowable types (those with __float__) self.assertEqual(dist((14.0, 1.0), (2.0, -4.0)), 13.0) self.assertEqual(dist((14, 1), (2, -4)), 13) self.assertEqual(dist((D(14), D(1)), (D(2), D(-4))), D(13)) self.assertEqual(dist((F(14, 32), F(1, 32)), (F(2, 32), F(-4, 32))), F(13, 32)) self.assertEqual(dist((True, True, False, True, False), (True, False, True, True, False)), sqrt(2.0)) # Test corner cases self.assertEqual(dist((13.25, 12.5, -3.25), (13.25, 12.5, -3.25)), 0.0) # Distance with self is zero self.assertEqual(dist((), ()), 0.0) # Zero-dimensional case self.assertEqual(1.0, # Convert negative zero to positive zero math.copysign(1.0, dist((-0.0,), (0.0,))) ) self.assertEqual(1.0, # Convert negative zero to positive zero math.copysign(1.0, dist((0.0,), (-0.0,))) ) self.assertEqual( # Handling of moving max to the end dist((1.5, 1.5, 0.5), (0, 0, 0)), dist((1.5, 0.5, 1.5), (0, 0, 0)) ) # Verify tuple subclasses are allowed class T(tuple): pass self.assertEqual(dist(T((1, 2, 3)), ((4, 2, -1))), 5.0) # Test handling of bad arguments with self.assertRaises(TypeError): # Reject keyword args dist(p=(1, 2, 3), q=(4, 5, 6)) with self.assertRaises(TypeError): # Too few args dist((1, 2, 3)) with self.assertRaises(TypeError): # Too many args dist((1, 2, 3), (4, 5, 6), (7, 8, 9)) with self.assertRaises(TypeError): # Scalars not allowed dist(1, 2) with self.assertRaises(TypeError): # Reject values without __float__ dist((1.1, 'string', 2.2), (1, 2, 3)) with self.assertRaises(ValueError): # Check dimension agree dist((1, 2, 3, 4), (5, 6, 7)) with self.assertRaises(ValueError): # Check dimension agree dist((1, 2, 3), (4, 5, 6, 7)) with self.assertRaises(TypeError): # Rejects invalid types dist("abc", "xyz") int_too_big_for_float = 10 ** (sys.float_info.max_10_exp + 5) with self.assertRaises((ValueError, OverflowError)): dist((1, int_too_big_for_float), (2, 3)) with self.assertRaises((ValueError, OverflowError)): dist((2, 3), (1, int_too_big_for_float)) # Verify that the one dimensional case is equivalent to abs() for i in range(20): p, q = random.random(), random.random() self.assertEqual(dist((p,), (q,)), abs(p - q)) # Test special values values = [NINF, -10.5, -0.0, 0.0, 10.5, INF, NAN] for p in itertools.product(values, repeat=3): for q in itertools.product(values, repeat=3): diffs = [px - qx for px, qx in zip(p, q)] if any(map(math.isinf, diffs)): # Any infinite difference gives positive infinity. self.assertEqual(dist(p, q), INF) elif any(map(math.isnan, diffs)): # If no infinity, any NaN gives a NaN. self.assertTrue(math.isnan(dist(p, q))) # Verify scaling for extremely large values fourthmax = FLOAT_MAX / 4.0 for n in range(32): p = (fourthmax,) * n q = (0.0,) * n self.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 testIsqrt(self): # Test a variety of inputs, large and small. test_values = ( list(range(1000)) + list(range(10**6 - 1000, 10**6 + 1000)) + [2**e + i for e in range(60, 200) for i in range(-40, 40)] + [3**9999, 10**5001] ) for value in test_values: with self.subTest(value=value): s = math.isqrt(value) self.assertIs(type(s), int) self.assertLessEqual(s*s, value) self.assertLess(value, (s+1)*(s+1)) # Negative values with self.assertRaises(ValueError): math.isqrt(-1) # Integer-like things s = math.isqrt(True) self.assertIs(type(s), int) self.assertEqual(s, 1) s = math.isqrt(False) self.assertIs(type(s), int) self.assertEqual(s, 0) class IntegerLike(object): def __init__(self, value): self.value = value def __index__(self): return self.value s = math.isqrt(IntegerLike(1729)) self.assertIs(type(s), int) self.assertEqual(s, 41) with self.assertRaises(ValueError): math.isqrt(IntegerLike(-3)) # Non-integer-like things bad_values = [ 3.5, "a string", decimal.Decimal("3.5"), 3.5j, 100.0, -4.0, ] for value in bad_values: with self.subTest(value=value): with self.assertRaises(TypeError): math.isqrt(value) def test_lcm(self): lcm = math.lcm self.assertEqual(lcm(0, 0), 0) self.assertEqual(lcm(1, 0), 0) self.assertEqual(lcm(-1, 0), 0) self.assertEqual(lcm(0, 1), 0) self.assertEqual(lcm(0, -1), 0) self.assertEqual(lcm(7, 1), 7) self.assertEqual(lcm(7, -1), 7) self.assertEqual(lcm(-23, 15), 345) self.assertEqual(lcm(120, 84), 840) self.assertEqual(lcm(84, -120), 840) self.assertEqual(lcm(1216342683557601535506311712, 436522681849110124616458784), 16592536571065866494401400422922201534178938447014944) x = 43461045657039990237 y = 10645022458251153277 for c in (652560, 57655923087165495981): a = x * c b = y * c d = x * y * c self.assertEqual(lcm(a, b), d) self.assertEqual(lcm(b, a), d) self.assertEqual(lcm(-a, b), d) self.assertEqual(lcm(b, -a), d) self.assertEqual(lcm(a, -b), d) self.assertEqual(lcm(-b, a), d) self.assertEqual(lcm(-a, -b), d) self.assertEqual(lcm(-b, -a), d) self.assertEqual(lcm(), 1) self.assertEqual(lcm(120), 120) self.assertEqual(lcm(-120), 120) self.assertEqual(lcm(120, 84, 102), 14280) self.assertEqual(lcm(120, 0, 84), 0) self.assertRaises(TypeError, lcm, 120.0) self.assertRaises(TypeError, lcm, 120.0, 84) self.assertRaises(TypeError, lcm, 120, 84.0) self.assertRaises(TypeError, lcm, 120, 0, 84.0) self.assertEqual(lcm(MyIndexable(120), MyIndexable(84)), 840) 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=math.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: def __trunc__(self): return 23 class FloatTrunc(float): def __trunc__(self): return 23 class TestNoTrunc: pass self.assertEqual(math.trunc(TestTrunc()), 23) self.assertEqual(math.trunc(FloatTrunc()), 23) self.assertRaises(TypeError, math.trunc) self.assertRaises(TypeError, math.trunc, 1, 2) self.assertRaises(TypeError, math.trunc, FloatLike(23.5)) 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)) def test_prod(self): prod = math.prod self.assertEqual(prod([]), 1) self.assertEqual(prod([], start=5), 5) self.assertEqual(prod(list(range(2,8))), 5040) self.assertEqual(prod(iter(list(range(2,8)))), 5040) self.assertEqual(prod(range(1, 10), start=10), 3628800) self.assertEqual(prod([1, 2, 3, 4, 5]), 120) self.assertEqual(prod([1.0, 2.0, 3.0, 4.0, 5.0]), 120.0) self.assertEqual(prod([1, 2, 3, 4.0, 5.0]), 120.0) self.assertEqual(prod([1.0, 2.0, 3.0, 4, 5]), 120.0) # Test overflow in fast-path for integers self.assertEqual(prod([1, 1, 2**32, 1, 1]), 2**32) # Test overflow in fast-path for floats self.assertEqual(prod([1.0, 1.0, 2**32, 1, 1]), float(2**32)) self.assertRaises(TypeError, prod) self.assertRaises(TypeError, prod, 42) self.assertRaises(TypeError, prod, ['a', 'b', 'c']) self.assertRaises(TypeError, prod, ['a', 'b', 'c'], '') self.assertRaises(TypeError, prod, [b'a', b'c'], b'') values = [bytearray(b'a'), bytearray(b'b')] self.assertRaises(TypeError, prod, values, bytearray(b'')) self.assertRaises(TypeError, prod, [[1], [2], [3]]) self.assertRaises(TypeError, prod, [{2:3}]) self.assertRaises(TypeError, prod, [{2:3}]*2, {2:3}) self.assertRaises(TypeError, prod, [[1], [2], [3]], []) with self.assertRaises(TypeError): prod([10, 20], [30, 40]) # start is a keyword-only argument self.assertEqual(prod([0, 1, 2, 3]), 0) self.assertEqual(prod([1, 0, 2, 3]), 0) self.assertEqual(prod([1, 2, 3, 0]), 0) def _naive_prod(iterable, start=1): for elem in iterable: start *= elem return start # Big integers iterable = range(1, 10000) self.assertEqual(prod(iterable), _naive_prod(iterable)) iterable = range(-10000, -1) self.assertEqual(prod(iterable), _naive_prod(iterable)) iterable = range(-1000, 1000) self.assertEqual(prod(iterable), 0) # Big floats iterable = [float(x) for x in range(1, 1000)] self.assertEqual(prod(iterable), _naive_prod(iterable)) iterable = [float(x) for x in range(-1000, -1)] self.assertEqual(prod(iterable), _naive_prod(iterable)) iterable = [float(x) for x in range(-1000, 1000)] self.assertIsNaN(prod(iterable)) # Float tests self.assertIsNaN(prod([1, 2, 3, float("nan"), 2, 3])) self.assertIsNaN(prod([1, 0, float("nan"), 2, 3])) self.assertIsNaN(prod([1, float("nan"), 0, 3])) self.assertIsNaN(prod([1, float("inf"), float("nan"),3])) self.assertIsNaN(prod([1, float("-inf"), float("nan"),3])) self.assertIsNaN(prod([1, float("nan"), float("inf"),3])) self.assertIsNaN(prod([1, float("nan"), float("-inf"),3])) self.assertEqual(prod([1, 2, 3, float('inf'),-3,4]), float('-inf')) self.assertEqual(prod([1, 2, 3, float('-inf'),-3,4]), float('inf')) self.assertIsNaN(prod([1,2,0,float('inf'), -3, 4])) self.assertIsNaN(prod([1,2,0,float('-inf'), -3, 4])) self.assertIsNaN(prod([1, 2, 3, float('inf'), -3, 0, 3])) self.assertIsNaN(prod([1, 2, 3, float('-inf'), -3, 0, 2])) # Type preservation self.assertEqual(type(prod([1, 2, 3, 4, 5, 6])), int) self.assertEqual(type(prod([1, 2.0, 3, 4, 5, 6])), float) self.assertEqual(type(prod(range(1, 10000))), int) self.assertEqual(type(prod(range(1, 10000), start=1.0)), float) self.assertEqual(type(prod([1, decimal.Decimal(2.0), 3, 4, 5, 6])), decimal.Decimal) def testPerm(self): perm = math.perm factorial = math.factorial # Test if factorial definition is satisfied for n in range(100): for k in range(n + 1): self.assertEqual(perm(n, k), factorial(n) // factorial(n - k)) # Test for Pascal's identity for n in range(1, 100): for k in range(1, n): self.assertEqual(perm(n, k), perm(n - 1, k - 1) * k + perm(n - 1, k)) # Test corner cases for n in range(1, 100): self.assertEqual(perm(n, 0), 1) self.assertEqual(perm(n, 1), n) self.assertEqual(perm(n, n), factorial(n)) # Test one argument form for n in range(20): self.assertEqual(perm(n), factorial(n)) self.assertEqual(perm(n, None), factorial(n)) # Raises TypeError if any argument is non-integer or argument count is # not 1 or 2 self.assertRaises(TypeError, perm, 10, 1.0) self.assertRaises(TypeError, perm, 10, decimal.Decimal(1.0)) self.assertRaises(TypeError, perm, 10, "1") self.assertRaises(TypeError, perm, 10.0, 1) self.assertRaises(TypeError, perm, decimal.Decimal(10.0), 1) self.assertRaises(TypeError, perm, "10", 1) self.assertRaises(TypeError, perm) self.assertRaises(TypeError, perm, 10, 1, 3) self.assertRaises(TypeError, perm) # Raises Value error if not k or n are negative numbers self.assertRaises(ValueError, perm, -1, 1) self.assertRaises(ValueError, perm, -2**1000, 1) self.assertRaises(ValueError, perm, 1, -1) self.assertRaises(ValueError, perm, 1, -2**1000) # Returns zero if k is greater than n self.assertEqual(perm(1, 2), 0) self.assertEqual(perm(1, 2**1000), 0) n = 2**1000 self.assertEqual(perm(n, 0), 1) self.assertEqual(perm(n, 1), n) self.assertEqual(perm(n, 2), n * (n-1)) if support.check_impl_detail(cpython=True): self.assertRaises(OverflowError, perm, n, n) for n, k in (True, True), (True, False), (False, False): self.assertEqual(perm(n, k), 1) self.assertIs(type(perm(n, k)), int) self.assertEqual(perm(IntSubclass(5), IntSubclass(2)), 20) self.assertEqual(perm(MyIndexable(5), MyIndexable(2)), 20) for k in range(3): self.assertIs(type(perm(IntSubclass(5), IntSubclass(k))), int) self.assertIs(type(perm(MyIndexable(5), MyIndexable(k))), int) def testComb(self): comb = math.comb factorial = math.factorial # Test if factorial definition is satisfied for n in range(100): for k in range(n + 1): self.assertEqual(comb(n, k), factorial(n) // (factorial(k) * factorial(n - k))) # Test for Pascal's identity for n in range(1, 100): for k in range(1, n): self.assertEqual(comb(n, k), comb(n - 1, k - 1) + comb(n - 1, k)) # Test corner cases for n in range(100): self.assertEqual(comb(n, 0), 1) self.assertEqual(comb(n, n), 1) for n in range(1, 100): self.assertEqual(comb(n, 1), n) self.assertEqual(comb(n, n - 1), n) # Test Symmetry for n in range(100): for k in range(n // 2): self.assertEqual(comb(n, k), comb(n, n - k)) # Raises TypeError if any argument is non-integer or argument count is # not 2 self.assertRaises(TypeError, comb, 10, 1.0) self.assertRaises(TypeError, comb, 10, decimal.Decimal(1.0)) self.assertRaises(TypeError, comb, 10, "1") self.assertRaises(TypeError, comb, 10.0, 1) self.assertRaises(TypeError, comb, decimal.Decimal(10.0), 1) self.assertRaises(TypeError, comb, "10", 1) self.assertRaises(TypeError, comb, 10) self.assertRaises(TypeError, comb, 10, 1, 3) self.assertRaises(TypeError, comb) # Raises Value error if not k or n are negative numbers self.assertRaises(ValueError, comb, -1, 1) self.assertRaises(ValueError, comb, -2**1000, 1) self.assertRaises(ValueError, comb, 1, -1) self.assertRaises(ValueError, comb, 1, -2**1000) # Returns zero if k is greater than n self.assertEqual(comb(1, 2), 0) self.assertEqual(comb(1, 2**1000), 0) n = 2**1000 self.assertEqual(comb(n, 0), 1) self.assertEqual(comb(n, 1), n) self.assertEqual(comb(n, 2), n * (n-1) // 2) self.assertEqual(comb(n, n), 1) self.assertEqual(comb(n, n-1), n) self.assertEqual(comb(n, n-2), n * (n-1) // 2) if support.check_impl_detail(cpython=True): self.assertRaises(OverflowError, comb, n, n//2) for n, k in (True, True), (True, False), (False, False): self.assertEqual(comb(n, k), 1) self.assertIs(type(comb(n, k)), int) self.assertEqual(comb(IntSubclass(5), IntSubclass(2)), 10) self.assertEqual(comb(MyIndexable(5), MyIndexable(2)), 10) for k in range(3): self.assertIs(type(comb(IntSubclass(5), IntSubclass(k))), int) self.assertIs(type(comb(MyIndexable(5), MyIndexable(k))), int) @requires_IEEE_754 def test_nextafter(self): # around 2^52 and 2^63 self.assertEqual(math.nextafter(4503599627370496.0, -INF), 4503599627370495.5) self.assertEqual(math.nextafter(4503599627370496.0, INF), 4503599627370497.0) self.assertEqual(math.nextafter(9223372036854775808.0, 0.0), 9223372036854774784.0) self.assertEqual(math.nextafter(-9223372036854775808.0, 0.0), -9223372036854774784.0) # around 1.0 self.assertEqual(math.nextafter(1.0, -INF), float.fromhex('0x1.fffffffffffffp-1')) self.assertEqual(math.nextafter(1.0, INF), float.fromhex('0x1.0000000000001p+0')) # x == y: y is returned self.assertEqual(math.nextafter(2.0, 2.0), 2.0) self.assertEqualSign(math.nextafter(-0.0, +0.0), +0.0) self.assertEqualSign(math.nextafter(+0.0, -0.0), -0.0) # around 0.0 smallest_subnormal = sys.float_info.min * sys.float_info.epsilon self.assertEqual(math.nextafter(+0.0, INF), smallest_subnormal) self.assertEqual(math.nextafter(-0.0, INF), smallest_subnormal) self.assertEqual(math.nextafter(+0.0, -INF), -smallest_subnormal) self.assertEqual(math.nextafter(-0.0, -INF), -smallest_subnormal) self.assertEqualSign(math.nextafter(smallest_subnormal, +0.0), +0.0) self.assertEqualSign(math.nextafter(-smallest_subnormal, +0.0), -0.0) self.assertEqualSign(math.nextafter(smallest_subnormal, -0.0), +0.0) self.assertEqualSign(math.nextafter(-smallest_subnormal, -0.0), -0.0) # around infinity largest_normal = sys.float_info.max self.assertEqual(math.nextafter(INF, 0.0), largest_normal) self.assertEqual(math.nextafter(-INF, 0.0), -largest_normal) self.assertEqual(math.nextafter(largest_normal, INF), INF) self.assertEqual(math.nextafter(-largest_normal, -INF), -INF) # NaN self.assertIsNaN(math.nextafter(NAN, 1.0)) self.assertIsNaN(math.nextafter(1.0, NAN)) self.assertIsNaN(math.nextafter(NAN, NAN)) @requires_IEEE_754 def test_ulp(self): self.assertEqual(math.ulp(1.0), sys.float_info.epsilon) # use int ** int rather than float ** int to not rely on pow() accuracy self.assertEqual(math.ulp(2 ** 52), 1.0) self.assertEqual(math.ulp(2 ** 53), 2.0) self.assertEqual(math.ulp(2 ** 64), 4096.0) # min and max self.assertEqual(math.ulp(0.0), sys.float_info.min * sys.float_info.epsilon) self.assertEqual(math.ulp(FLOAT_MAX), FLOAT_MAX - math.nextafter(FLOAT_MAX, -INF)) # special cases self.assertEqual(math.ulp(INF), INF) self.assertIsNaN(math.ulp(math.nan)) # negative number: ulp(-x) == ulp(x) for x in (0.0, 1.0, 2 ** 52, 2 ** 64, INF): with self.subTest(x=x): self.assertEqual(math.ulp(-x), math.ulp(x)) def test_issue39871(self): # A SystemError should not be raised if the first arg to atan2(), # copysign(), or remainder() cannot be converted to a float. class F: def __float__(self): self.converted = True 1/0 for func in math.atan2, math.copysign, math.remainder: y = F() with self.assertRaises(TypeError): func("not a number", y) # There should not have been any attempt to convert the second # argument to a float. self.assertFalse(getattr(y, "converted", False)) # Custom assertions. def assertIsNaN(self, value): if not math.isnan(value): self.fail("Expected a NaN, got {!r}.".format(value)) def assertEqualSign(self, x, y): """Similar to assertEqual(), but compare also the sign with copysign(). Function useful to compare signed zeros. """ self.assertEqual(x, y) self.assertEqual(math.copysign(1.0, x), math.copysign(1.0, y)) 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()