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Issue #26040: Improve test_math and test_cmath coverage and rigour. Thanks Jeff Allen.

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This commit is contained in:
Mark Dickinson 2016-09-03 19:30:22 +01:00
parent 06cf601e4f
commit 96f774d824
3 changed files with 320 additions and 85 deletions
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261
Lib/test/test_math.py
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@ -29,6 +29,7 @@ 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
@ -36,25 +37,39 @@ def to_ulps(x):
floats.
The results from this function will only make sense on platforms
where C doubles are represented in IEEE 754 binary64 format.
where native doubles are represented in IEEE 754 binary64 format.
Note: 0.0 and -0.0 are converted to 0 and -1, respectively.
"""
n = struct.unpack('<q', struct.pack('<d', x))[0]
if n < 0:
n = ~(n+2**63)
return n
def 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."""
def ulp(x):
"""Return the value of the least significant bit of a
float x, such that the first float bigger than x is x+ulp(x).
Then, given an expected result x and a tolerance of n ulps,
the result y should be such that abs(y-x) <= n * ulp(x).
The results from this function will only make sense on platforms
where native doubles are represented in IEEE 754 binary64 format.
"""
x = abs(float(x))
if math.isnan(x) or math.isinf(x):
return x
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)
# Find next float up from x.
n = struct.unpack('<q', struct.pack('<d', x))[0]
x_next = struct.unpack('<d', struct.pack('<q', n + 1))[0]
if math.isinf(x_next):
# Corner case: x was the largest finite float. Then it's
# not an exact power of two, so we can take the difference
# between x and the previous float.
x_prev = struct.unpack('<d', struct.pack('<q', n - 1))[0]
return x - x_prev
else:
return x_next - x
# Here's a pure Python version of the math.factorial algorithm, for
# documentation and comparison purposes.
@ -106,24 +121,23 @@ def py_factorial(n):
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."""
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.
# need to special case infinities, since inf - inf gives nan
if math.isinf(expected) and got == expected:
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
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)
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
@ -150,6 +164,7 @@ def parse_mtestfile(fname):
yield (id, fn, float(arg), float(exp), flags)
def parse_testfile(fname):
"""Parse a file with test values
@ -171,8 +186,53 @@ def parse_testfile(fname):
yield (id, fn,
float(arg_real), float(arg_imag),
float(exp_real), float(exp_imag),
flags
)
flags)
def result_check(expected, got, ulp_tol=5, abs_tol=0.0):
# Common logic of MathTests.(ftest, test_testcases, test_mtestcases)
"""Compare arguments expected and got, as floats, if either
is a float, using a tolerance expressed in multiples of
ulp(expected) or absolutely (if given and greater).
As a convenience, when neither argument is a float, and for
non-finite floats, exact equality is demanded. Also, nan==nan
as far as this function is concerned.
Returns None on success and an error message on failure.
"""
# Check exactly equal (applies also to strings representing exceptions)
if got == expected:
return None
failure = "not equal"
# Turn mixed float and int comparison (e.g. floor()) to all-float
if isinstance(expected, float) and isinstance(got, int):
got = float(got)
elif isinstance(got, float) and isinstance(expected, int):
expected = float(expected)
if isinstance(expected, float) and isinstance(got, float):
if math.isnan(expected) and math.isnan(got):
# Pass, since both nan
failure = None
elif math.isinf(expected) or math.isinf(got):
# We already know they're not equal, drop through to failure
pass
else:
# Both are finite floats (now). Are they close enough?
failure = ulp_abs_check(expected, got, ulp_tol, abs_tol)
# arguments are not equal, and if numeric, are too far apart
if failure is not None:
fail_fmt = "expected {!r}, got {!r}"
fail_msg = fail_fmt.format(expected, got)
fail_msg += ' ({})'.format(failure)
return fail_msg
else:
return None
# Class providing an __index__ method.
class MyIndexable(object):
@ -184,18 +244,23 @@ class MyIndexable(object):
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 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):
self.ftest('pi', math.pi, 3.1415926)
self.ftest('e', math.e, 2.7182818)
# 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):
@ -378,9 +443,9 @@ class MathTests(unittest.TestCase):
def testCos(self):
self.assertRaises(TypeError, math.cos)
self.ftest('cos(-pi/2)', math.cos(-math.pi/2), 0)
self.ftest('cos(-pi/2)', math.cos(-math.pi/2), 0, abs_tol=ulp(1))
self.ftest('cos(0)', math.cos(0), 1)
self.ftest('cos(pi/2)', math.cos(math.pi/2), 0)
self.ftest('cos(pi/2)', math.cos(math.pi/2), 0, abs_tol=ulp(1))
self.ftest('cos(pi)', math.cos(math.pi), -1)
try:
self.assertTrue(math.isnan(math.cos(INF)))
@ -970,7 +1035,8 @@ class MathTests(unittest.TestCase):
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(1)+tanh(-1)', math.tanh(1)+math.tanh(-1), 0,
abs_tol=ulp(1))
self.ftest('tanh(inf)', math.tanh(INF), 1)
self.ftest('tanh(-inf)', math.tanh(NINF), -1)
self.assertTrue(math.isnan(math.tanh(NAN)))
@ -1084,30 +1150,48 @@ class MathTests(unittest.TestCase):
@requires_IEEE_754
def test_testfile(self):
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, or if
# flags is nonempty
if ai != 0. or ei != 0. or flags:
# 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
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 as exc:
message = (("Unexpected ValueError: %s\n " +
"in test %s:%s(%r)\n") % (exc.args[0], id, fn, ar))
self.fail(message)
except ValueError:
result = 'ValueError'
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)
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}): expected {!r}, got {!r}"
fail_fmt = "{}: {}({!r}): {}"
failures = []
for id, fn, arg, expected, flags in parse_mtestfile(math_testcases):
@ -1125,41 +1209,48 @@ class MathTests(unittest.TestCase):
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)
# Default tolerances
ulp_tol, abs_tol = 5, 0.0
else:
accuracy_failure = ulps_check(expected, got, 20)
if accuracy_failure is None:
continue
# 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
if isinstance(got, str) and isinstance(expected, str):
if got == expected:
continue
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
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)
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 ' +