"""

# Constants, depending only on the floating-point format in use.

# We use an extra 2 bits of precision for rounding purposes.

PRECISION = sys.float_info.mant_dig + 2

SHIFT_MAX = sys.float_info.max_exp - PRECISION

Q_MAX = 1 << PRECISION

ROUND_HALF_TO_EVEN_CORRECTION = [0, -1, -2, 1, 0, -1, 2, 1]

# Reduce to the case where n is positive.

if n == 0:

return 0.0

elif n < 0:

return -int_to_float(-n)

# Convert n to a 'floating-point' number q * 2**shift, where q is an

# integer with 'PRECISION' significant bits. When shifting n to create q,

# the least significant bit of q is treated as 'sticky'. That is, the

# least significant bit of q is set if either the corresponding bit of n

# was already set, or any one of the bits of n lost in the shift was set.

shift = n.bit_length() - PRECISION

q = n << -shift if shift < 0 else (n >> shift) | bool(n & ~(-1 << shift))

# Round half to even (actually rounds to the nearest multiple of 4,

# rounding ties to a multiple of 8).

q += ROUND_HALF_TO_EVEN_CORRECTION[q & 7]

# Detect overflow.

if shift + (q == Q_MAX) > SHIFT_MAX:

raise OverflowError("integer too large to convert to float")

# Checks: q is exactly representable, and q**2**shift doesn't overflow.

assert q % 4 == 0 and q // 4 <= 2**(sys.float_info.mant_dig)

assert q * 2**shift <= sys.float_info.max

# Some circularity here, since float(q) is doing an int-to-float

# conversion. But here q is of bounded size, and is exactly representable

# as a float. In a low-level C-like language, this operation would be a

# simple cast (e.g., from unsigned long long to double).

"""Correctly-rounded true division for integers."""

negative = a^b < 0

a, b = abs(a), abs(b)

# exceptions: division by zero, overflow

if not b:

raise ZeroDivisionError("division by zero")

if a >= DBL_MIN_OVERFLOW * b:

raise OverflowError("int/int too large to represent as a float")

# find integer d satisfying 2**(d - 1) <= a/b < 2**d

d = a.bit_length() - b.bit_length()

if d >= 0 and a >= 2**d * b or d < 0 and a * 2**-d >= b:

d += 1

# compute 2**-exp * a / b for suitable exp

exp = max(d, DBL_MIN_EXP) - DBL_MANT_DIG

a, b = a << max(-exp, 0), b << max(exp, 0)

q, r = divmod(a, b)

# round-half-to-even: fractional part is r/b, which is > 0.5 iff

# 2*r > b, and == 0.5 iff 2*r == b.

if 2*r > b or 2*r == b and q % 2 == 1:

q += 1

result = math.ldexp(q, exp)

# Get quasi-random long consisting of ndigits digits (in base BASE).

# quasi == the most-significant digit will not be 0, and the number

# is constructed to contain long strings of 0 and 1 bits. These are

# more likely than random bits to provoke digit-boundary errors.

# The sign of the number is also random.

def getran(self, ndigits):

self.assertGreater(ndigits, 0)

nbits_hi = ndigits * SHIFT

nbits_lo = nbits_hi - SHIFT + 1

answer = 0

nbits = 0

r = int(random.random() * (SHIFT * 2)) | 1 # force 1 bits to start

while nbits < nbits_lo:

bits = (r >> 1) + 1

bits = min(bits, nbits_hi - nbits)

self.assertTrue(1 <= bits <= SHIFT)

nbits = nbits + bits

answer = answer << bits

if r & 1:

answer = answer | ((1 << bits) - 1)

r = int(random.random() * (SHIFT * 2))

self.assertTrue(nbits_lo <= nbits <= nbits_hi)

if random.random() < 0.5:

answer = -answer

return answer

# Get random long consisting of ndigits random digits (relative to base

# BASE). The sign bit is also random.

def getran2(ndigits):

answer = 0

for i in range(ndigits):

answer = (answer << SHIFT) | random.randint(0, MASK)

if random.random() < 0.5:

answer = -answer

return answer

def check_division(self, x, y):

eq = self.assertEqual

with self.subTest(x=x, y=y):

q, r = divmod(x, y)

q2, r2 = x//y, x%y

pab, pba = x*y, y*x

eq(pab, pba, "multiplication does not commute")

eq(q, q2, "divmod returns different quotient than /")

eq(r, r2, "divmod returns different mod than %")

eq(x, q*y + r, "x != q*y + r after divmod")

if y > 0:

self.assertTrue(0 <= r < y, "bad mod from divmod")

else:

self.assertTrue(y < r <= 0, "bad mod from divmod")

def test_division(self):

digits = list(range(1, MAXDIGITS+1)) + list(range(KARATSUBA_CUTOFF,

KARATSUBA_CUTOFF + 14))

digits.append(KARATSUBA_CUTOFF * 3)

for lenx in digits:

x = self.getran(lenx)

for leny in digits:

y = self.getran(leny) or 1

self.check_division(x, y)

# specific numbers chosen to exercise corner cases of the

# current long division implementation

# 30-bit cases involving a quotient digit estimate of BASE+1

self.check_division(1231948412290879395966702881,

1147341367131428698)

self.check_division(815427756481275430342312021515587883,

707270836069027745)

self.check_division(627976073697012820849443363563599041,

643588798496057020)

self.check_division(1115141373653752303710932756325578065,

1038556335171453937726882627)

# 30-bit cases that require the post-subtraction correction step

self.check_division(922498905405436751940989320930368494,

949985870686786135626943396)

self.check_division(768235853328091167204009652174031844,

1091555541180371554426545266)

# 15-bit cases involving a quotient digit estimate of BASE+1

self.check_division(20172188947443, 615611397)

self.check_division(1020908530270155025, 950795710)

self.check_division(128589565723112408, 736393718)

self.check_division(609919780285761575, 18613274546784)

# 15-bit cases that require the post-subtraction correction step

self.check_division(710031681576388032, 26769404391308)

self.check_division(1933622614268221, 30212853348836)

def test_karatsuba(self):

digits = list(range(1, 5)) + list(range(KARATSUBA_CUTOFF,

KARATSUBA_CUTOFF + 10))

digits.extend([KARATSUBA_CUTOFF * 10, KARATSUBA_CUTOFF * 100])

bits = [digit * SHIFT for digit in digits]

# Test products of long strings of 1 bits -- (2**x-1)*(2**y-1) ==

# 2**(x+y) - 2**x - 2**y + 1, so the proper result is easy to check.

for abits in bits:

a = (1 << abits) - 1

for bbits in bits:

if bbits < abits:

continue

with self.subTest(abits=abits, bbits=bbits):

b = (1 << bbits) - 1

x = a * b

y = ((1 << (abits + bbits)) -

(1 << abits) -

(1 << bbits) +

1)

self.assertEqual(x, y)

def check_bitop_identities_1(self, x):

eq = self.assertEqual

with self.subTest(x=x):

eq(x & 0, 0)

eq(x | 0, x)

eq(x ^ 0, x)

eq(x & -1, x)

eq(x | -1, -1)

eq(x ^ -1, ~x)

eq(x, ~~x)

eq(x & x, x)

eq(x | x, x)

eq(x ^ x, 0)

eq(x & ~x, 0)

eq(x | ~x, -1)

eq(x ^ ~x, -1)

eq(-x, 1 + ~x)

eq(-x, ~(x-1))

for n in range(2*SHIFT):

p2 = 2 ** n

with self.subTest(x=x, n=n, p2=p2):

eq(x << n >> n, x)

eq(x // p2, x >> n)

eq(x * p2, x << n)

eq(x & -p2, x >> n << n)

eq(x & -p2, x & ~(p2 - 1))

def check_bitop_identities_2(self, x, y):

eq = self.assertEqual

with self.subTest(x=x, y=y):

eq(x & y, y & x)

eq(x | y, y | x)

eq(x ^ y, y ^ x)

eq(x ^ y ^ x, y)

eq(x & y, ~(~x | ~y))

eq(x | y, ~(~x & ~y))

eq(x ^ y, (x | y) & ~(x & y))

eq(x ^ y, (x & ~y) | (~x & y))

eq(x ^ y, (x | y) & (~x | ~y))

def check_bitop_identities_3(self, x, y, z):

eq = self.assertEqual

with self.subTest(x=x, y=y, z=z):

eq((x & y) & z, x & (y & z))

eq((x | y) | z, x | (y | z))

eq((x ^ y) ^ z, x ^ (y ^ z))

eq(x & (y | z), (x & y) | (x & z))

eq(x | (y & z), (x | y) & (x | z))

def test_bitop_identities(self):

for x in special:

self.check_bitop_identities_1(x)

digits = range(1, MAXDIGITS+1)

for lenx in digits:

x = self.getran(lenx)

self.check_bitop_identities_1(x)

for leny in digits:

y = self.getran(leny)

self.check_bitop_identities_2(x, y)

self.check_bitop_identities_3(x, y, self.getran((lenx + leny)//2))

def slow_format(self, x, base):

digits = []

sign = 0

if x < 0:

sign, x = 1, -x

while x:

x, r = divmod(x, base)

digits.append(int(r))

digits.reverse()

digits = digits or [0]

return '-'[:sign] + \

{2: '0b', 8: '0o', 10: '', 16: '0x'}[base] + \

"".join("0123456789abcdef"[i] for i in digits)

def check_format_1(self, x):

for base, mapper in (2, bin), (8, oct), (10, str), (10, repr), (16, hex):

got = mapper(x)

with self.subTest(x=x, mapper=mapper.__name__):

expected = self.slow_format(x, base)

self.assertEqual(got, expected)

with self.subTest(got=got):

self.assertEqual(int(got, 0), x)

def test_format(self):

for x in special:

self.check_format_1(x)

for i in range(10):

for lenx in range(1, MAXDIGITS+1):

x = self.getran(lenx)

self.check_format_1(x)

def test_long(self):

# Check conversions from string

LL = [

('1' + '0'*20, 10**20),

('1' + '0'*100, 10**100)

]

for s, v in LL:

for sign in "", "+", "-":

for prefix in "", " ", "\t", " \t\t ":

ss = prefix + sign + s

vv = v

if sign == "-" and v is not ValueError:

vv = -v

try:

self.assertEqual(int(ss), vv)

except ValueError:

pass

# trailing L should no longer be accepted...

self.assertRaises(ValueError, int, '123L')

self.assertRaises(ValueError, int, '123l')

self.assertRaises(ValueError, int, '0L')

self.assertRaises(ValueError, int, '-37L')

self.assertRaises(ValueError, int, '0x32L', 16)

self.assertRaises(ValueError, int, '1L', 21)

# ... but it's just a normal digit if base >= 22

self.assertEqual(int('1L', 22), 43)

# tests with base 0

self.assertEqual(int('000', 0), 0)

self.assertEqual(int('0o123', 0), 83)

self.assertEqual(int('0x123', 0), 291)

self.assertEqual(int('0b100', 0), 4)

self.assertEqual(int(' 0O123 ', 0), 83)

self.assertEqual(int(' 0X123 ', 0), 291)

self.assertEqual(int(' 0B100 ', 0), 4)

self.assertEqual(int('0', 0), 0)

self.assertEqual(int('+0', 0), 0)

self.assertEqual(int('-0', 0), 0)

self.assertEqual(int('00', 0), 0)

self.assertRaises(ValueError, int, '08', 0)

self.assertRaises(ValueError, int, '-012395', 0)

# invalid bases

invalid_bases = [-909,

2**31-1, 2**31, -2**31, -2**31-1,

2**63-1, 2**63, -2**63, -2**63-1,

2**100, -2**100,

]

for base in invalid_bases:

self.assertRaises(ValueError, int, '42', base)

# Invalid unicode string

# See bpo-34087

self.assertRaises(ValueError, int, '\u3053\u3093\u306b\u3061\u306f')

def test_conversion(self):

class JustLong:

# test that __long__ no longer used in 3.x

def __long__(self):

return 42

self.assertRaises(TypeError, int, JustLong())

def check_float_conversion(self, n):

# Check that int -> float conversion behaviour matches

# that of the pure Python version above.

try:

actual = float(n)

except OverflowError:

actual = 'overflow'

try:

expected = int_to_float(n)

except OverflowError:

expected = 'overflow'

msg = ("Error in conversion of integer {} to float. "

"Got {}, expected {}.".format(n, actual, expected))

self.assertEqual(actual, expected, msg)

@support.requires_IEEE_754

def test_float_conversion(self):

exact_values = [0, 1, 2,

2**53-3,

2**53-2,

2**53-1,

2**53,

2**53+2,

2**54-4,

2**54-2,

2**54,

2**54+4]

for x in exact_values:

self.assertEqual(float(x), x)

self.assertEqual(float(-x), -x)

# test round-half-even

for x, y in [(1, 0), (2, 2), (3, 4), (4, 4), (5, 4), (6, 6), (7, 8)]:

for p in range(15):

self.assertEqual(int(float(2**p*(2**53+x))), 2**p*(2**53+y))

for x, y in [(0, 0), (1, 0), (2, 0), (3, 4), (4, 4), (5, 4), (6, 8),

(7, 8), (8, 8), (9, 8), (10, 8), (11, 12), (12, 12),

(13, 12), (14, 16), (15, 16)]:

for p in range(15):

self.assertEqual(int(float(2**p*(2**54+x))), 2**p*(2**54+y))

# behaviour near extremes of floating-point range

int_dbl_max = int(DBL_MAX)

top_power = 2**DBL_MAX_EXP

halfway = (int_dbl_max + top_power)//2

self.assertEqual(float(int_dbl_max), DBL_MAX)

self.assertEqual(float(int_dbl_max+1), DBL_MAX)

self.assertEqual(float(halfway-1), DBL_MAX)

self.assertRaises(OverflowError, float, halfway)

self.assertEqual(float(1-halfway), -DBL_MAX)

self.assertRaises(OverflowError, float, -halfway)

self.assertRaises(OverflowError, float, top_power-1)

self.assertRaises(OverflowError, float, top_power)

self.assertRaises(OverflowError, float, top_power+1)

self.assertRaises(OverflowError, float, 2*top_power-1)

self.assertRaises(OverflowError, float, 2*top_power)

self.assertRaises(OverflowError, float, top_power*top_power)

for p in range(100):

x = 2**p * (2**53 + 1) + 1

y = 2**p * (2**53 + 2)

self.assertEqual(int(float(x)), y)

x = 2**p * (2**53 + 1)

y = 2**p * 2**53

self.assertEqual(int(float(x)), y)

# Compare builtin float conversion with pure Python int_to_float

# function above.

test_values = [

int_dbl_max-1, int_dbl_max, int_dbl_max+1,

halfway-1, halfway, halfway + 1,

top_power-1, top_power, top_power+1,

2*top_power-1, 2*top_power, top_power*top_power,

]

test_values.extend(exact_values)

for p in range(-4, 8):

for x in range(-128, 128):

test_values.append(2**(p+53) + x)

for value in test_values:

self.check_float_conversion(value)

self.check_float_conversion(-value)

@support.requires_IEEE_754

@support.bigmemtest(2**32, memuse=0.2)

def test_float_conversion_huge_integer(self, size):

v = 1 << size

self.assertRaises(OverflowError, float, v)

def test_float_overflow(self):

for x in -2.0, -1.0, 0.0, 1.0, 2.0:

self.assertEqual(float(int(x)), x)

shuge = '12345' * 120

huge = 1 << 30000

mhuge = -huge

namespace = {'huge': huge, 'mhuge': mhuge, 'shuge': shuge, 'math': math}

for test in ["float(huge)", "float(mhuge)",

"complex(huge)", "complex(mhuge)",

"complex(huge, 1)", "complex(mhuge, 1)",

"complex(1, huge)", "complex(1, mhuge)",

"1. + huge", "huge + 1.", "1. + mhuge", "mhuge + 1.",

"1. - huge", "huge - 1.", "1. - mhuge", "mhuge - 1.",

"1. * huge", "huge * 1.", "1. * mhuge", "mhuge * 1.",

"1. // huge", "huge // 1.", "1. // mhuge", "mhuge // 1.",

"1. / huge", "huge / 1.", "1. / mhuge", "mhuge / 1.",

"1. ** huge", "huge ** 1.", "1. ** mhuge", "mhuge ** 1.",

"math.sin(huge)", "math.sin(mhuge)",

"math.sqrt(huge)", "math.sqrt(mhuge)", # should do better

# math.floor() of an int returns an int now

##"math.floor(huge)", "math.floor(mhuge)",

]:

self.assertRaises(OverflowError, eval, test, namespace)

# XXX Perhaps float(shuge) can raise OverflowError on some box?

# The comparison should not.

self.assertNotEqual(float(shuge), int(shuge),

"float(shuge) should not equal int(shuge)")

def test_logs(self):

LOG10E = math.log10(math.e)

for exp in list(range(10)) + [100, 1000, 10000]:

value = 10 ** exp

log10 = math.log10(value)

self.assertAlmostEqual(log10, exp)

# log10(value) == exp, so log(value) == log10(value)/log10(e) ==

# exp/LOG10E

expected = exp / LOG10E

log = math.log(value)

self.assertAlmostEqual(log, expected)

for bad in -(1 << 10000), -2, 0:

self.assertRaises(ValueError, math.log, bad)

self.assertRaises(ValueError, math.log10, bad)

def test_mixed_compares(self):

eq = self.assertEqual

# We're mostly concerned with that mixing floats and ints does the

# right stuff, even when ints are too large to fit in a float.

# The safest way to check the results is to use an entirely different

# method, which we do here via a skeletal rational class (which

# represents all Python ints and floats exactly).

class Rat:

def __init__(self, value):

if isinstance(value, int):

self.n = value

self.d = 1

elif isinstance(value, float):

# Convert to exact rational equivalent.

f, e = math.frexp(abs(value))

assert f == 0 or 0.5 <= f < 1.0

# |value| = f * 2**e exactly

# Suck up CHUNK bits at a time; 28 is enough so that we suck

# up all bits in 2 iterations for all known binary double-

# precision formats, and small enough to fit in an int.

CHUNK = 28

top = 0

# invariant: |value| = (top + f) * 2**e exactly

while f:

f = math.ldexp(f, CHUNK)

digit = int(f)

assert digit >> CHUNK == 0

top = (top << CHUNK) | digit

f -= digit

assert 0.0 <= f < 1.0

e -= CHUNK

# Now |value| = top * 2**e exactly.

if e >= 0:

n = top << e

d = 1

else:

n = top

d = 1 << -e

if value < 0:

n = -n

self.n = n

self.d = d

assert float(n) / float(d) == value

else:

raise TypeError("can't deal with %r" % value)

def _cmp__(self, other):

if not isinstance(other, Rat):

other = Rat(other)

x, y = self.n * other.d, self.d * other.n

return (x > y) - (x < y)

def __eq__(self, other):

return self._cmp__(other) == 0

def __ge__(self, other):

return self._cmp__(other) >= 0

def __gt__(self, other):

return self._cmp__(other) > 0

def __le__(self, other):

return self._cmp__(other) <= 0

def __lt__(self, other):

return self._cmp__(other) < 0

cases = [0, 0.001, 0.99, 1.0, 1.5, 1e20, 1e200]

# 2**48 is an important boundary in the internals. 2**53 is an

# important boundary for IEEE double precision.

for t in 2.0**48, 2.0**50, 2.0**53:

cases.extend([t - 1.0, t - 0.3, t, t + 0.3, t + 1.0,

int(t-1), int(t), int(t+1)])

cases.extend([0, 1, 2, sys.maxsize, float(sys.maxsize)])

# 1 << 20000 should exceed all double formats. int(1e200) is to

# check that we get equality with 1e200 above.

t = int(1e200)

cases.extend([0, 1, 2, 1 << 20000, t-1, t, t+1])

cases.extend([-x for x in cases])

for x in cases:

Rx = Rat(x)

for y in cases:

Ry = Rat(y)

Rcmp = (Rx > Ry) - (Rx < Ry)

with self.subTest(x=x, y=y, Rcmp=Rcmp):

xycmp = (x > y) - (x < y)

eq(Rcmp, xycmp)

eq(x == y, Rcmp == 0)

eq(x != y, Rcmp != 0)

eq(x < y, Rcmp < 0)

eq(x <= y, Rcmp <= 0)

eq(x > y, Rcmp > 0)

eq(x >= y, Rcmp >= 0)

@support.requires_IEEE_754

@support.bigmemtest(2**32, memuse=0.2)

def test_mixed_compares_huge_integer(self, size):

v = 1 << size

f = sys.float_info.max

self.assertIs(f == v, False)

self.assertIs(f != v, True)

self.assertIs(f < v, True)

self.assertIs(f <= v, True)

self.assertIs(f > v, False)

self.assertIs(f >= v, False)

f = float('inf')

self.assertIs(f == v, False)

self.assertIs(f != v, True)

self.assertIs(f < v, False)

self.assertIs(f <= v, False)

self.assertIs(f > v, True)

self.assertIs(f >= v, True)

f = float('nan')

self.assertIs(f == v, False)

self.assertIs(f != v, True)

self.assertIs(f < v, False)

self.assertIs(f <= v, False)

self.assertIs(f > v, False)

self.assertIs(f >= v, False)

del v

v = (-1) << size

f = -sys.float_info.max

self.assertIs(f == v, False)

self.assertIs(f != v, True)

self.assertIs(f < v, False)

self.assertIs(f <= v, False)

self.assertIs(f > v, True)

self.assertIs(f >= v, True)

f = float('-inf')

self.assertIs(f == v, False)

self.assertIs(f != v, True)

self.assertIs(f < v, True)

self.assertIs(f <= v, True)

self.assertIs(f > v, False)

self.assertIs(f >= v, False)

f = float('nan')

self.assertIs(f == v, False)

self.assertIs(f != v, True)

self.assertIs(f < v, False)

self.assertIs(f <= v, False)

self.assertIs(f > v, False)

self.assertIs(f >= v, False)

def test__format__(self):

self.assertEqual(format(123456789, 'd'), '123456789')

self.assertEqual(format(123456789, 'd'), '123456789')

self.assertEqual(format(123456789, ','), '123,456,789')

self.assertEqual(format(123456789, '_'), '123_456_789')

# sign and aligning are interdependent

self.assertEqual(format(1, "-"), '1')

self.assertEqual(format(-1, "-"), '-1')

self.assertEqual(format(1, "-3"), ' 1')

self.assertEqual(format(-1, "-3"), ' -1')

self.assertEqual(format(1, "+3"), ' +1')

self.assertEqual(format(-1, "+3"), ' -1')

self.assertEqual(format(1, " 3"), ' 1')

self.assertEqual(format(-1, " 3"), ' -1')

self.assertEqual(format(1, " "), ' 1')

self.assertEqual(format(-1, " "), '-1')

# hex

self.assertEqual(format(3, "x"), "3")

self.assertEqual(format(3, "X"), "3")

self.assertEqual(format(1234, "x"), "4d2")

self.assertEqual(format(-1234, "x"), "-4d2")

self.assertEqual(format(1234, "8x"), " 4d2")

self.assertEqual(format(-1234, "8x"), " -4d2")

self.assertEqual(format(1234, "x"), "4d2")

self.assertEqual(format(-1234, "x"), "-4d2")

self.assertEqual(format(-3, "x"), "-3")

self.assertEqual(format(-3, "X"), "-3")

self.assertEqual(format(int('be', 16), "x"), "be")

self.assertEqual(format(int('be', 16), "X"), "BE")

self.assertEqual(format(-int('be', 16), "x"), "-be")

self.assertEqual(format(-int('be', 16), "X"), "-BE")

self.assertRaises(ValueError, format, 1234567890, ',x')

self.assertEqual(format(1234567890, '_x'), '4996_02d2')

self.assertEqual(format(1234567890, '_X'), '4996_02D2')

# octal

self.assertEqual(format(3, "o"), "3")

self.assertEqual(format(-3, "o"), "-3")

self.assertEqual(format(1234, "o"), "2322")

self.assertEqual(format(-1234, "o"), "-2322")

self.assertEqual(format(1234, "-o"), "2322")

self.assertEqual(format(-1234, "-o"), "-2322")

self.assertEqual(format(1234, " o"), " 2322")

self.assertEqual(format(-1234, " o"), "-2322")

self.assertEqual(format(1234, "+o"), "+2322")

self.assertEqual(format(-1234, "+o"), "-2322")

self.assertRaises(ValueError, format, 1234567890, ',o')

self.assertEqual(format(1234567890, '_o'), '111_4540_1322')

# binary

self.assertEqual(format(3, "b"), "11")

self.assertEqual(format(-3, "b"), "-11")

self.assertEqual(format(1234, "b"), "10011010010")

self.assertEqual(format(-1234, "b"), "-10011010010")

self.assertEqual(format(1234, "-b"), "10011010010")

self.assertEqual(format(-1234, "-b"), "-10011010010")

self.assertEqual(format(1234, " b"), " 10011010010")

self.assertEqual(format(-1234, " b"), "-10011010010")

self.assertEqual(format(1234, "+b"), "+10011010010")

self.assertEqual(format(-1234, "+b"), "-10011010010")

self.assertRaises(ValueError, format, 1234567890, ',b')

self.assertEqual(format(12345, '_b'), '11_0000_0011_1001')

# make sure these are errors

self.assertRaises(ValueError, format, 3, "1.3") # precision disallowed

self.assertRaises(ValueError, format, 3, "_c") # underscore,

self.assertRaises(ValueError, format, 3, ",c") # comma, and

self.assertRaises(ValueError, format, 3, "+c") # sign not allowed

# with 'c'

self.assertRaisesRegex(ValueError, 'Cannot specify both', format, 3, '_,')

self.assertRaisesRegex(ValueError, 'Cannot specify both', format, 3, ',_')

self.assertRaisesRegex(ValueError, 'Cannot specify both', format, 3, '_,d')

self.assertRaisesRegex(ValueError, 'Cannot specify both', format, 3, ',_d')

self.assertRaisesRegex(ValueError, "Cannot specify ',' with 's'", format, 3, ',s')

self.assertRaisesRegex(ValueError, "Cannot specify '_' with 's'", format, 3, '_s')

# ensure that only int and float type specifiers work

for format_spec in ([chr(x) for x in range(ord('a'), ord('z')+1)] +

[chr(x) for x in range(ord('A'), ord('Z')+1)]):

if not format_spec in 'bcdoxXeEfFgGn%':

self.assertRaises(ValueError, format, 0, format_spec)

self.assertRaises(ValueError, format, 1, format_spec)

self.assertRaises(ValueError, format, -1, format_spec)

self.assertRaises(ValueError, format, 2**100, format_spec)

self.assertRaises(ValueError, format, -(2**100), format_spec)

# ensure that float type specifiers work; format converts

# the int to a float

for format_spec in 'eEfFgG%':

for value in [0, 1, -1, 100, -100, 1234567890, -1234567890]:

self.assertEqual(format(value, format_spec),

format(float(value), format_spec))

def test_nan_inf(self):

self.assertRaises(OverflowError, int, float('inf'))

self.assertRaises(OverflowError, int, float('-inf'))

self.assertRaises(ValueError, int, float('nan'))

def test_mod_division(self):

with self.assertRaises(ZeroDivisionError):

_ = 1 % 0

self.assertEqual(13 % 10, 3)

self.assertEqual(-13 % 10, 7)

self.assertEqual(13 % -10, -7)

self.assertEqual(-13 % -10, -3)

self.assertEqual(12 % 4, 0)

self.assertEqual(-12 % 4, 0)

self.assertEqual(12 % -4, 0)

self.assertEqual(-12 % -4, 0)

def test_true_division(self):

huge = 1 << 40000

mhuge = -huge

self.assertEqual(huge / huge, 1.0)

self.assertEqual(mhuge / mhuge, 1.0)

self.assertEqual(huge / mhuge, -1.0)

self.assertEqual(mhuge / huge, -1.0)

self.assertEqual(1 / huge, 0.0)

self.assertEqual(1 / huge, 0.0)

self.assertEqual(1 / mhuge, 0.0)

self.assertEqual(1 / mhuge, 0.0)

self.assertEqual((666 * huge + (huge >> 1)) / huge, 666.5)

self.assertEqual((666 * mhuge + (mhuge >> 1)) / mhuge, 666.5)

self.assertEqual((666 * huge + (huge >> 1)) / mhuge, -666.5)

self.assertEqual((666 * mhuge + (mhuge >> 1)) / huge, -666.5)

self.assertEqual(huge / (huge << 1), 0.5)

self.assertEqual((1000000 * huge) / huge, 1000000)

namespace = {'huge': huge, 'mhuge': mhuge}

for overflow in ["float(huge)", "float(mhuge)",

"huge / 1", "huge / 2", "huge / -1", "huge / -2",

"mhuge / 100", "mhuge / 200"]:

self.assertRaises(OverflowError, eval, overflow, namespace)

for underflow in ["1 / huge", "2 / huge", "-1 / huge", "-2 / huge",

"100 / mhuge", "200 / mhuge"]:

result = eval(underflow, namespace)

self.assertEqual(result, 0.0,

"expected underflow to 0 from %r" % underflow)

for zero in ["huge / 0", "mhuge / 0"]:

self.assertRaises(ZeroDivisionError, eval, zero, namespace)

def test_floordiv(self):

with self.assertRaises(ZeroDivisionError):

_ = 1 // 0

self.assertEqual(2 // 3, 0)

self.assertEqual(2 // -3, -1)

self.assertEqual(-2 // 3, -1)

self.assertEqual(-2 // -3, 0)

self.assertEqual(-11 // -3, 3)

self.assertEqual(-11 // 3, -4)

self.assertEqual(11 // -3, -4)

self.assertEqual(11 // 3, 3)

self.assertEqual(-12 // -3, 4)

self.assertEqual(-12 // 3, -4)

self.assertEqual(12 // -3, -4)

self.assertEqual(12 // 3, 4)

def check_truediv(self, a, b, skip_small=True):

"""Verify that the result of a/b is correctly rounded, by

# skip check for small a and b: in this case, the current

# implementation converts the arguments to float directly and

# then applies a float division. This can give doubly-rounded

# results on x87-using machines (particularly 32-bit Linux).

if skip_small and max(abs(a), abs(b)) < 2**DBL_MANT_DIG:

return

try:

# use repr so that we can distinguish between -0.0 and 0.0

expected = repr(truediv(a, b))

except OverflowError:

expected = 'overflow'

except ZeroDivisionError:

expected = 'zerodivision'

try:

got = repr(a / b)

except OverflowError:

got = 'overflow'