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import sys
try:
from _operator import index
except ImportError:
pass # for tests only
def factorial(x):
"""factorial(x) -> Integral
"Find x!. Raise a ValueError if x is negative or non-integral."""
if '__index__' not in dir(x):
raise TypeError("'%s' object cannot be interpreted as an integer" % type(x).__name__)
if x > sys.maxsize:
raise OverflowError("Too large for a factorial")
if x <= 100:
if x < 0:
raise ValueError("x must be >= 0")
res = 1
for i in range(2, x + 1):
res *= i
return res
# Experimentally this gap seems good
gap = max(100, x >> 7)
def _fac_odd(low, high):
if low + gap >= high:
t = 1
for i in range(low, high, 2):
t *= i
return t
mid = ((low + high) >> 1) | 1
return _fac_odd(low, mid) * _fac_odd(mid, high)
def _fac1(x):
if x <= 2:
return 1, 1, x - 1
x2 = x >> 1
f, g, shift = _fac1(x2)
g *= _fac_odd((x2 + 1) | 1, x + 1)
return (f * g, g, shift + x2)
res, _, shift = _fac1(x)
return res << shift
def remainder(x, y):
"""Difference between x and the closest integer multiple of y.
Return x - n*y where n*y is the closest integer multiple of y.
In the case where x is exactly halfway between two multiples of
y, the nearest even value of n is used. The result is always exact."""
from math import copysign, fabs, fmod, isfinite, isinf, isnan, nan
try:
x = float(x)
except ValueError:
raise TypeError("must be real number, not %s" % (type(x).__name__, ))
y = float(y)
# Deal with most common case first.
if isfinite(x) and isfinite(y):
if y == 0.0:
# return nan
# Merging the logic from math_2 in CPython's mathmodule.c
# nan returned and x and y both not nan -> domain error
raise ValueError("math domain error")
absx = fabs(x)
absy = fabs(y)
m = fmod(absx, absy)
# Warning: some subtlety here. What we *want* to know at this point is
# whether the remainder m is less than, equal to, or greater than half
# of absy. However, we can't do that comparison directly because we
# can't be sure that 0.5*absy is representable (the mutiplication
# might incur precision loss due to underflow). So instead we compare
# m with the complement c = absy - m: m < 0.5*absy if and only if m <
# c, and so on. The catch is that absy - m might also not be
# representable, but it turns out that it doesn't matter:
# - if m > 0.5*absy then absy - m is exactly representable, by
# Sterbenz's lemma, so m > c
# - if m == 0.5*absy then again absy - m is exactly representable
# and m == c
# - if m < 0.5*absy then either (i) 0.5*absy is exactly representable,
# in which case 0.5*absy < absy - m, so 0.5*absy <= c and hence m <
# c, or (ii) absy is tiny, either subnormal or in the lowest normal
# binade. Then absy - m is exactly representable and again m < c.
c = absy - m
if m < c:
r = m
elif m > c:
r = -c
else:
# Here absx is exactly halfway between two multiples of absy,
# and we need to choose the even multiple. x now has the form
# absx = n * absy + m
# for some integer n (recalling that m = 0.5*absy at this point).
# If n is even we want to return m; if n is odd, we need to
# return -m.
# So
# 0.5 * (absx - m) = (n/2) * absy
# and now reducing modulo absy gives us:
# | m, if n is odd
# fmod(0.5 * (absx - m), absy) = |
# | 0, if n is even
# Now m - 2.0 * fmod(...) gives the desired result: m
# if n is even, -m if m is odd.
# Note that all steps in fmod(0.5 * (absx - m), absy)
# will be computed exactly, with no rounding error
# introduced.
assert m == c
r = m - 2.0 * fmod(0.5 * (absx - m), absy)
return copysign(1.0, x) * r
# Special values.
if isnan(x):
return x
if isnan(y):
return y
if isinf(x):
# return nan
# Merging the logic from math_2 in CPython's mathmodule.c
# nan returned and x and y both not nan -> domain error
raise ValueError("math domain error")
assert isinf(y)
return x
def isqrt(n):
"""
Return the integer part of the square root of the input.
"""
n = index(n)
if n < 0:
raise ValueError("isqrt() argument must be nonnegative")
if n == 0:
return 0
c = (n.bit_length() - 1) // 2
a = 1
d = 0
for s in reversed(range(c.bit_length())):
# Loop invariant: (a-1)**2 < (n >> 2*(c - d)) < (a+1)**2
e = d
d = c >> s
a = (a << d - e - 1) + (n >> 2*c - e - d + 1) // a
return a - (a*a > n)
def prod(iterable, /, *, start=1):
"""
Calculate the product of all the elements in the input iterable.
The default start value for the product is 1.
When the iterable is empty, return the start value. This function is
intended specifically for use with numeric values and may reject
non-numeric types.
"""
res = start
for x in iterable:
res = res * x
return res
def comb(n, k, /):
"""
Number of ways to choose k items from n items without repetition and without order.
Evaluates to n! / (k! * (n - k)!) when k <= n and evaluates
to zero when k > n.
Also called the binomial coefficient because it is equivalent
to the coefficient of k-th term in polynomial expansion of the
expression (1 + x)**n.
Raises TypeError if either of the arguments are not integers.
Raises ValueError if either of the arguments are negative.
"""
n = index(n)
k = index(k)
if n < 0:
raise ValueError("n must be a non-negative integer")
if k < 0:
raise ValueError("k must be a non-negative integer")
if k > n:
return 0
k = min(k, n-k)
if k == 0:
return 1
num, den = n, 1
for i in range(1, k):
num *= n - i
den *= i + 1
# reduce the fraction occasionally, but not after every multiplication
if i & 15 == 0:
num //= den
den = 1
return num // den
def perm(n, k=None, /):
"""
Number of ways to choose k items from n items without repetition and with order.
Evaluates to n! / (n - k)! when k <= n and evaluates
to zero when k > n.
If k is not specified or is None, then k defaults to n
and the function returns n!.
Raises TypeError if either of the arguments are not integers.
Raises ValueError if either of the arguments are negative.
"""
n = index(n)
if k is None:
k = n
else:
k = index(k)
if n < 0:
raise ValueError("n must be a non-negative integer")
if k < 0:
raise ValueError("k must be a non-negative integer")
if k > n:
return 0
if k <= 100:
res = 1
for x in range(n, n - k, -1):
res *= x
return res
gap = max(100, k >> 7)
def _prod_range(low, high):
if low + gap >= high:
t = 1
for i in range(low, high):
t *= i
return t
mid = (low + high) >> 1
return _prod_range(low, mid) * _prod_range(mid, high)
return _prod_range(n - k + 1, n + 1)
def lcm(*integers):
import math
if not integers:
return 1
if len(integers) == 1:
return abs(index(integers[0]))
if len(integers) == 2:
a, b = integers
a, b = index(a), index(b)
if a == 0 or b == 0:
return 0
return abs(a // math.gcd(a, b) * b)
res = index(integers[0])
if res == 0:
return res
for i in range(1, len(integers)):
v = index(integers[i])
if v == 0:
return 0
res = abs(res // math.gcd(res, v) * v)
return res
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