def test_ff_eval_apply():
x, y = symbols('x,y')
n, k = symbols('n k', integer=True)
m = Symbol('m', integer=True, nonnegative=True)
assert ff(nan, y) is nan
assert ff(x, nan) is nan
assert unchanged(ff, x, y)
assert ff(oo, 0) == 1
assert ff(-oo, 0) == 1
assert ff(oo, 6) is oo
assert ff(-oo, 7) is -oo
assert ff(-oo, 6) is oo
assert ff(oo, -6) is oo
assert ff(-oo, -7) is oo
assert ff(x, 0) == 1
assert ff(x, 1) == x
assert ff(x, 2) == x * (x - 1)
assert ff(x, 3) == x * (x - 1) * (x - 2)
assert ff(x, 5) == x * (x - 1) * (x - 2) * (x - 3) * (x - 4)
assert ff(x, -1) == 1 / (x + 1)
assert ff(x, -2) == 1 / ((x + 1) * (x + 2))
assert ff(x, -3) == 1 / ((x + 1) * (x + 2) * (x + 3))
assert ff(100, 100) == factorial(100)
assert ff(2 * x**2 - 5 * x,
2) == (2 * x**2 - 5 * x) * (2 * x**2 - 5 * x - 1)
assert isinstance(ff(2 * x**2 - 5 * x, 2), Mul)
assert ff(x**2 + 3 * x,
-2) == 1 / ((x**2 + 3 * x + 1) * (x**2 + 3 * x + 2))
assert ff(Poly(2 * x**2 - 5 * x, x),
2) == Poly(4 * x**4 - 28 * x**3 + 59 * x**2 - 35 * x, x)
assert isinstance(ff(Poly(2 * x**2 - 5 * x, x), 2), Poly)
raises(ValueError, lambda: ff(Poly(2 * x**2 - 5 * x, x, y), 2))
assert ff(Poly(x**2 + 3 * x, x),
-2) == 1 / (x**4 + 12 * x**3 + 49 * x**2 + 78 * x + 40)
raises(ValueError, lambda: ff(Poly(x**2 + 3 * x, x, y), -2))
assert ff(x, m).is_integer is None
assert ff(n, k).is_integer is None
assert ff(n, m).is_integer is True
assert ff(n, k + pi).is_integer is False
assert ff(n, m + pi).is_integer is False
assert ff(pi, m).is_integer is False
assert isinstance(ff(x, x), ff)
assert ff(n, n) == factorial(n)
assert ff(x, k).rewrite(rf) == rf(x - k + 1, k)
assert ff(x, k).rewrite(gamma) == (-1)**k * gamma(k - x) / gamma(-x)
assert ff(n, k).rewrite(factorial) == factorial(n) / factorial(n - k)
assert ff(x, k).rewrite(binomial) == factorial(k) * binomial(x, k)
assert ff(x, y).rewrite(factorial) == ff(x, y)
assert ff(x, y).rewrite(binomial) == ff(x, y)
import random
from mpmath import ff as mpmath_ff
for i in range(100):
x = -500 + 500 * random.random()
k = -500 + 500 * random.random()
assert (abs(mpmath_ff(x, k) - ff(x, k)) < 10**(-15))
def test_ff_eval_apply():
x, y = symbols('x,y')
n, k = symbols('n k', integer=True)
m = Symbol('m', integer=True, nonnegative=True)
assert ff(nan, y) is nan
assert ff(x, nan) is nan
assert unchanged(ff, x, y)
assert ff(oo, 0) == 1
assert ff(-oo, 0) == 1
assert ff(oo, 6) is oo
assert ff(-oo, 7) is -oo
assert ff(-oo, 6) is oo
assert ff(oo, -6) is oo
assert ff(-oo, -7) is oo
assert ff(x, 0) == 1
assert ff(x, 1) == x
assert ff(x, 2) == x * (x - 1)
assert ff(x, 3) == x * (x - 1) * (x - 2)
assert ff(x, 5) == x * (x - 1) * (x - 2) * (x - 3) * (x - 4)
assert ff(x, -1) == 1 / (x + 1)
assert ff(x, -2) == 1 / ((x + 1) * (x + 2))
assert ff(x, -3) == 1 / ((x + 1) * (x + 2) * (x + 3))
assert ff(100, 100) == factorial(100)
assert ff(2 * x**2 - 5 * x,
2) == (2 * x**2 - 5 * x) * (2 * x**2 - 5 * x - 1)
assert isinstance(ff(2 * x**2 - 5 * x, 2), Mul)
assert ff(x**2 + 3 * x,
-2) == 1 / ((x**2 + 3 * x + 1) * (x**2 + 3 * x + 2))
assert ff(Poly(2 * x**2 - 5 * x, x),
2) == Poly(4 * x**4 - 28 * x**3 + 59 * x**2 - 35 * x, x)
assert isinstance(ff(Poly(2 * x**2 - 5 * x, x), 2), Poly)
raises(ValueError, lambda: ff(Poly(2 * x**2 - 5 * x, x, y), 2))
assert ff(Poly(x**2 + 3 * x, x),
-2) == 1 / (x**4 + 12 * x**3 + 49 * x**2 + 78 * x + 40)
raises(ValueError, lambda: ff(Poly(x**2 + 3 * x, x, y), -2))
assert ff(x, m).is_integer is None
assert ff(n, k).is_integer is None
assert ff(n, m).is_integer is True
assert ff(n, k + pi).is_integer is False
assert ff(n, m + pi).is_integer is False
assert ff(pi, m).is_integer is False
assert isinstance(ff(x, x), ff)
assert ff(n, n) == factorial(n)
def check(x, k, o, n):
a, b = Dummy(), Dummy()
r = lambda x, k: o(a, b).rewrite(n).subs({a: x, b: k})
for i in range(-5, 5):
for j in range(-5, 5):
assert o(i, j) == r(i, j), (o, n)
check(x, k, ff, rf)
check(x, k, ff, gamma)
check(n, k, ff, factorial)
check(x, k, ff, binomial)
check(x, y, ff, factorial)
check(x, y, ff, binomial)
assert ff(x, k).rewrite(rf) == rf(x - k + 1, k)
assert ff(x, k).rewrite(gamma) == Piecewise(
(gamma(x + 1) / gamma(-k + x + 1), x >= 0),
((-1)**k * gamma(k - x) / gamma(-x), True))
assert ff(5, k).rewrite(gamma) == 120 / gamma(6 - k)
assert ff(n, k).rewrite(factorial) == Piecewise(
(factorial(n) / factorial(-k + n), n >= 0),
((-1)**k * factorial(k - n - 1) / factorial(-n - 1), True))
assert ff(5, k).rewrite(factorial) == 120 / factorial(5 - k)
assert ff(x, k).rewrite(binomial) == factorial(k) * binomial(x, k)
assert ff(x, y).rewrite(factorial) == ff(x, y)
assert ff(x, y).rewrite(binomial) == ff(x, y)
import random
from mpmath import ff as mpmath_ff
for i in range(100):
x = -500 + 500 * random.random()
k = -500 + 500 * random.random()
a = mpmath_ff(x, k)
b = ff(x, k)
assert (abs(a - b) < abs(a) * 10**(-15))
def test_ff_eval_apply():
x, y = symbols('x,y')
n, k = symbols('n k', integer=True)
m = Symbol('m', integer=True, nonnegative=True)
assert ff(nan, y) == nan
assert ff(x, nan) == nan
assert ff(x, y) == ff(x, y)
assert ff(oo, 0) == 1
assert ff(-oo, 0) == 1
assert ff(oo, 6) == oo
assert ff(-oo, 7) == -oo
assert ff(oo, -6) == oo
assert ff(-oo, -7) == oo
assert ff(x, 0) == 1
assert ff(x, 1) == x
assert ff(x, 2) == x*(x - 1)
assert ff(x, 3) == x*(x - 1)*(x - 2)
assert ff(x, 5) == x*(x - 1)*(x - 2)*(x - 3)*(x - 4)
assert ff(x, -1) == 1/(x + 1)
assert ff(x, -2) == 1/((x + 1)*(x + 2))
assert ff(x, -3) == 1/((x + 1)*(x + 2)*(x + 3))
assert ff(100, 100) == factorial(100)
assert ff(2*x**2 - 5*x, 2) == (2*x**2 - 5*x)*(2*x**2 - 5*x - 1)
assert isinstance(ff(2*x**2 - 5*x, 2), Mul)
assert ff(x**2 + 3*x, -2) == 1/((x**2 + 3*x + 1)*(x**2 + 3*x + 2))
assert ff(Poly(2*x**2 - 5*x, x), 2) == Poly(4*x**4 - 28*x**3 + 59*x**2 - 35*x, x)
assert isinstance(ff(Poly(2*x**2 - 5*x, x), 2), Poly)
raises(ValueError, lambda: ff(Poly(2*x**2 - 5*x, x, y), 2))
assert ff(Poly(x**2 + 3*x, x), -2) == 1/(x**4 + 12*x**3 + 49*x**2 + 78*x + 40)
raises(ValueError, lambda: ff(Poly(x**2 + 3*x, x, y), -2))
assert ff(x, m).is_integer is None
assert ff(n, k).is_integer is None
assert ff(n, m).is_integer is True
assert ff(n, k + pi).is_integer is False
assert ff(n, m + pi).is_integer is False
assert ff(pi, m).is_integer is False
assert isinstance(ff(x, x), ff)
assert ff(n, n) == factorial(n)
assert ff(x, k).rewrite(rf) == rf(x - k + 1, k)
assert ff(x, k).rewrite(gamma) == (-1)**k*gamma(k - x) / gamma(-x)
assert ff(n, k).rewrite(factorial) == factorial(n) / factorial(n - k)
assert ff(x, k).rewrite(binomial) == factorial(k) * binomial(x, k)
import random
from mpmath import ff as mpmath_ff
for i in range(100):
x = -500 + 500 * random.random()
k = -500 + 500 * random.random()
assert (abs(mpmath_ff(x, k) - ff(x, k)) < 10**(-15))