def test_rf_eval_apply():
x, y = symbols('x,y')
n, k = symbols('n k', integer=True)
m = Symbol('m', integer=True, nonnegative=True)
assert rf(nan, y) == nan
assert rf(x, nan) == nan
assert rf(x, y) == rf(x, y)
assert rf(oo, 0) == 1
assert rf(-oo, 0) == 1
assert rf(oo, 6) == oo
assert rf(-oo, 7) == -oo
assert rf(oo, -6) == oo
assert rf(-oo, -7) == oo
assert rf(x, 0) == 1
assert rf(x, 1) == x
assert rf(x, 2) == x*(x + 1)
assert rf(x, 3) == x*(x + 1)*(x + 2)
assert rf(x, 5) == x*(x + 1)*(x + 2)*(x + 3)*(x + 4)
assert rf(x, -1) == 1/(x - 1)
assert rf(x, -2) == 1/((x - 1)*(x - 2))
assert rf(x, -3) == 1/((x - 1)*(x - 2)*(x - 3))
assert rf(1, 100) == factorial(100)
assert rf(x**2 + 3*x, 2) == (x**2 + 3*x)*(x**2 + 3*x + 1)
assert isinstance(rf(x**2 + 3*x, 2), Mul)
assert rf(x**3 + x, -2) == 1/((x**3 + x - 1)*(x**3 + x - 2))
assert rf(Poly(x**2 + 3*x, x), 2) == Poly(x**4 + 8*x**3 + 19*x**2 + 12*x, x)
assert isinstance(rf(Poly(x**2 + 3*x, x), 2), Poly)
raises(ValueError, lambda: rf(Poly(x**2 + 3*x, x, y), 2))
assert rf(Poly(x**3 + x, x), -2) == 1/(x**6 - 9*x**5 + 35*x**4 - 75*x**3 + 94*x**2 - 66*x + 20)
raises(ValueError, lambda: rf(Poly(x**3 + x, x, y), -2))
assert rf(x, m).is_integer is None
assert rf(n, k).is_integer is None
assert rf(n, m).is_integer is True
assert rf(n, k + pi).is_integer is False
assert rf(n, m + pi).is_integer is False
assert rf(pi, m).is_integer is False
assert rf(x, k).rewrite(ff) == ff(x + k - 1, k)
assert rf(x, k).rewrite(binomial) == factorial(k)*binomial(x + k - 1, k)
assert rf(n, k).rewrite(factorial) == \
factorial(n + k - 1) / factorial(n - 1)
import random
from mpmath import rf as mpmath_rf
for i in range(100):
x = -500 + 500 * random.random()
k = -500 + 500 * random.random()
assert (abs(mpmath_rf(x, k) - rf(x, k)) < 10**(-15))
def test_rf_eval_apply():
x, y = symbols('x,y')
n, k = symbols('n k', integer=True)
m = Symbol('m', integer=True, nonnegative=True)
assert rf(nan, y) == nan
assert rf(x, nan) == nan
assert rf(x, y) == rf(x, y)
assert rf(oo, 0) == 1
assert rf(-oo, 0) == 1
assert rf(oo, 6) == oo
assert rf(-oo, 7) == -oo
assert rf(oo, -6) == oo
assert rf(-oo, -7) == oo
assert rf(x, 0) == 1
assert rf(x, 1) == x
assert rf(x, 2) == x*(x + 1)
assert rf(x, 3) == x*(x + 1)*(x + 2)
assert rf(x, 5) == x*(x + 1)*(x + 2)*(x + 3)*(x + 4)
assert rf(x, -1) == 1/(x - 1)
assert rf(x, -2) == 1/((x - 1)*(x - 2))
assert rf(x, -3) == 1/((x - 1)*(x - 2)*(x - 3))
assert rf(1, 100) == factorial(100)
assert rf(x**2 + 3*x, 2) == (x**2 + 3*x)*(x**2 + 3*x + 1)
assert isinstance(rf(x**2 + 3*x, 2), Mul)
assert rf(x**3 + x, -2) == 1/((x**3 + x - 1)*(x**3 + x - 2))
assert rf(Poly(x**2 + 3*x, x), 2) == Poly(x**4 + 8*x**3 + 19*x**2 + 12*x, x)
assert isinstance(rf(Poly(x**2 + 3*x, x), 2), Poly)
raises(ValueError, lambda: rf(Poly(x**2 + 3*x, x, y), 2))
assert rf(Poly(x**3 + x, x), -2) == 1/(x**6 - 9*x**5 + 35*x**4 - 75*x**3 + 94*x**2 - 66*x + 20)
raises(ValueError, lambda: rf(Poly(x**3 + x, x, y), -2))
assert rf(x, m).is_integer is None
assert rf(n, k).is_integer is None
assert rf(n, m).is_integer is True
assert rf(n, k + pi).is_integer is False
assert rf(n, m + pi).is_integer is False
assert rf(pi, m).is_integer is False
assert rf(x, k).rewrite(ff) == ff(x + k - 1, k)
assert rf(x, k).rewrite(binomial) == factorial(k)*binomial(x + k - 1, k)
assert rf(n, k).rewrite(factorial) == \
factorial(n + k - 1) / factorial(n - 1)
import random
from mpmath import rf as mpmath_rf
for i in range(100):
x = -500 + 500 * random.random()
k = -500 + 500 * random.random()
assert (abs(mpmath_rf(x, k) - rf(x, k)) < 10**(-15))
def test_rf_eval_apply():
x, y = symbols('x,y')
n, k = symbols('n k', integer=True)
m = Symbol('m', integer=True, nonnegative=True)
assert rf(nan, y) is nan
assert rf(x, nan) is nan
assert unchanged(rf, x, y)
assert rf(oo, 0) == 1
assert rf(-oo, 0) == 1
assert rf(oo, 6) is oo
assert rf(-oo, 7) is -oo
assert rf(-oo, 6) is oo
assert rf(oo, -6) is oo
assert rf(-oo, -7) is oo
assert rf(-1, pi) == 0
assert rf(-5, 1 + I) == 0
assert unchanged(rf, -3, k)
assert unchanged(rf, x, Symbol('k', integer=False))
assert rf(-3, Symbol('k', integer=False)) == 0
assert rf(Symbol('x', negative=True, integer=True),
Symbol('k', integer=False)) == 0
assert rf(x, 0) == 1
assert rf(x, 1) == x
assert rf(x, 2) == x * (x + 1)
assert rf(x, 3) == x * (x + 1) * (x + 2)
assert rf(x, 5) == x * (x + 1) * (x + 2) * (x + 3) * (x + 4)
assert rf(x, -1) == 1 / (x - 1)
assert rf(x, -2) == 1 / ((x - 1) * (x - 2))
assert rf(x, -3) == 1 / ((x - 1) * (x - 2) * (x - 3))
assert rf(1, 100) == factorial(100)
assert rf(x**2 + 3 * x, 2) == (x**2 + 3 * x) * (x**2 + 3 * x + 1)
assert isinstance(rf(x**2 + 3 * x, 2), Mul)
assert rf(x**3 + x, -2) == 1 / ((x**3 + x - 1) * (x**3 + x - 2))
assert rf(Poly(x**2 + 3 * x, x),
2) == Poly(x**4 + 8 * x**3 + 19 * x**2 + 12 * x, x)
assert isinstance(rf(Poly(x**2 + 3 * x, x), 2), Poly)
raises(ValueError, lambda: rf(Poly(x**2 + 3 * x, x, y), 2))
assert rf(Poly(x**3 + x, x),
-2) == 1 / (x**6 - 9 * x**5 + 35 * x**4 - 75 * x**3 + 94 * x**2 -
66 * x + 20)
raises(ValueError, lambda: rf(Poly(x**3 + x, x, y), -2))
assert rf(x, m).is_integer is None
assert rf(n, k).is_integer is None
assert rf(n, m).is_integer is True
assert rf(n, k + pi).is_integer is False
assert rf(n, m + pi).is_integer is False
assert rf(pi, m).is_integer is False
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, i, j)
check(x, k, rf, ff)
check(x, k, rf, binomial)
check(n, k, rf, factorial)
check(x, y, rf, factorial)
check(x, y, rf, binomial)
assert rf(x, k).rewrite(ff) == ff(x + k - 1, k)
assert rf(x, k).rewrite(gamma) == Piecewise(
(gamma(k + x) / gamma(x), x > 0),
((-1)**k * gamma(1 - x) / gamma(-k - x + 1), True))
assert rf(5, k).rewrite(gamma) == gamma(k + 5) / 24
assert rf(x, k).rewrite(binomial) == factorial(k) * binomial(x + k - 1, k)
assert rf(n, k).rewrite(factorial) == Piecewise(
(factorial(k + n - 1) / factorial(n - 1), n > 0),
((-1)**k * factorial(-n) / factorial(-k - n), True))
assert rf(5, k).rewrite(factorial) == factorial(k + 4) / 24
assert rf(x, y).rewrite(factorial) == rf(x, y)
assert rf(x, y).rewrite(binomial) == rf(x, y)
import random
from mpmath import rf as mpmath_rf
for i in range(100):
x = -500 + 500 * random.random()
k = -500 + 500 * random.random()
assert (abs(mpmath_rf(x, k) - rf(x, k)) < 10**(-15))