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from sympy.core.numbers import (Rational, pi)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.codegen.cfunctions import (
expm1, log1p, exp2, log2, fma, log10, Sqrt, Cbrt, hypot
)
from sympy.core.function import expand_log
def test_expm1():
# Eval
assert expm1(0) == 0
x = Symbol('x', real=True)
# Expand and rewrite
assert expm1(x).expand(func=True) - exp(x) == -1
assert expm1(x).rewrite('tractable') - exp(x) == -1
assert expm1(x).rewrite('exp') - exp(x) == -1
# Precision
assert not ((exp(1e-10).evalf() - 1) - 1e-10 - 5e-21) < 1e-22 # for comparison
assert abs(expm1(1e-10).evalf() - 1e-10 - 5e-21) < 1e-22
# Properties
assert expm1(x).is_real
assert expm1(x).is_finite
# Diff
assert expm1(42*x).diff(x) - 42*exp(42*x) == 0
assert expm1(42*x).diff(x) - expm1(42*x).expand(func=True).diff(x) == 0
def test_log1p():
# Eval
assert log1p(0) == 0
d = S(10)
assert expand_log(log1p(d**-1000) - log(d**1000 + 1) + log(d**1000)) == 0
x = Symbol('x', real=True)
# Expand and rewrite
assert log1p(x).expand(func=True) - log(x + 1) == 0
assert log1p(x).rewrite('tractable') - log(x + 1) == 0
assert log1p(x).rewrite('log') - log(x + 1) == 0
# Precision
assert not abs(log(1e-99 + 1).evalf() - 1e-99) < 1e-100 # for comparison
assert abs(expand_log(log1p(1e-99)).evalf() - 1e-99) < 1e-100
# Properties
assert log1p(-2**Rational(-1, 2)).is_real
assert not log1p(-1).is_finite
assert log1p(pi).is_finite
assert not log1p(x).is_positive
assert log1p(Symbol('y', positive=True)).is_positive
assert not log1p(x).is_zero
assert log1p(Symbol('z', zero=True)).is_zero
assert not log1p(x).is_nonnegative
assert log1p(Symbol('o', nonnegative=True)).is_nonnegative
# Diff
assert log1p(42*x).diff(x) - 42/(42*x + 1) == 0
assert log1p(42*x).diff(x) - log1p(42*x).expand(func=True).diff(x) == 0
def test_exp2():
# Eval
assert exp2(2) == 4
x = Symbol('x', real=True)
# Expand
assert exp2(x).expand(func=True) - 2**x == 0
# Diff
assert exp2(42*x).diff(x) - 42*exp2(42*x)*log(2) == 0
assert exp2(42*x).diff(x) - exp2(42*x).diff(x) == 0
def test_log2():
# Eval
assert log2(8) == 3
assert log2(pi) != log(pi)/log(2) # log2 should *save* (CPU) instructions
x = Symbol('x', real=True)
assert log2(x) != log(x)/log(2)
assert log2(2**x) == x
# Expand
assert log2(x).expand(func=True) - log(x)/log(2) == 0
# Diff
assert log2(42*x).diff() - 1/(log(2)*x) == 0
assert log2(42*x).diff() - log2(42*x).expand(func=True).diff(x) == 0
def test_fma():
x, y, z = symbols('x y z')
# Expand
assert fma(x, y, z).expand(func=True) - x*y - z == 0
expr = fma(17*x, 42*y, 101*z)
# Diff
assert expr.diff(x) - expr.expand(func=True).diff(x) == 0
assert expr.diff(y) - expr.expand(func=True).diff(y) == 0
assert expr.diff(z) - expr.expand(func=True).diff(z) == 0
assert expr.diff(x) - 17*42*y == 0
assert expr.diff(y) - 17*42*x == 0
assert expr.diff(z) - 101 == 0
def test_log10():
x = Symbol('x')
# Expand
assert log10(x).expand(func=True) - log(x)/log(10) == 0
# Diff
assert log10(42*x).diff(x) - 1/(log(10)*x) == 0
assert log10(42*x).diff(x) - log10(42*x).expand(func=True).diff(x) == 0
def test_Cbrt():
x = Symbol('x')
# Expand
assert Cbrt(x).expand(func=True) - x**Rational(1, 3) == 0
# Diff
assert Cbrt(42*x).diff(x) - 42*(42*x)**(Rational(1, 3) - 1)/3 == 0
assert Cbrt(42*x).diff(x) - Cbrt(42*x).expand(func=True).diff(x) == 0
def test_Sqrt():
x = Symbol('x')
# Expand
assert Sqrt(x).expand(func=True) - x**S.Half == 0
# Diff
assert Sqrt(42*x).diff(x) - 42*(42*x)**(S.Half - 1)/2 == 0
assert Sqrt(42*x).diff(x) - Sqrt(42*x).expand(func=True).diff(x) == 0
def test_hypot():
x, y = symbols('x y')
# Expand
assert hypot(x, y).expand(func=True) - (x**2 + y**2)**S.Half == 0
# Diff
assert hypot(17*x, 42*y).diff(x).expand(func=True) - hypot(17*x, 42*y).expand(func=True).diff(x) == 0
assert hypot(17*x, 42*y).diff(y).expand(func=True) - hypot(17*x, 42*y).expand(func=True).diff(y) == 0
assert hypot(17*x, 42*y).diff(x).expand(func=True) - 2*17*17*x*((17*x)**2 + (42*y)**2)**Rational(-1, 2)/2 == 0
assert hypot(17*x, 42*y).diff(y).expand(func=True) - 2*42*42*y*((17*x)**2 + (42*y)**2)**Rational(-1, 2)/2 == 0
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