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# This file is a part of Julia. License is MIT: https://julialang.org/license
# sinh, cosh, tanh, asinh, acosh, and atanh are heavily based on FDLIBM code:
# e_sinh.c, e_sinhf, e_cosh.c, e_coshf, s_tanh.c, s_tanhf.c, s_asinh.c,
# s_asinhf.c, e_acosh.c, e_coshf.c, e_atanh.c, and e_atanhf.c
# that are made available under the following licence:
# ====================================================
# Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
#
# Developed at SunSoft, a Sun Microsystems, Inc. business.
# Permission to use, copy, modify, and distribute this
# software is freely granted, provided that this notice
# is preserved.
# ====================================================
_ldexp_exp(x::Float64, i::Int32) = ccall(("__ldexp_exp", libm), Float64, (Float64, Int32), x, i)
_ldexp_exp(x::Float32, i::Int32) = ccall(("__ldexp_expf",libm), Float32, (Float32, Int32), x, i)
_ldexp_exp(x::Real, i::Int32) = _ldexp_exp(float(x), i)
# Hyperbolic functions
# sinh methods
H_SMALL_X(::Type{Float64}) = 2.0^-28
H_MEDIUM_X(::Type{Float64}) = 22.0
H_SMALL_X(::Type{Float32}) = 2f-12
H_MEDIUM_X(::Type{Float32}) = 9f0
H_LARGE_X(::Type{Float64}) = 709.7822265633563 # nextfloat(709.7822265633562)
H_OVERFLOW_X(::Type{Float64}) = 710.475860073944 # nextfloat(710.4758600739439)
H_LARGE_X(::Type{Float32}) = 88.72283f0
H_OVERFLOW_X(::Type{Float32}) = 89.415985f0
function sinh(x::T) where T <: Union{Float32, Float64}
# Method
# mathematically sinh(x) is defined to be (exp(x)-exp(-x))/2
# 1. Replace x by |x| (sinh(-x) = -sinh(x)).
# 2. Find the branch and the expression to calculate and return it
# a) 0 <= x < H_SMALL_X
# return x
# b) H_SMALL_X <= x < H_MEDIUM_X
# return sinh(x) = (E + E/(E+1))/2, where E=expm1(x)
# c) H_MEDIUM_X <= x < H_LARGE_X
# return sinh(x) = exp(x)/2
# d) H_LARGE_X <= x < H_OVERFLOW_X
# return sinh(x) = exp(x/2)/2 * exp(x/2)
# e) H_OVERFLOW_X <= x
# return sinh(x) = T(Inf)
#
# Notes:
# only sinh(0) = 0 is exact for finite x.
isnan(x) && return x
absx = abs(x)
h = T(0.5)
if x < 0
h = -h
end
# in a) or b)
if absx < H_MEDIUM_X(T)
# in a)
if absx < H_SMALL_X(T)
return x
end
t = expm1(absx)
if absx < T(1)
return h*(T(2)*t - t*t/(t + T(1)))
end
return h*(t + t/(t + T(1)))
end
# in c)
if absx < H_LARGE_X(T)
return h*exp(absx)
end
# in d)
if absx < H_OVERFLOW_X(T)
return h*T(2)*_ldexp_exp(absx, Int32(-1))
end
# in e)
return copysign(T(Inf), x)
end
sinh(x::Real) = sinh(float(x))
# cosh methods
COSH_SMALL_X(::Type{Float32}) = 0.00024414062f0
COSH_SMALL_X(::Type{Float64}) = 2.7755602085408512e-17
function cosh(x::T) where T <: Union{Float32, Float64}
# Method
# mathematically cosh(x) is defined to be (exp(x)+exp(-x))/2
# 1. Replace x by |x| (cosh(x) = cosh(-x)).
# 2. Find the branch and the expression to calculate and return it
# a) x <= COSH_SMALL_X
# return T(1)
# b) COSH_SMALL_X <= x <= ln2/2
# return 1+expm1(|x|)^2/(2*exp(|x|))
# c) ln2/2 <= x <= H_MEDIUM_X
# return (exp(|x|)+1/exp(|x|)/2
# d) H_MEDIUM_X <= x < H_LARGE_X
# return cosh(x) = exp(x)/2
# e) H_LARGE_X <= x < H_OVERFLOW_X
# return cosh(x) = exp(x/2)/2 * exp(x/2)
# f) H_OVERFLOW_X <= x
# return cosh(x) = T(Inf)
isnan(x) && return x
absx = abs(x)
h = T(0.5)
# in a) or b)
if absx < log(T(2))/2
# in a)
if absx < COSH_SMALL_X(T)
return T(1)
end
t = expm1(absx)
w = T(1) + t
return T(1) + (t*t)/(w + w)
end
# in c)
if absx < H_MEDIUM_X(T)
t = exp(absx)
return h*t + h/t
end
# in d)
if absx < H_LARGE_X(T)
return h*exp(absx)
end
# in e)
if absx < H_OVERFLOW_X(T)
return _ldexp_exp(absx, Int32(-1))
end
# in f)
return T(Inf)
end
cosh(x::Real) = cosh(float(x))
# tanh methods
TANH_LARGE_X(::Type{Float64}) = 22.0
TANH_LARGE_X(::Type{Float32}) = 9.0f0
function tanh(x::T) where T<:Union{Float32, Float64}
# Method
# mathematically tanh(x) is defined to be (exp(x)-exp(-x))/(exp(x)+exp(-x))
# 1. reduce x to non-negative by tanh(-x) = -tanh(x).
# 2. Find the branch and the expression to calculate and return it
# a) 0 <= x < H_SMALL_X
# return x
# b) H_SMALL_X <= x < 1
# -expm1(-2x)/(expm1(-2x) + 2)
# c) 1 <= x < TANH_LARGE_X
# 1 - 2/(expm1(2x) + 2)
# d) TANH_LARGE_X <= x
# return 1
if isnan(x)
return x
elseif isinf(x)
return copysign(T(1), x)
end
absx = abs(x)
if absx < TANH_LARGE_X(T)
# in a)
if absx < H_SMALL_X(T)
return x
end
if absx >= T(1)
# in c)
t = expm1(T(2)*absx)
z = T(1) - T(2)/(t + T(2))
else
# in b)
t = expm1(-T(2)*absx)
z = -t/(t + T(2))
end
else
# in d)
z = T(1)
end
return copysign(z, x)
end
tanh(x::Real) = tanh(float(x))
# Inverse hyperbolic functions
AH_LN2(::Type{Float64}) = 6.93147180559945286227e-01
AH_LN2(::Type{Float32}) = 6.9314718246f-01
# asinh methods
function asinh(x::T) where T <: Union{Float32, Float64}
# Method
# mathematically asinh(x) = sign(x)*log(|x| + sqrt(x*x + 1))
# is the principle value of the inverse hyperbolic sine
# 1. Find the branch and the expression to calculate and return it
# a) |x| < 2^-28
# return x
# b) |x| < 2
# return sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
# c) 2 <= |x| < 2^28
# return sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1)))
# d) |x| >= 2^28
# return sign(x)*(log(x)+ln2))
if isnan(x) || isinf(x)
return x
end
absx = abs(x)
if absx < T(2)
# in a)
if absx < T(2)^-28
return x
end
# in b)
t = x*x
w = log1p(absx + t/(T(1) + sqrt(T(1) + t)))
elseif absx < T(2)^28
# in c)
t = absx
w = log(T(2)*t + T(1)/(sqrt(x*x + T(1)) + t))
else
# in d)
w = log(absx) + AH_LN2(T)
end
return copysign(w, x)
end
asinh(x::Real) = asinh(float(x))
# acosh methods
@noinline acosh_domain_error(x) = throw(DomainError(x, "acosh(x) is only defined for x ≥ 1."))
function acosh(x::T) where T <: Union{Float32, Float64}
# Method
# mathematically acosh(x) if defined to be log(x + sqrt(x*x-1))
# 1. Find the branch and the expression to calculate and return it
# a) x = 1
# return log1p(t+sqrt(2.0*t+t*t)) where t=x-1.
# b) 1 < x < 2
# return log1p(t+sqrt(2.0*t+t*t)) where t=x-1.
# c) 2 <= x <
# return log(2x-1/(sqrt(x*x-1)+x))
# d) x >= 2^28
# return log(x)+ln2
# Special cases:
# if x < 1 throw DomainError
isnan(x) && return x
if x < T(1)
return acosh_domain_error(x)
elseif x == T(1)
# in a)
return T(0)
elseif x < T(2)
# in b)
t = x - T(1)
return log1p(t + sqrt(T(2)*t + t*t))
elseif x < T(2)^28
# in c)
t = x*x
return log(T(2)*x - T(1)/(x+sqrt(t - T(1))))
else
# in d)
return log(x) + AH_LN2(T)
end
end
acosh(x::Real) = acosh(float(x))
# atanh methods
@noinline atanh_domain_error(x) = throw(DomainError(x, "atanh(x) is only defined for |x| ≤ 1."))
function atanh(x::T) where T <: Union{Float32, Float64}
# Method
# 1.Reduced x to positive by atanh(-x) = -atanh(x)
# 2. Find the branch and the expression to calculate and return it
# a) 0 <= x < 2^-28
# return x
# b) 2^-28 <= x < 0.5
# return 0.5*log1p(2x+2x*x/(1-x))
# c) 0.5 <= x < 1
# return 0.5*log1p(2x/1-x)
# d) x = 1
# return Inf
# Special cases:
# if |x| > 1 throw DomainError
isnan(x) && return x
absx = abs(x)
if absx > 1
atanh_domain_error(x)
end
if absx < T(2)^-28
# in a)
return x
end
if absx < T(0.5)
# in b)
t = absx+absx
t = T(0.5)*log1p(t+t*absx/(T(1)-absx))
elseif absx < T(1)
# in c)
t = T(0.5)*log1p((absx + absx)/(T(1)-absx))
elseif absx == T(1)
# in d)
return copysign(T(Inf), x)
end
return copysign(t, x)
end
atanh(x::Real) = atanh(float(x))
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