# 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))
