math.jl

# This file is a part of Julia. License is MIT: http://julialang.org/license

module Math

export sin, cos, tan, sinh, cosh, tanh, asin, acos, atan,
       asinh, acosh, atanh, sec, csc, cot, asec, acsc, acot,
       sech, csch, coth, asech, acsch, acoth,
       sinpi, cospi, sinc, cosc,
       cosd, cotd, cscd, secd, sind, tand,
       acosd, acotd, acscd, asecd, asind, atand, atan2,
       rad2deg, deg2rad,
       log, log2, log10, log1p, exponent, exp, exp2, exp10, expm1,
       cbrt, sqrt, erf, erfc, erfcx, erfi, dawson,
       significand,
       lgamma, hypot, gamma, lfact, max, min, minmax, ldexp, frexp,
       clamp, clamp!, modf, ^, mod2pi,
       airy, airyai, airyprime, airyaiprime, airybi, airybiprime, airyx,
       besselj0, besselj1, besselj, besseljx,
       bessely0, bessely1, bessely, besselyx,
       hankelh1, hankelh2, hankelh1x, hankelh2x,
       besseli, besselix, besselk, besselkx, besselh, besselhx,
       beta, lbeta, eta, zeta, polygamma, invdigamma, digamma, trigamma,
       erfinv, erfcinv, @evalpoly

import Base: log, exp, sin, cos, tan, sinh, cosh, tanh, asin,
             acos, atan, asinh, acosh, atanh, sqrt, log2, log10,
             max, min, minmax, ^, exp2, muladd,
             exp10, expm1, log1p,
             sign_mask, exponent_mask, exponent_one, exponent_half,
             significand_mask, significand_bits, exponent_bits, exponent_bias


import Core.Intrinsics: sqrt_llvm, box, unbox, powi_llvm

# non-type specific math functions

"""
    clamp(x, lo, hi)

Return `x` if `lo <= x <= hi`. If `x < lo`, return `lo`. If `x > hi`, return `hi`. Arguments
are promoted to a common type. Operates elementwise over `x` if `x` is an array.

```jldoctest
julia> clamp([pi, 1.0, big(10.)], 2., 9.)
3-element Array{BigFloat,1}:
 3.141592653589793238462643383279502884197169399375105820974944592307816406286198
 2.000000000000000000000000000000000000000000000000000000000000000000000000000000
 9.000000000000000000000000000000000000000000000000000000000000000000000000000000
```
"""
clamp{X,L,H}(x::X, lo::L, hi::H) =
    ifelse(x > hi, convert(promote_type(X,L,H), hi),
           ifelse(x < lo,
                  convert(promote_type(X,L,H), lo),
                  convert(promote_type(X,L,H), x)))

clamp{T}(x::AbstractArray{T,1}, lo, hi) = [clamp(xx, lo, hi) for xx in x]
clamp{T}(x::AbstractArray{T,2}, lo, hi) =
    [clamp(x[i,j], lo, hi) for i in indices(x,1), j in indices(x,2)]

clamp{T}(x::AbstractArray{T}, lo, hi) =
    reshape([clamp(xx, lo, hi) for xx in x], size(x))

"""
    clamp!(array::AbstractArray, lo, hi)

Restrict values in `array` to the specified range, in-place.
See also [`clamp`](:func:`clamp`).
"""
function clamp!{T}(x::AbstractArray{T}, lo, hi)
    @inbounds for i in eachindex(x)
        x[i] = clamp(x[i], lo, hi)
    end
    x
end

# evaluate p[1] + x * (p[2] + x * (....)), i.e. a polynomial via Horner's rule
macro horner(x, p...)
    ex = esc(p[end])
    for i = length(p)-1:-1:1
        ex = :(muladd(t, $ex, $(esc(p[i]))))
    end
    Expr(:block, :(t = $(esc(x))), ex)
end

# Evaluate p[1] + z*p[2] + z^2*p[3] + ... + z^(n-1)*p[n].  This uses
# Horner's method if z is real, but for complex z it uses a more
# efficient algorithm described in Knuth, TAOCP vol. 2, section 4.6.4,
# equation (3).
macro evalpoly(z, p...)
    a = :($(esc(p[end])))
    b = :($(esc(p[end-1])))
    as = []
    for i = length(p)-2:-1:1
        ai = Symbol("a", i)
        push!(as, :($ai = $a))
        a = :(muladd(r, $ai, $b))
        b = :($(esc(p[i])) - s * $ai) # see issue #15985 on fused mul-subtract
    end
    ai = :a0
    push!(as, :($ai = $a))
    C = Expr(:block,
             :(x = real(tt)),
             :(y = imag(tt)),
             :(r = x + x),
             :(s = muladd(x, x, y*y)),
             as...,
             :(muladd($ai, tt, $b)))
    R = Expr(:macrocall, Symbol("@horner"), :tt, map(esc, p)...)
    :(let tt = $(esc(z))
          isa(tt, Complex) ? $C : $R
      end)
end

rad2deg(z::AbstractFloat) = z * (180 / oftype(z, pi))
deg2rad(z::AbstractFloat) = z * (oftype(z, pi) / 180)
rad2deg(z::Real) = rad2deg(float(z))
deg2rad(z::Real) = deg2rad(float(z))
@vectorize_1arg Real rad2deg
@vectorize_1arg Real deg2rad

log{T<:Number}(b::T, x::T) = log(x)/log(b)
log(b::Number, x::Number) = log(promote(b,x)...)
@vectorize_2arg Number log

# type specific math functions

const libm = Base.libm_name
const openspecfun = "libopenspecfun"

# functions with no domain error
for f in (:cbrt, :sinh, :cosh, :tanh, :atan, :asinh, :exp, :erf, :erfc, :exp2, :expm1)
    @eval begin
        ($f)(x::Float64) = ccall(($(string(f)),libm), Float64, (Float64,), x)
        ($f)(x::Float32) = ccall(($(string(f,"f")),libm), Float32, (Float32,), x)
        ($f)(x::Real) = ($f)(float(x))
        @vectorize_1arg Number $f
    end
end

# fallback definitions to prevent infinite loop from $f(x::Real) def above
cbrt(x::AbstractFloat) = x^(1//3)
exp2(x::AbstractFloat) = 2^x
for f in (:sinh, :cosh, :tanh, :atan, :asinh, :exp, :erf, :erfc, :expm1)
    @eval ($f)(x::AbstractFloat) = error("not implemented for ", typeof(x))
end

# functions with special cases for integer arguments
@inline function exp2(x::Base.BitInteger)
    if x > 1023
        Inf64
    elseif x <= -1023
        # if -1073 < x <= -1023 then Result will be a subnormal number
        # Hex literal with padding must be used to work on 32bit machine
        reinterpret(Float64, 0x0000_0000_0000_0001 << ((x + 1074)) % UInt)
    else
        # We will cast everything to Int64 to avoid errors in case of Int128
        # If x is a Int128, and is outside the range of Int64, then it is not -1023<x<=1023
        reinterpret(Float64, (exponent_bias(Float64) + (x % Int64)) << (significand_bits(Float64)) % UInt)
    end
end

# TODO: GNU libc has exp10 as an extension; should openlibm?
exp10(x::Float64) = 10.0^x
exp10(x::Float32) = 10.0f0^x
exp10(x::Integer) = exp10(float(x))
@vectorize_1arg Number exp10

# utility for converting NaN return to DomainError
@inline nan_dom_err(f, x) = isnan(f) & !isnan(x) ? throw(DomainError()) : f

# functions that return NaN on non-NaN argument for domain error
for f in (:sin, :cos, :tan, :asin, :acos, :acosh, :atanh, :log, :log2, :log10,
          :lgamma, :log1p)
    @eval begin
        ($f)(x::Float64) = nan_dom_err(ccall(($(string(f)),libm), Float64, (Float64,), x), x)
        ($f)(x::Float32) = nan_dom_err(ccall(($(string(f,"f")),libm), Float32, (Float32,), x), x)
        ($f)(x::Real) = ($f)(float(x))
        @vectorize_1arg Number $f
    end
end

sqrt(x::Float64) = box(Float64,sqrt_llvm(unbox(Float64,x)))
sqrt(x::Float32) = box(Float32,sqrt_llvm(unbox(Float32,x)))
sqrt(x::Real) = sqrt(float(x))
@vectorize_1arg Number sqrt

"""
    hypot(x, y)

Compute the hypotenuse ``\\sqrt{x^2+y^2}`` avoiding overflow and underflow.
"""
hypot(x::Number, y::Number) = hypot(promote(x, y)...)
function hypot{T<:Number}(x::T, y::T)
    ax = abs(x)
    ay = abs(y)
    if ax < ay
        ax, ay = ay, ax
    end
    if ax == 0
        r = ay / one(ax)
    else
        r = ay / ax
    end

    rr = ax * sqrt(1 + r * r)

    # Use type of rr to make sure that return type is the same for
    # all branches
    if isnan(r)
        isinf(ax) && return oftype(rr, Inf)
        isinf(ay) && return oftype(rr, Inf)
        return oftype(rr, r)
    else
        return rr
    end
end
@vectorize_2arg Number hypot

"""
    hypot(x...)

Compute the hypotenuse ``\\sqrt{\\sum x_i^2}`` avoiding overflow and underflow.
"""
hypot(x::Number...) = vecnorm(x)

atan2(y::Real, x::Real) = atan2(promote(float(y),float(x))...)
atan2{T<:AbstractFloat}(y::T, x::T) = Base.no_op_err("atan2", T)

atan2(y::Float64, x::Float64) = ccall((:atan2,libm), Float64, (Float64, Float64,), y, x)
atan2(y::Float32, x::Float32) = ccall((:atan2f,libm), Float32, (Float32, Float32), y, x)
@vectorize_2arg Number atan2

max{T<:AbstractFloat}(x::T, y::T) = ifelse((y > x) | (signbit(y) < signbit(x)),
                                    ifelse(isnan(y), x, y), ifelse(isnan(x), y, x))

@vectorize_2arg Real max

min{T<:AbstractFloat}(x::T, y::T) = ifelse((y < x) | (signbit(y) > signbit(x)),
                                    ifelse(isnan(y), x, y), ifelse(isnan(x), y, x))
@vectorize_2arg Real min

minmax{T<:AbstractFloat}(x::T, y::T) = ifelse(isnan(x-y), ifelse(isnan(x), (y, y), (x, x)),
                                       ifelse((y < x) | (signbit(y) > signbit(x)), (y, x),
                                       ifelse((y > x) | (signbit(y) < signbit(x)), (x, y),
                                       ifelse(x == x, (x, x), (y, y)))))

ldexp(x::Float64,e::Integer) = ccall((:scalbn,libm),  Float64, (Float64,Int32), x, Int32(e))
ldexp(x::Float32,e::Integer) = ccall((:scalbnf,libm), Float32, (Float32,Int32), x, Int32(e))
# TODO: vectorize ldexp

function exponent{T<:AbstractFloat}(x::T)
    xu = reinterpret(Unsigned,x)
    xe = xu & exponent_mask(T)
    k = Int(xe >> significand_bits(T))
    if xe == 0 # x is subnormal
        x == 0 && throw(DomainError())
        xu &= significand_mask(T)
        m = leading_zeros(xu)-exponent_bits(T)
        k = 1-m
    elseif xe == exponent_mask(T) # NaN or Inf
        throw(DomainError())
    end
    k - exponent_bias(T)
end
@vectorize_1arg Real exponent

function significand{T<:AbstractFloat}(x::T)
    xu = reinterpret(Unsigned,x)
    xe = xu & exponent_mask(T)
    if xe == 0 # x is subnormal
        x == 0 && return x
        xs = xu & sign_mask(T)
        xu $= xs
        m = leading_zeros(xu)-exponent_bits(T)
        xu <<= m
        xu $= xs
    elseif xe == exponent_mask(T) # NaN or Inf
        return x
    end
    xu = (xu & ~exponent_mask(T)) | exponent_one(T)
    reinterpret(T,xu)
end
@vectorize_1arg Real significand

function frexp{T<:AbstractFloat}(x::T)
    xu = reinterpret(Unsigned,x)
    xe = xu & exponent_mask(T)
    k = Int(xe >> significand_bits(T))
    if xe == 0 # x is subnormal
        x == 0 && return x, 0
        xs = xu & sign_mask(T)
        xu $= xs
        m = leading_zeros(xu)-exponent_bits(T)
        xu <<= m
        xu $= xs
        k = 1-m
    elseif xe == exponent_mask(T) # NaN or Inf
        return x,0
    end
    k -= (exponent_bias(T)-1)
    xu = (xu & ~exponent_mask(T)) | exponent_half(T)
    reinterpret(T,xu), k
end

function frexp{T<:AbstractFloat}(A::Array{T})
    F = similar(A)
    E = Array{Int}(size(A))
    for (iF, iE, iA) in zip(eachindex(F), eachindex(E), eachindex(A))
        F[iF], E[iE] = frexp(A[iA])
    end
    return (F, E)
end

modf(x) = rem(x,one(x)), trunc(x)

const _modff_temp = Ref{Float32}()
function modf(x::Float32)
    f = ccall((:modff,libm), Float32, (Float32,Ptr{Float32}), x, _modff_temp)
    f, _modff_temp[]
end

const _modf_temp = Ref{Float64}()
function modf(x::Float64)
    f = ccall((:modf,libm), Float64, (Float64,Ptr{Float64}), x, _modf_temp)
    f, _modf_temp[]
end

^(x::Float64, y::Float64) = nan_dom_err(ccall((:pow,libm),  Float64, (Float64,Float64), x, y), x+y)
^(x::Float32, y::Float32) = nan_dom_err(ccall((:powf,libm), Float32, (Float32,Float32), x, y), x+y)

^(x::Float64, y::Integer) =
    box(Float64, powi_llvm(unbox(Float64,x), unbox(Int32,Int32(y))))
^(x::Float32, y::Integer) =
    box(Float32, powi_llvm(unbox(Float32,x), unbox(Int32,Int32(y))))

function angle_restrict_symm(theta)
    const P1 = 4 * 7.8539812564849853515625e-01
    const P2 = 4 * 3.7748947079307981766760e-08
    const P3 = 4 * 2.6951514290790594840552e-15

    y = 2*floor(theta/(2*pi))
    r = ((theta - y*P1) - y*P2) - y*P3
    if (r > pi)
        r -= (2*pi)
    end
    return r
end

## mod2pi-related calculations ##

function add22condh(xh::Float64, xl::Float64, yh::Float64, yl::Float64)
    # as above, but only compute and return high double
    r = xh+yh
    s = (abs(xh) > abs(yh)) ? (xh-r+yh+yl+xl) : (yh-r+xh+xl+yl)
    zh = r+s
    return zh
end

function ieee754_rem_pio2(x::Float64)
    # rem_pio2 essentially computes x mod pi/2 (ie within a quarter circle)
    # and returns the result as
    # y between + and - pi/4 (for maximal accuracy (as the sign bit is exploited)), and
    # n, where n specifies the integer part of the division, or, at any rate,
    # in which quadrant we are.
    # The invariant fulfilled by the returned values seems to be
    #  x = y + n*pi/2 (where y = y1+y2 is a double-double and y2 is the "tail" of y).
    # Note: for very large x (thus n), the invariant might hold only modulo 2pi
    # (in other words, n might be off by a multiple of 4, or a multiple of 100)

    # this is just wrapping up
    # https://github.com/JuliaLang/openspecfun/blob/master/rem_pio2/e_rem_pio2.c

    y = [0.0,0.0]
    n = ccall((:__ieee754_rem_pio2, openspecfun), Cint, (Float64,Ptr{Float64}), x, y)
    return (n,y)
end

# multiples of pi/2, as double-double (ie with "tail")
const pi1o2_h  = 1.5707963267948966     # convert(Float64, pi * BigFloat(1/2))
const pi1o2_l  = 6.123233995736766e-17  # convert(Float64, pi * BigFloat(1/2) - pi1o2_h)

const pi2o2_h  = 3.141592653589793      # convert(Float64, pi * BigFloat(1))
const pi2o2_l  = 1.2246467991473532e-16 # convert(Float64, pi * BigFloat(1) - pi2o2_h)

const pi3o2_h  = 4.71238898038469       # convert(Float64, pi * BigFloat(3/2))
const pi3o2_l  = 1.8369701987210297e-16 # convert(Float64, pi * BigFloat(3/2) - pi3o2_h)

const pi4o2_h  = 6.283185307179586      # convert(Float64, pi * BigFloat(2))
const pi4o2_l  = 2.4492935982947064e-16 # convert(Float64, pi * BigFloat(2) - pi4o2_h)

"""
    mod2pi(x)

Modulus after division by `2π`, returning in the range ``[0,2π)``.

This function computes a floating point representation of the modulus after division by
numerically exact `2π`, and is therefore not exactly the same as `mod(x,2π)`, which would
compute the modulus of `x` relative to division by the floating-point number `2π`.
"""
function mod2pi(x::Float64) # or modtau(x)
# with r = mod2pi(x)
# a) 0 <= r < 2π  (note: boundary open or closed - a bit fuzzy, due to rem_pio2 implementation)
# b) r-x = k*2π with k integer

# note: mod(n,4) is 0,1,2,3; while mod(n-1,4)+1 is 1,2,3,4.
# We use the latter to push negative y in quadrant 0 into the positive (one revolution, + 4*pi/2)

    if x < pi4o2_h
        if 0.0 <= x return x end
        if x > -pi4o2_h
            return add22condh(x,0.0,pi4o2_h,pi4o2_l)
        end
    end

    (n,y) = ieee754_rem_pio2(x)

    if iseven(n)
        if n & 2 == 2 # add pi
            return add22condh(y[1],y[2],pi2o2_h,pi2o2_l)
        else # add 0 or 2pi
            if y[1] > 0.0
                return y[1]
            else # else add 2pi
                return add22condh(y[1],y[2],pi4o2_h,pi4o2_l)
            end
        end
    else # add pi/2 or 3pi/2
        if n & 2 == 2 # add 3pi/2
            return add22condh(y[1],y[2],pi3o2_h,pi3o2_l)
        else # add pi/2
            return add22condh(y[1],y[2],pi1o2_h,pi1o2_l)
        end
    end
end

mod2pi(x::Float32) = Float32(mod2pi(Float64(x)))
mod2pi(x::Int32) = mod2pi(Float64(x))
function mod2pi(x::Int64)
  fx = Float64(x)
  fx == x || throw(ArgumentError("Int64 argument to mod2pi is too large: $x"))
  mod2pi(fx)
end

# generic fallback; for number types, promotion.jl does promotion
muladd(x,y,z) = x*y+z

# Float16 definitions

for func in (:sin,:cos,:tan,:asin,:acos,:atan,:sinh,:cosh,:tanh,:asinh,:acosh,
             :atanh,:exp,:log,:log2,:log10,:sqrt,:lgamma,:log1p,:erf,:erfc)
    @eval begin
        $func(a::Float16) = Float16($func(Float32(a)))
        $func(a::Complex32) = Complex32($func(Complex64(a)))
    end
end

for func in (:atan2,:hypot)
    @eval begin
        $func(a::Float16,b::Float16) = Float16($func(Float32(a),Float32(b)))
    end
end

ldexp(a::Float16, b::Integer) = Float16(ldexp(Float32(a), b))

# More special functions
include("special/trig.jl")
include("special/bessel.jl")
include("special/erf.jl")
include("special/gamma.jl")

module JuliaLibm
include("special/log.jl")
end

end # module