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,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))
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
# 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 = Float32[0]
function modf(x::Float32)
f = ccall((:modff,libm), Float32, (Float32,Ptr{Float32}), x, _modff_temp)
f, _modff_temp[1]
end
const _modf_temp = Float64[0]
function modf(x::Float64)
f = ccall((:modf,libm), Float64, (Float64,Ptr{Float64}), x, _modf_temp)
f, _modf_temp[1]
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
# 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