Revision cf55683b764816b6ec4c6e5f0414ac26fb30af79 authored by Rafael Fourquet on 09 June 2017, 07:36:25 UTC, committed by Rafael Fourquet on 21 June 2017, 11:36:04 UTC
1 parent a52a78d
trig.jl
# This file is a part of Julia. Except for the *_kernel functions (see below),
# license is MIT: https://julialang.org/license
struct DoubleFloat64
hi::Float64
lo::Float64
end
struct DoubleFloat32
hi::Float64
end
# *_kernel functions are only valid for |x| < pi/4 = 0.7854
# translated from openlibm code: k_sin.c, k_cos.c, k_sinf.c, k_cosf.c
# which are made available under the following licence:
## Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
##
## Developed at SunPro, a Sun Microsystems, Inc. business.
## Permission to use, copy, modify, and distribute this
## software is freely granted, provided that this notice
## is preserved.
function sin_kernel(x::DoubleFloat64)
S1 = -1.66666666666666324348e-01
S2 = 8.33333333332248946124e-03
S3 = -1.98412698298579493134e-04
S4 = 2.75573137070700676789e-06
S5 = -2.50507602534068634195e-08
S6 = 1.58969099521155010221e-10
z = x.hi*x.hi
w = z*z
r = S2+z*(S3+z*S4) + z*w*(S5+z*S6)
v = z*x.hi
x.hi-((z*(0.5*x.lo-v*r)-x.lo)-v*S1)
end
function cos_kernel(x::DoubleFloat64)
C1 = 4.16666666666666019037e-02
C2 = -1.38888888888741095749e-03
C3 = 2.48015872894767294178e-05
C4 = -2.75573143513906633035e-07
C5 = 2.08757232129817482790e-09
C6 = -1.13596475577881948265e-11
z = x.hi*x.hi
w = z*z
r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6))
hz = 0.5*z
w = 1.0-hz
w + (((1.0-w)-hz) + (z*r-x.hi*x.lo))
end
function sin_kernel(x::DoubleFloat32)
S1 = -0.16666666641626524
S2 = 0.008333329385889463
S3 = -0.00019839334836096632
S4 = 2.718311493989822e-6
z = x.hi*x.hi
w = z*z
r = S3+z*S4
s = z*x.hi
Float32((x.hi + s*(S1+z*S2)) + s*w*r)
end
function cos_kernel(x::DoubleFloat32)
C0 = -0.499999997251031
C1 = 0.04166662332373906
C2 = -0.001388676377460993
C3 = 2.439044879627741e-5
z = x.hi*x.hi
w = z*z
r = C2+z*C3
Float32(((1.0+z*C0) + w*C1) + (w*z)*r)
end
# fallback methods
sin_kernel(x::Real) = sin(x)
cos_kernel(x::Real) = cos(x)
# multiply in extended precision
function mulpi_ext(x::Float64)
m = 3.141592653589793
m_hi = 3.1415926218032837
m_lo = 3.178650954705639e-8
x_hi = reinterpret(Float64, reinterpret(UInt64,x) & 0xffff_ffff_f800_0000)
x_lo = x-x_hi
y_hi = m*x
y_lo = x_hi * m_lo + (x_lo* m_hi + ((x_hi*m_hi-y_hi) + x_lo*m_lo))
DoubleFloat64(y_hi,y_lo)
end
mulpi_ext(x::Float32) = DoubleFloat32(pi*Float64(x))
mulpi_ext(x::Rational) = mulpi_ext(float(x))
mulpi_ext(x::Real) = pi*x # Fallback
"""
sinpi(x)
Compute ``\\sin(\\pi x)`` more accurately than `sin(pi*x)`, especially for large `x`.
"""
function sinpi(x::T) where T<:AbstractFloat
if !isfinite(x)
isnan(x) && return x
throw(DomainError())
end
ax = abs(x)
s = maxintfloat(T)/2
ax >= s && return copysign(zero(T),x) # integer-valued
# reduce to interval [-1,1]
# assumes RoundNearest rounding mode
t = 3*s
rx = x-((x+t)-t) # zeros may be incorrectly signed
arx = abs(rx)
if (arx == 0) | (arx == 1)
copysign(zero(T),x)
elseif arx < 0.25
sin_kernel(mulpi_ext(rx))
elseif arx < 0.75
y = mulpi_ext(T(0.5) - arx)
copysign(cos_kernel(y),rx)
else
y = mulpi_ext(copysign(one(T),rx) - rx)
sin_kernel(y)
end
end
# Integers and Rationals
function sinpi(x::T) where T<:Union{Integer,Rational}
Tf = float(T)
if !isfinite(x)
throw(DomainError())
end
# until we get an IEEE remainder function (#9283)
rx = rem(x,2)
if rx > 1
rx -= 2
elseif rx < -1
rx += 2
end
arx = abs(rx)
if (arx == 0) | (arx == 1)
copysign(zero(Tf),x)
elseif arx < 0.25
sin_kernel(mulpi_ext(rx))
elseif arx < 0.75
y = mulpi_ext(T(0.5) - arx)
copysign(cos_kernel(y),rx)
else
y = mulpi_ext(copysign(one(T),rx) - rx)
sin_kernel(y)
end
end
"""
cospi(x)
Compute ``\\cos(\\pi x)`` more accurately than `cos(pi*x)`, especially for large `x`.
"""
function cospi(x::T) where T<:AbstractFloat
if !isfinite(x)
isnan(x) && return x
throw(DomainError())
end
ax = abs(x)
s = maxintfloat(T)
ax >= s && return one(T) # even integer-valued
# reduce to interval [-1,1], then [0,1]
# assumes RoundNearest rounding mode
rx = abs(ax-((ax+s)-s))
if rx <= 0.25
cos_kernel(mulpi_ext(rx))
elseif rx < 0.75
y = mulpi_ext(T(0.5) - rx)
sin_kernel(y)
else
y = mulpi_ext(one(T) - rx)
-cos_kernel(y)
end
end
# Integers and Rationals
function cospi(x::T) where T<:Union{Integer,Rational}
if !isfinite(x)
throw(DomainError())
end
ax = abs(x)
# until we get an IEEE remainder function (#9283)
rx = rem(ax,2)
if rx > 1
rx = 2-rx
end
if rx <= 0.25
cos_kernel(mulpi_ext(rx))
elseif rx < 0.75
y = mulpi_ext(T(0.5) - rx)
sin_kernel(y)
else
y = mulpi_ext(one(T) - rx)
-cos_kernel(y)
end
end
sinpi(x::Integer) = x >= 0 ? zero(float(x)) : -zero(float(x))
cospi(x::Integer) = isodd(x) ? -one(float(x)) : one(float(x))
sinpi(x::Real) = sinpi(float(x))
cospi(x::Real) = cospi(float(x))
function sinpi(z::Complex{T}) where T
F = float(T)
zr, zi = reim(z)
if isinteger(zr)
# zr = ...,-2,-1,0,1,2,...
# sin(pi*zr) == ±0
# cos(pi*zr) == ±1
# cosh(pi*zi) > 0
s = copysign(zero(F),zr)
c_pos = isa(zr,Integer) ? iseven(zr) : isinteger(zr/2)
sh = sinh(pi*zi)
Complex(s, c_pos ? sh : -sh)
elseif isinteger(2*zr)
# zr = ...,-1.5,-0.5,0.5,1.5,2.5,...
# sin(pi*zr) == ±1
# cos(pi*zr) == +0
# sign(sinh(pi*zi)) == sign(zi)
s_pos = isinteger((2*zr-1)/4)
ch = cosh(pi*zi)
Complex(s_pos ? ch : -ch, isnan(zi) ? zero(F) : copysign(zero(F),zi))
elseif !isfinite(zr)
if zi == 0 || isinf(zi)
Complex(F(NaN), F(zi))
else
Complex(F(NaN), F(NaN))
end
else
pizi = pi*zi
Complex(sinpi(zr)*cosh(pizi), cospi(zr)*sinh(pizi))
end
end
function cospi(z::Complex{T}) where T
F = float(T)
zr, zi = reim(z)
if isinteger(zr)
# zr = ...,-2,-1,0,1,2,...
# sin(pi*zr) == ±0
# cos(pi*zr) == ±1
# sign(sinh(pi*zi)) == sign(zi)
# cosh(pi*zi) > 0
s = copysign(zero(F),zr)
c_pos = isa(zr,Integer) ? iseven(zr) : isinteger(zr/2)
ch = cosh(pi*zi)
Complex(c_pos ? ch : -ch, isnan(zi) ? s : -flipsign(s,zi))
elseif isinteger(2*zr)
# zr = ...,-1.5,-0.5,0.5,1.5,2.5,...
# sin(pi*zr) == ±1
# cos(pi*zr) == +0
# sign(sinh(pi*zi)) == sign(zi)
s_pos = isinteger((2*zr-1)/4)
sh = sinh(pi*zi)
Complex(zero(F), s_pos ? -sh : sh)
elseif !isfinite(zr)
if zi == 0
Complex(F(NaN), isnan(zr) ? zero(F) : -flipsign(F(zi),zr))
elseif isinf(zi)
Complex(F(Inf), F(NaN))
else
Complex(F(NaN), F(NaN))
end
else
pizi = pi*zi
Complex(cospi(zr)*cosh(pizi), -sinpi(zr)*sinh(pizi))
end
end
"""
sinc(x)
Compute ``\\sin(\\pi x) / (\\pi x)`` if ``x \\neq 0``, and ``1`` if ``x = 0``.
"""
sinc(x::Number) = x==0 ? one(x) : oftype(x,sinpi(x)/(pi*x))
sinc(x::Integer) = x==0 ? one(x) : zero(x)
sinc(x::Complex{<:AbstractFloat}) = x==0 ? one(x) : oftype(x, sinpi(x)/(pi*x))
sinc(x::Complex) = sinc(float(x))
sinc(x::Real) = x==0 ? one(x) : isinf(x) ? zero(x) : sinpi(x)/(pi*x)
"""
cosc(x)
Compute ``\\cos(\\pi x) / x - \\sin(\\pi x) / (\\pi x^2)`` if ``x \\neq 0``, and ``0`` if
``x = 0``. This is the derivative of `sinc(x)`.
"""
cosc(x::Number) = x==0 ? zero(x) : oftype(x,(cospi(x)-sinpi(x)/(pi*x))/x)
cosc(x::Integer) = cosc(float(x))
cosc(x::Complex{<:AbstractFloat}) = x==0 ? zero(x) : oftype(x,(cospi(x)-sinpi(x)/(pi*x))/x)
cosc(x::Complex) = cosc(float(x))
cosc(x::Real) = x==0 || isinf(x) ? zero(x) : (cospi(x)-sinpi(x)/(pi*x))/x
for (finv, f) in ((:sec, :cos), (:csc, :sin), (:cot, :tan),
(:sech, :cosh), (:csch, :sinh), (:coth, :tanh),
(:secd, :cosd), (:cscd, :sind), (:cotd, :tand))
@eval begin
($finv)(z::T) where {T<:Number} = one(T) / (($f)(z))
end
end
for (tfa, tfainv, hfa, hfainv, fn) in ((:asec, :acos, :asech, :acosh, "secant"),
(:acsc, :asin, :acsch, :asinh, "cosecant"),
(:acot, :atan, :acoth, :atanh, "cotangent"))
tname = string(tfa)
hname = string(hfa)
@eval begin
@doc """
$($tname)(x)
Compute the inverse $($fn) of `x`, where the output is in radians. """ ($tfa)(y::T) where {T<:Number} = ($tfainv)(one(T) / y)
@doc """
$($hname)(x)
Compute the inverse hyperbolic $($fn) of `x`. """ ($hfa)(y::T) where {T<:Number} = ($hfainv)(one(T) / y)
end
end
# multiply in extended precision
function deg2rad_ext(x::Float64)
m = 0.017453292519943295
m_hi = 0.01745329238474369
m_lo = 1.3519960527851425e-10
u = 134217729.0*x # 0x1p27 + 1
x_hi = u-(u-x)
x_lo = x-x_hi
y_hi = m*x
y_lo = x_hi * m_lo + (x_lo* m_hi + ((x_hi*m_hi-y_hi) + x_lo*m_lo))
DoubleFloat64(y_hi,y_lo)
end
deg2rad_ext(x::Float32) = DoubleFloat32(deg2rad(Float64(x)))
deg2rad_ext(x::Real) = deg2rad(x) # Fallback
function sind(x::Real)
if isinf(x)
return throw(DomainError())
elseif isnan(x)
return oftype(x,NaN)
end
rx = copysign(float(rem(x,360)),x)
arx = abs(rx)
if rx == zero(rx)
return rx
elseif arx < oftype(rx,45)
return sin_kernel(deg2rad_ext(rx))
elseif arx <= oftype(rx,135)
y = deg2rad_ext(oftype(rx,90) - arx)
return copysign(cos_kernel(y),rx)
elseif arx == oftype(rx,180)
return copysign(zero(rx),rx)
elseif arx < oftype(rx,225)
y = deg2rad_ext((oftype(rx,180) - arx)*sign(rx))
return sin_kernel(y)
elseif arx <= oftype(rx,315)
y = deg2rad_ext(oftype(rx,270) - arx)
return -copysign(cos_kernel(y),rx)
else
y = deg2rad_ext(rx - copysign(oftype(rx,360),rx))
return sin_kernel(y)
end
end
function cosd(x::Real)
if isinf(x)
return throw(DomainError())
elseif isnan(x)
return oftype(x,NaN)
end
rx = abs(float(rem(x,360)))
if rx <= oftype(rx,45)
return cos_kernel(deg2rad_ext(rx))
elseif rx < oftype(rx,135)
y = deg2rad_ext(oftype(rx,90) - rx)
return sin_kernel(y)
elseif rx <= oftype(rx,225)
y = deg2rad_ext(oftype(rx,180) - rx)
return -cos_kernel(y)
elseif rx < oftype(rx,315)
y = deg2rad_ext(rx - oftype(rx,270))
return sin_kernel(y)
else
y = deg2rad_ext(oftype(rx,360) - rx)
return cos_kernel(y)
end
end
tand(x::Real) = sind(x) / cosd(x)
for (fd, f, fn) in ((:sind, :sin, "sine"), (:cosd, :cos, "cosine"), (:tand, :tan, "tangent"))
name = string(fd)
@eval begin
@doc """
$($name)(x)
Compute $($fn) of `x`, where `x` is in degrees. """ ($fd)(z) = ($f)(deg2rad(z))
end
end
for (fd, f, fn) in ((:asind, :asin, "sine"), (:acosd, :acos, "cosine"), (:atand, :atan, "tangent"),
(:asecd, :asec, "secant"), (:acscd, :acsc, "cosecant"), (:acotd, :acot, "cotangent"))
name = string(fd)
@eval begin
@doc """
$($name)(x)
Compute the inverse $($fn) of `x`, where the output is in degrees. """ ($fd)(y) = rad2deg(($f)(y))
end
end
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