diff --git a/src/erf.jl b/src/erf.jl index c878cf5d..92865360 100644 --- a/src/erf.jl +++ b/src/erf.jl @@ -12,9 +12,6 @@ for f in (:erf, :erfc) @eval begin $f(x::Number) = $internalf(float(x)) - $internalf(x::Float64) = ccall(($libopenlibmf, libopenlibm), Float64, (Float64,), x) - $internalf(x::Float32) = ccall(($libopenlibmf0, libopenlibm), Float32, (Float32,), x) - $internalf(x::Float16) = Float16($internalf(Float32(x))) $internalf(z::Complex{Float64}) = Complex{Float64}(ccall(($openspecfunf, libopenspecfun), Complex{Float64}, (Complex{Float64}, Float64), z, zero(Float64))) $internalf(z::Complex{Float32}) = Complex{Float32}(ccall(($openspecfunf, libopenspecfun), Complex{Float64}, (Complex{Float64}, Float64), Complex{Float64}(z), Float64(eps(Float32)))) @@ -28,6 +25,10 @@ for f in (:erf, :erfc) end end + _erfc(x::Float64) = ccall((:erfc, libopenlibm), Float64, (Float64,), x) + _erfc(x::Float32) = ccall((:erfcf, libopenlibm), Float32, (Float32,), x) + _erfc(x::Float16) = Float16(_erfc(Float32(x))) + for f in (:erfcx, :erfi, :dawson, :faddeeva) internalf = Symbol(:_, f) openspecfunfsym = Symbol(:Faddeeva_, f === :dawson ? :Dawson : f === :faddeeva ? :w : f) @@ -96,10 +97,253 @@ See also: [`erfinv(x)`](@ref erfinv), [`erfcinv(x)`](@ref erfcinv). # Implementation by -- `Float32`/`Float64`: C standard math library - [libm](https://en.wikipedia.org/wiki/C_mathematical_functions#libm). +- `Float32`/`Float64`: Julia implementation of https://github.com/ARM-software/optimized-routines/blob/master/math/erf.c - `BigFloat`: C library for multiple-precision floating-point [MPFR](https://www.mpfr.org/) """ + +# Fast erf implementation using a mix of +# rational and polynomial approximations. +# Highest measured error is 1.01 ULPs at 0x1.39956ac43382fp+0. +function _erf(x::Float64) + # Minimax approximation of erf + PA=(0x1.06eba8214db68p-3, -0x1.812746b037948p-2, 0x1.ce2f21a03872p-4,-0x1.b82ce30e6548p-6, 0x1.565bcc360a2f2p-8, -0x1.c02d812bc979ap-11,0x1.f99bddfc1ebe9p-14, -0x1.f42c457cee912p-17, 0x1.b0e414ec20ee9p-20,-0x1.18c47fd143c5ep-23) + # Rational approximation on [0x1p-28, 0.84375] + NA=(0x1.06eba8214db68p-3, -0x1.4cd7d691cb913p-2, -0x1.d2a51dbd7194fp-6,-0x1.7a291236668e4p-8, -0x1.8ead6120016acp-16) + DA=(0x1.97779cddadc09p-2, 0x1.0a54c5536cebap-4, 0x1.4d022c4d36b0fp-8,0x1.15dc9221c1a1p-13, -0x1.09c4342a2612p-18) + # Rational approximation on [0.84375, 1.25] + NB=( -0x1.359b8bef77538p-9, 0x1.a8d00ad92b34dp-2, -0x1.7d240fbb8c3f1p-2, 0x1.45fca805120e4p-2, -0x1.c63983d3e28ecp-4, 0x1.22a36599795ebp-5, -0x1.1bf380a96073fp-9 ) + DB=( 0x1.b3e6618eee323p-4, 0x1.14af092eb6f33p-1, 0x1.2635cd99fe9a7p-4, 0x1.02660e763351fp-3, 0x1.bedc26b51dd1cp-7, 0x1.88b545735151dp-7 ) + + # Generated using Sollya::remez(f(c*x+d), deg, [(a-d)/c;(b-d)/c], 1, 1e-16), [|D ...|] with deg=15 a=1.25 b=2 c=1 d=1.25 + PC=( 0x1.3bcd133aa0ffcp-4, -0x1.e4652fadcb702p-3, 0x1.2ebf3dcca0446p-2, -0x1.571d01c62d66p-3, 0x1.93a9a8f5b3413p-8, 0x1.8281cbcc2cd52p-5, -0x1.5cffd86b4de16p-6, -0x1.db4ccf595053ep-9, 0x1.757cbf8684edap-8, -0x1.ce7dfd2a9e56ap-11, -0x1.99ee3bc5a3263p-11, 0x1.3c57cf9213f5fp-12, 0x1.60692996bf254p-14, -0x1.6e44cb7c1fa2ap-14, 0x1.9d4484ac482b2p-16, -0x1.578c9e375d37p-19) + # Generated using Sollya::remez(f(c*x+d), deg, [(a-d)/c;(b-d)/c], 1, 1e-16), [|D ...|] with deg=17 a=2 b=3.25 c=2 d=2 + PD=( 0x1.328f5ec350e5p-8, -0x1.529b9e8cf8e99p-5, 0x1.529b9e8cd9e71p-3, -0x1.8b0ae3a023bf2p-2, 0x1.1a2c592599d82p-1, -0x1.ace732477e494p-2, -0x1.e1a06a27920ffp-6, 0x1.bae92a6d27af6p-2, -0x1.a15470fcf5ce7p-2, 0x1.bafe45d18e213p-6, 0x1.0d950680d199ap-2, -0x1.8c9481e8f22e3p-3, -0x1.158450ed5c899p-4, 0x1.c01f2973b44p-3, -0x1.73ed2827546a7p-3, 0x1.47733687d1ff7p-4, -0x1.2dec70d00b8e1p-6, 0x1.a947ab83cd4fp-10 ) + # Generated using Sollya::remez(f(c*x+d), deg, [(a-d)/c;(b-d)/c], 1, 1e-16), [|D ...|] with deg=13 a=3.25 b=4 c=1 d=3.25 + PE=( 0x1.20c13035539e4p-18, -0x1.e9b5e8d16df7ep-16, 0x1.8de3cd4733bf9p-14, -0x1.9aa48beb8382fp-13, 0x1.2c7d713370a9fp-12, -0x1.490b12110b9e2p-12, 0x1.1459c5d989d23p-12, -0x1.64b28e9f1269p-13, 0x1.57c76d9d05cf8p-14, -0x1.bf271d9951cf8p-16, 0x1.db7ea4d4535c9p-19, 0x1.91c2e102d5e49p-20, -0x1.e9f0826c2149ep-21, 0x1.60eebaea236e1p-23 ) + # Generated using Sollya::remez(f(c*x+d), deg, [(a-d)/c;(b-d)/c], 1, 1e-16), [|D ...|] with deg=16 a=4 b=5.90625 c=2 d=4 + PF=( 0x1.08ddd130d1fa6p-26, -0x1.10b146f59ff06p-22, 0x1.10b135328b7b2p-19, -0x1.6039988e7575fp-17, 0x1.497d365e19367p-15, -0x1.da48d9afac83ep-14, 0x1.1024c9b1fbb48p-12, -0x1.fc962e7066272p-12, 0x1.87297282d4651p-11, -0x1.f057b255f8c59p-11, 0x1.0228d0eee063p-10, -0x1.b1b21b84ec41cp-11, 0x1.1ead8ae9e1253p-11, -0x1.1e708fba37fccp-12, 0x1.9559363991edap-14, -0x1.68c827b783d9cp-16, 0x1.2ec4adeccf4a2p-19 ) + + C = 0x1.b0ac16p-1 + + TwoOverSqrtPiMinusOne=0x1.06eba8214db69p-3 + + + # # top 32 bits + ix::UInt32=reinterpret(UInt64,x)>>32 + # # top 32, without sign bit + ia::UInt32=ix & 0x7fffffff + # # sign + # sign::UInt32=ix>>31 + + sign::Bool=x<0 + + + + if (ia < 0x3feb0000) + # a = |x| < 0.84375. + + x2 = x * x + + if (ia < 0x3fe00000) + ## a < 0.5 - Use polynomial approximation. + r1 = fma(x2, PA[2], PA[1]) + r2 = fma(x2, PA[4], PA[3]) + r3 = fma(x2, PA[6], PA[5]) + r4 = fma(x2, PA[8], PA[7]) + r5 = fma(x2, PA[10], PA[9]) + + x4 = x2 * x2 + r = r5 + r = fma(x4, r, r4) + r = fma(x4, r, r3) + r = fma(x4, r, r2) + r = fma(x4, r, r1) + return fma(r, x, x) ## This fma is crucial for accuracy. + else + ## 0.5 <= a < 0.84375 - Use rational approximation. + + r1n = fma(x2, NA[2], NA[1]) + x4 = x2 * x2 + r2n = fma(x2, NA[4], NA[3]) + x8 = x4 * x4 + r1d = fma(x2, DA[1], 1.0) + r2d = fma(x2, DA[3], DA[2]) + r3d = fma(x2, DA[5], DA[4]) + P = r1n + x4 * r2n + x8 * NA[5] + + Q = r1d + x4 * r2d + x8 * r3d + return fma(P / Q, x, x) + end + elseif (ia < 0x3ff40000) + ## 0.84375 <= |x| < 1.25. + + a = abs(x) - 1.0 + r1n = fma(a, NB[2], NB[1]) + a2 = a * a + r1d = fma(a, DB[1], 1.0) + a4 = a2 * a2 + r2n = fma(a, NB[4], NB[3]) + a6 = a4 * a2 + r2d = fma(a, DB[3], DB[2]) + r3n = fma(a, NB[6], NB[5]) + r3d = fma(a, DB[5], DB[4]) + r4n = NB[7] + r4d = DB[6] + + P = r1n + a2 * r2n + a4 * r3n + a6 * r4n + Q = r1d + a2 * r2d + a4 * r3d + a6 * r4d + if (sign) + return -C - P / Q + else + return C + P / Q + end + elseif (ia < 0x40000000) + ## 1.25 <= |x| < 2.0. + a = abs(x) + a = a - 1.25 + + r1 = fma(a, PC[2], PC[1]) + r2 = fma(a, PC[4], PC[3]) + r3 = fma(a, PC[6], PC[5]) + r4 = fma(a, PC[8], PC[7]) + r5 = fma(a, PC[10], PC[9]) + r6 = fma(a, PC[12], PC[11]) + r7 = fma(a, PC[14], PC[13]) + r8 = fma(a, PC[16], PC[15]) + + + a2 = a * a + + r = r8 + r = fma(a2, r, r7) + r = fma(a2, r, r6) + r = fma(a2, r, r5) + r = fma(a2, r, r4) + r = fma(a2, r, r3) + r = fma(a2, r, r2) + r = fma(a2, r, r1) + + if (sign) + return -1.0 + r + else + return 1.0 - r + end + elseif (ia < 0x400a0000) + ## 2 <= |x| < 3.25. + a = abs(x) + a = fma(0.5, a, -1.0) + + r1 = fma(a, PD[2], PD[1]) + r2 = fma(a, PD[4], PD[3]) + r3 = fma(a, PD[6], PD[5]) + r4 = fma(a, PD[8], PD[7]) + r5 = fma(a, PD[10], PD[9]) + r6 = fma(a, PD[12], PD[11]) + r7 = fma(a, PD[14], PD[13]) + r8 = fma(a, PD[16], PD[15]) + r9 = fma(a, PD[18], PD[17]) + + a2 = a * a + + r = r9 + r = fma(a2, r, r8) + r = fma(a2, r, r7) + r = fma(a2, r, r6) + r = fma(a2, r, r5) + r = fma(a2, r, r4) + r = fma(a2, r, r3) + r = fma(a2, r, r2) + r = fma(a2, r, r1) + + if (sign) + return -1.0 + r + else + return 1.0 - r + end + elseif (ia < 0x40100000) + ## 3.25 <= |x| < 4.0. + a = abs(x) + a = a - 3.25 + + r1 = fma(a, PE[2], PE[1]) + r2 = fma(a, PE[4], PE[3]) + r3 = fma(a, PE[6], PE[5]) + r4 = fma(a, PE[8], PE[7]) + r5 = fma(a, PE[10], PE[9]) + r6 = fma(a, PE[12], PE[11]) + r7 = fma(a, PE[14], PE[13]) + + + a2 = a * a + + r = r7 + r = fma(a2, r, r6) + r = fma(a2, r, r5) + r = fma(a2, r, r4) + r = fma(a2, r, r3) + r = fma(a2, r, r2) + r = fma(a2, r, r1) + + if (sign) + return -1.0 + r + else + return 1.0 - r + end + elseif (ia < 0x4017a000) + ## 4 <= |x| < 5.90625. + a = abs(x) + a = fma(0.5, a, -2.0) + + r1 = fma(a, PF[2], PF[1]) + r2 = fma(a, PF[4], PF[3]) + r3 = fma(a, PF[6], PF[5]) + r4 = fma(a, PF[8], PF[7]) + r5 = fma(a, PF[10], PF[9]) + r6 = fma(a, PF[12], PF[11]) + r7 = fma(a, PF[14], PF[13]) + r8 = fma(a, PF[16], PF[15]) + + r9 = PF[17] + + a2 = a * a + + r = r9 + r = fma(a2, r, r8) + r = fma(a2, r, r7) + r = fma(a2, r, r6) + r = fma(a2, r, r5) + r = fma(a2, r, r4) + r = fma(a2, r, r3) + r = fma(a2, r, r2) + r = fma(a2, r, r1) + + if (sign) + return -1.0 + r + else + return 1.0 - r + end + else + + if(isnan(x)) + return NaN + end + + if (sign) + return -1.0 + else + return 1.0 + end + + end + + +end + +_erf(x::Float32)=Float32(_erf(Float64(x))) + +_erf(x::Float16)=Float16(_erf(Float64(x))) + + function erf end """ erf(x, y) diff --git a/test/erf.jl b/test/erf.jl index 29127343..70f1bff0 100644 --- a/test/erf.jl +++ b/test/erf.jl @@ -2,7 +2,23 @@ @testset "real argument" begin for T in (Float16, Float32, Float64) @test @inferred(erf(T(1))) isa T + @test erf(T(0.25)) ≈ T(0.27632639016823696) rtol=2*eps(T) + @test erf(T(0.75)) ≈ T(0.7111556336535151) rtol=2*eps(T) @test erf(T(1)) ≈ T(0.84270079294971486934) rtol=2*eps(T) + @test erf(T(1.5)) ≈ T(0.9661051464753108) rtol=2*eps(T) + @test erf(T(2.5)) ≈ T(0.9995930479825551) rtol=2*eps(T) + @test erf(T(3.5)) ≈ T(0.9999992569016276) rtol=2*eps(T) + @test erf(T(4.5)) ≈ T(0.9999999998033839) rtol=2*eps(T) + @test erf(T(6)) ≈ T(1.0) rtol=2*eps(T) + + @test erf(T(-0.25)) ≈ T(-0.27632639016823696) rtol=2*eps(T) + @test erf(T(-0.75)) ≈ T(-0.7111556336535151) rtol=2*eps(T) + @test erf(T(-1)) ≈ T(-0.84270079294971486934) rtol=2*eps(T) + @test erf(T(-1.5)) ≈ T(-0.9661051464753108) rtol=2*eps(T) + @test erf(T(-2.5)) ≈ T(-0.9995930479825551) rtol=2*eps(T) + @test erf(T(-3.5)) ≈ T(-0.9999992569016276) rtol=2*eps(T) + @test erf(T(-4.5)) ≈ T(-0.9999999998033839) rtol=2*eps(T) + @test erf(T(-6)) ≈ T(-1.0) rtol=2*eps(T) @test @inferred(erfc(T(1))) isa T @test erfc(T(1)) ≈ T(0.15729920705028513066) rtol=2*eps(T)