gistfile1.txt

julia> versioninfo()
Julia Version 0.7.0-DEV.1208
Commit 952dc93489* (2017-08-02 23:54 UTC)
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: Intel(R) Core(TM) i7-4510U CPU @ 2.00GHz
  WORD_SIZE: 64
  BLAS: libopenblas (USE64BITINT DYNAMIC_ARCH NO_AFFINITY Haswell)
  LAPACK: libopenblas64_
  LIBM: libopenlibm
  LLVM: libLLVM-3.9.1 (ORCJIT, haswell)
Environment:

using Base.Math.@horner, Base.sqrt_llvm
# asin methods
ASIN_X_MIN_THRESHOLD(::Type{Float32}) = 2.0f0^-12
ASIN_X_MIN_THRESHOLD(::Type{Float64}) = sqrt(eps(Float64))

asin_p(t::Float64) =
    t*@horner(t,
    1.66666666666666657415e-01,
    -3.25565818622400915405e-01,
    2.01212532134862925881e-01,
    -4.00555345006794114027e-02,
    7.91534994289814532176e-04,
    3.47933107596021167570e-05)

asin_q(t::Float64) =
    @horner(t,
    1.0,
    -2.40339491173441421878e+00,
    2.02094576023350569471e+00,
    -6.88283971605453293030e-01,
    7.70381505559019352791e-02)

asin_p(t::Float32) =
    t*@horner(t,
    1.6666586697f-01,
    -4.2743422091f-02,
    -8.6563630030f-03)

asin_q(t::Float32) = @horner(t, 1.0f0, -7.0662963390f-01)

@inline function asin_kernel(t::Float64, x::Float64)
    pio2_lo = 6.12323399573676603587e-17
    s = sqrt_llvm(t)
    p = asin_p(t) # numerator polynomial
    q = asin_q(t) # denominator polynomial
    if abs(x) >= 0.975 # |x| > 0.975
        Rx = p/q
        return flipsign(pi/2 - (2.0*(s + s*Rx) - pio2_lo), x)
    else
        s0 = reinterpret(Float64, (reinterpret(UInt64, s) >> 32) << 32)
        c = (t - s0*s0)/(s + s0)
        Rx = p/q
        p = 2.0*s*Rx - (pio2_lo - 2.0*c)
        q = pi/4 - 2.0*s0
        return flipsign(pi/4 - (p-q), x)
    end
end
@inline function asin_kernel(t::Float32, x::Float32)
    s = sqrt_llvm(Float64(t))
    p = asin_p(t) # numerator polynomial
    q = asin_q(t) # denominator polynomial
    Rx = p/q # rational approximation
    flipsign(Float32(pi/2 - 2*(s + s*Rx)), x)
end

@noinline asin_domain_error(x) = throw(DomainError(x, "asin(x) is not defined for |x|>1."))
function asin1(x::T) where T<:Union{Float32, Float64}
    absx = abs(x)
    if absx >= T(1.0) # |x|>= 1
        if absx == T(1.0)
            return flipsign(T(pi)/2, x)
        end
        asin_domain_error(x)
    elseif absx < T(1.0)/2
        # if |x| sufficiently small, |x| is a good approximation
        if absx < ASIN_X_MIN_THRESHOLD(T)
            return x
        end
        # else if |x|<0.5 we use a rational approximation R(x)=p(x)/q(x) such that
        # tan(x) ≈ x+x*R(x)
        x² = x*x
        Rx = asin_p(x²)/asin_q(x²) # rational approximation
        return muladd(x, Rx, x)
    end
    # else 1/2 <= |x| < 1
    t = (T(1.0) - absx)/2
    return asin_kernel(t, x)
end

function asin2(x::T) where T<:Union{Float32, Float64}
    absx = abs(x)
    if absx >= T(1.0) # |x|>= 1
        if absx == T(1.0)
            return flipsign(T(pi)/2, x)
        end
        throw(DomainError("SDF"))
    elseif absx < T(1.0)/2
        # if |x| sufficiently small, |x| is a good approximation
        if absx < ASIN_X_MIN_THRESHOLD(T)
            return x
        end
        # else if |x|<0.5 we use a rational approximation R(x)=p(x)/q(x) such that
        # tan(x) ≈ x+x*R(x)
        x² = x*x
        Rx = asin_p(x²)/asin_q(x²) # rational approximation
        return muladd(x, Rx, x)
    end
    # else 1/2 <= |x| < 1
    t = (T(1.0) - absx)/2
    return asin_kernel(t, x)
end


julia> @benchmark asin2(.532)
BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     14.000 ns (0.00% GC)
  median time:      15.000 ns (0.00% GC)
  mean time:        16.310 ns (0.00% GC)
  maximum time:     67.000 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     1000

julia> @benchmark asin1(.532)
BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     13.000 ns (0.00% GC)
  median time:      15.000 ns (0.00% GC)
  mean time:        15.408 ns (0.00% GC)
  maximum time:     112.000 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     1000

julia> @benchmark asin2(.532f0)
BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     9.000 ns (0.00% GC)
  median time:      10.000 ns (0.00% GC)
  mean time:        10.001 ns (0.00% GC)
  maximum time:     98.000 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     1000

julia> @benchmark asin1(.532f0)
BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     7.000 ns (0.00% GC)
  median time:      8.000 ns (0.00% GC)
  mean time:        8.361 ns (0.00% GC)
  maximum time:     200.000 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     1000

julia> @benchmark asin1(.9f0)
BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     7.000 ns (0.00% GC)
  median time:      8.000 ns (0.00% GC)
  mean time:        8.397 ns (0.00% GC)
  maximum time:     47.000 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     1000

julia> @benchmark asin2(.9f0)
BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     9.000 ns (0.00% GC)
  median time:      10.000 ns (0.00% GC)
  mean time:        10.607 ns (0.00% GC)
  maximum time:     83.000 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     1000


julia> @benchmark asin2(.9)
BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     15.000 ns (0.00% GC)
  median time:      15.000 ns (0.00% GC)
  mean time:        15.975 ns (0.00% GC)
  maximum time:     57.000 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     1000

julia> @benchmark asin1(.9)
BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     13.000 ns (0.00% GC)
  median time:      14.000 ns (0.00% GC)
  mean time:        15.044 ns (0.00% GC)
  maximum time:     69.000 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     1000

Different Machine

julia> @benchmark asin1(.75)
BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     11.303 ns (0.00% GC)
  median time:      11.324 ns (0.00% GC)
  mean time:        11.357 ns (0.00% GC)
  maximum time:     30.260 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     999

julia> @benchmark asin2(.75)
BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     12.912 ns (0.00% GC)
  median time:      12.939 ns (0.00% GC)
  mean time:        12.987 ns (0.00% GC)
  maximum time:     48.831 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     999

julia> @benchmark asin1(.75f0)
BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     7.111 ns (0.00% GC)
  median time:      7.127 ns (0.00% GC)
  mean time:        7.158 ns (0.00% GC)
  maximum time:     26.212 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     999

julia> @benchmark asin2(.75f0)
BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     8.412 ns (0.00% GC)
  median time:      8.734 ns (0.00% GC)
  mean time:        8.798 ns (0.00% GC)
  maximum time:     28.603 ns (0.00% GC)
  --------------
  samples:          10000
  evals/sample:     999