Jmat.Real.dawson

Percentage Accurate: 54.2% → 100.0%

Time: 13.0s

Alternatives: 4

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := t\_0 \cdot \left(x \cdot x\right)\\ t_2 := t\_1 \cdot \left(x \cdot x\right)\\ t_3 := t\_2 \cdot \left(x \cdot x\right)\\ \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t\_0\right) + 0.0072644182 \cdot t\_1\right) + 0.0005064034 \cdot t\_2\right) + 0.0001789971 \cdot t\_3}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot t\_0\right) + 0.0694555761 \cdot t\_1\right) + 0.0140005442 \cdot t\_2\right) + 0.0008327945 \cdot t\_3\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(t\_3 \cdot \left(x \cdot x\right)\right)} \cdot x \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x)))
        (t_1 (* t_0 (* x x)))
        (t_2 (* t_1 (* x x)))
        (t_3 (* t_2 (* x x))))
   (*
    (/
     (+
      (+
       (+
        (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 t_0))
        (* 0.0072644182 t_1))
       (* 0.0005064034 t_2))
      (* 0.0001789971 t_3))
     (+
      (+
       (+
        (+
         (+ (+ 1.0 (* 0.7715471019 (* x x))) (* 0.2909738639 t_0))
         (* 0.0694555761 t_1))
        (* 0.0140005442 t_2))
       (* 0.0008327945 t_3))
      (* (* 2.0 0.0001789971) (* t_3 (* x x)))))
    x)))

C

double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = t_0 * (x * x);
	double t_2 = t_1 * (x * x);
	double t_3 = t_2 * (x * x);
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = (x * x) * (x * x)
    t_1 = t_0 * (x * x)
    t_2 = t_1 * (x * x)
    t_3 = t_2 * (x * x)
    code = ((((((1.0d0 + (0.1049934947d0 * (x * x))) + (0.0424060604d0 * t_0)) + (0.0072644182d0 * t_1)) + (0.0005064034d0 * t_2)) + (0.0001789971d0 * t_3)) / ((((((1.0d0 + (0.7715471019d0 * (x * x))) + (0.2909738639d0 * t_0)) + (0.0694555761d0 * t_1)) + (0.0140005442d0 * t_2)) + (0.0008327945d0 * t_3)) + ((2.0d0 * 0.0001789971d0) * (t_3 * (x * x))))) * x
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = t_0 * (x * x);
	double t_2 = t_1 * (x * x);
	double t_3 = t_2 * (x * x);
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
}

Python

def code(x):
	t_0 = (x * x) * (x * x)
	t_1 = t_0 * (x * x)
	t_2 = t_1 * (x * x)
	t_3 = t_2 * (x * x)
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	t_1 = Float64(t_0 * Float64(x * x))
	t_2 = Float64(t_1 * Float64(x * x))
	t_3 = Float64(t_2 * Float64(x * x))
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(0.0424060604 * t_0)) + Float64(0.0072644182 * t_1)) + Float64(0.0005064034 * t_2)) + Float64(0.0001789971 * t_3)) / Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.7715471019 * Float64(x * x))) + Float64(0.2909738639 * t_0)) + Float64(0.0694555761 * t_1)) + Float64(0.0140005442 * t_2)) + Float64(0.0008327945 * t_3)) + Float64(Float64(2.0 * 0.0001789971) * Float64(t_3 * Float64(x * x))))) * x)
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * (x * x);
	t_1 = t_0 * (x * x);
	t_2 = t_1 * (x * x);
	t_3 = t_2 * (x * x);
	tmp = ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(t$95$3 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Initial Program: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := t\_0 \cdot \left(x \cdot x\right)\\ t_2 := t\_1 \cdot \left(x \cdot x\right)\\ t_3 := t\_2 \cdot \left(x \cdot x\right)\\ \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t\_0\right) + 0.0072644182 \cdot t\_1\right) + 0.0005064034 \cdot t\_2\right) + 0.0001789971 \cdot t\_3}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot t\_0\right) + 0.0694555761 \cdot t\_1\right) + 0.0140005442 \cdot t\_2\right) + 0.0008327945 \cdot t\_3\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(t\_3 \cdot \left(x \cdot x\right)\right)} \cdot x \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x)))
        (t_1 (* t_0 (* x x)))
        (t_2 (* t_1 (* x x)))
        (t_3 (* t_2 (* x x))))
   (*
    (/
     (+
      (+
       (+
        (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 t_0))
        (* 0.0072644182 t_1))
       (* 0.0005064034 t_2))
      (* 0.0001789971 t_3))
     (+
      (+
       (+
        (+
         (+ (+ 1.0 (* 0.7715471019 (* x x))) (* 0.2909738639 t_0))
         (* 0.0694555761 t_1))
        (* 0.0140005442 t_2))
       (* 0.0008327945 t_3))
      (* (* 2.0 0.0001789971) (* t_3 (* x x)))))
    x)))

C

double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = t_0 * (x * x);
	double t_2 = t_1 * (x * x);
	double t_3 = t_2 * (x * x);
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = (x * x) * (x * x)
    t_1 = t_0 * (x * x)
    t_2 = t_1 * (x * x)
    t_3 = t_2 * (x * x)
    code = ((((((1.0d0 + (0.1049934947d0 * (x * x))) + (0.0424060604d0 * t_0)) + (0.0072644182d0 * t_1)) + (0.0005064034d0 * t_2)) + (0.0001789971d0 * t_3)) / ((((((1.0d0 + (0.7715471019d0 * (x * x))) + (0.2909738639d0 * t_0)) + (0.0694555761d0 * t_1)) + (0.0140005442d0 * t_2)) + (0.0008327945d0 * t_3)) + ((2.0d0 * 0.0001789971d0) * (t_3 * (x * x))))) * x
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = t_0 * (x * x);
	double t_2 = t_1 * (x * x);
	double t_3 = t_2 * (x * x);
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
}

Python

def code(x):
	t_0 = (x * x) * (x * x)
	t_1 = t_0 * (x * x)
	t_2 = t_1 * (x * x)
	t_3 = t_2 * (x * x)
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	t_1 = Float64(t_0 * Float64(x * x))
	t_2 = Float64(t_1 * Float64(x * x))
	t_3 = Float64(t_2 * Float64(x * x))
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(0.0424060604 * t_0)) + Float64(0.0072644182 * t_1)) + Float64(0.0005064034 * t_2)) + Float64(0.0001789971 * t_3)) / Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.7715471019 * Float64(x * x))) + Float64(0.2909738639 * t_0)) + Float64(0.0694555761 * t_1)) + Float64(0.0140005442 * t_2)) + Float64(0.0008327945 * t_3)) + Float64(Float64(2.0 * 0.0001789971) * Float64(t_3 * Float64(x * x))))) * x)
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * (x * x);
	t_1 = t_0 * (x * x);
	t_2 = t_1 * (x * x);
	t_3 = t_2 * (x * x);
	tmp = ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(t$95$3 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Alternative 1: 100.0% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)\\ t_1 := t\_0 \cdot \left(x\_m \cdot x\_m\right)\\ t_2 := t\_1 \cdot \left(x\_m \cdot x\_m\right)\\ t_3 := {\left(x\_m \cdot x\_m\right)}^{-1}\\ t_4 := t\_2 \cdot \left(x\_m \cdot x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 100:\\ \;\;\;\;\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x\_m \cdot x\_m\right)\right) + 0.0424060604 \cdot t\_0\right) + 0.0072644182 \cdot t\_1\right) + 0.0005064034 \cdot t\_2\right) + 0.0001789971 \cdot t\_4}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x\_m \cdot x\_m\right)\right) + 0.2909738639 \cdot t\_0\right) + 0.0694555761 \cdot t\_1\right) + 0.0140005442 \cdot t\_2\right) + 0.0008327945 \cdot t\_4\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(t\_4 \cdot \left(x\_m \cdot x\_m\right)\right)} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.15298196345929074}{{\left(x\_m \cdot x\_m\right)}^{2}}}{x\_m} + \frac{\mathsf{fma}\left({t\_3}^{3}, 11.259630434457211, t\_3 \cdot 0.2514179000665374\right)}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (* x_m x_m) (* x_m x_m)))
        (t_1 (* t_0 (* x_m x_m)))
        (t_2 (* t_1 (* x_m x_m)))
        (t_3 (pow (* x_m x_m) -1.0))
        (t_4 (* t_2 (* x_m x_m))))
   (*
    x_s
    (if (<= x_m 100.0)
      (*
       (/
        (+
         (+
          (+
           (+ (+ 1.0 (* 0.1049934947 (* x_m x_m))) (* 0.0424060604 t_0))
           (* 0.0072644182 t_1))
          (* 0.0005064034 t_2))
         (* 0.0001789971 t_4))
        (+
         (+
          (+
           (+
            (+ (+ 1.0 (* 0.7715471019 (* x_m x_m))) (* 0.2909738639 t_0))
            (* 0.0694555761 t_1))
           (* 0.0140005442 t_2))
          (* 0.0008327945 t_4))
         (* (* 2.0 0.0001789971) (* t_4 (* x_m x_m)))))
       x_m)
      (+
       (/ (+ 0.5 (/ 0.15298196345929074 (pow (* x_m x_m) 2.0))) x_m)
       (/
        (fma (pow t_3 3.0) 11.259630434457211 (* t_3 0.2514179000665374))
        x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = (x_m * x_m) * (x_m * x_m);
	double t_1 = t_0 * (x_m * x_m);
	double t_2 = t_1 * (x_m * x_m);
	double t_3 = pow((x_m * x_m), -1.0);
	double t_4 = t_2 * (x_m * x_m);
	double tmp;
	if (x_m <= 100.0) {
		tmp = ((((((1.0 + (0.1049934947 * (x_m * x_m))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_4)) / ((((((1.0 + (0.7715471019 * (x_m * x_m))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_4)) + ((2.0 * 0.0001789971) * (t_4 * (x_m * x_m))))) * x_m;
	} else {
		tmp = ((0.5 + (0.15298196345929074 / pow((x_m * x_m), 2.0))) / x_m) + (fma(pow(t_3, 3.0), 11.259630434457211, (t_3 * 0.2514179000665374)) / x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(x_m * x_m) * Float64(x_m * x_m))
	t_1 = Float64(t_0 * Float64(x_m * x_m))
	t_2 = Float64(t_1 * Float64(x_m * x_m))
	t_3 = Float64(x_m * x_m) ^ -1.0
	t_4 = Float64(t_2 * Float64(x_m * x_m))
	tmp = 0.0
	if (x_m <= 100.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x_m * x_m))) + Float64(0.0424060604 * t_0)) + Float64(0.0072644182 * t_1)) + Float64(0.0005064034 * t_2)) + Float64(0.0001789971 * t_4)) / Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.7715471019 * Float64(x_m * x_m))) + Float64(0.2909738639 * t_0)) + Float64(0.0694555761 * t_1)) + Float64(0.0140005442 * t_2)) + Float64(0.0008327945 * t_4)) + Float64(Float64(2.0 * 0.0001789971) * Float64(t_4 * Float64(x_m * x_m))))) * x_m);
	else
		tmp = Float64(Float64(Float64(0.5 + Float64(0.15298196345929074 / (Float64(x_m * x_m) ^ 2.0))) / x_m) + Float64(fma((t_3 ^ 3.0), 11.259630434457211, Float64(t_3 * 0.2514179000665374)) / x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 100.0], N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$4), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(t$95$4 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(0.5 + N[(0.15298196345929074 / N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] + N[(N[(N[Power[t$95$3, 3.0], $MachinePrecision] * 11.259630434457211 + N[(t$95$3 * 0.2514179000665374), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 100

   1. Initial program 68.5%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   if 100 < x

   1. Initial program 10.1%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \left(\frac{\frac{1307076337763}{8543989815576}}{{x}^{4}} + \left(\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.15298196345929074}{{\left(x \cdot x\right)}^{2}}}{x} + \frac{\mathsf{fma}\left({\left({\left(x \cdot x\right)}^{-1}\right)}^{3}, 11.259630434457211, {\left(x \cdot x\right)}^{-1} \cdot 0.2514179000665374\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 75.4% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := {\left(x\_m \cdot x\_m\right)}^{5}\\ t_1 := {\left(x\_m \cdot x\_m\right)}^{2}\\ t_2 := {\left(x\_m \cdot x\_m\right)}^{-1}\\ t_3 := {\left(x\_m \cdot x\_m\right)}^{3}\\ t_4 := \mathsf{fma}\left(0.0003579942 \cdot t\_0, x\_m \cdot x\_m, \mathsf{fma}\left(0.0008327945 \cdot {t\_1}^{2}, x\_m \cdot x\_m, \mathsf{fma}\left(0.0140005442 \cdot t\_3, x\_m \cdot x\_m, \mathsf{fma}\left(0.0694555761 \cdot t\_1, x\_m \cdot x\_m, \mathsf{fma}\left(t\_2 + 0.7715471019, x\_m \cdot x\_m, \left(\left(x\_m \cdot x\_m\right) \cdot 0.2909738639\right) \cdot \left(x\_m \cdot x\_m\right)\right)\right)\right)\right)\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 100:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(0.0005064034 \cdot t\_3, x\_m \cdot x\_m, \mathsf{fma}\left(0.0072644182 \cdot t\_1, x\_m \cdot x\_m, \mathsf{fma}\left(t\_2 + 0.1049934947, x\_m \cdot x\_m, t\_1 \cdot 0.0424060604\right)\right)\right)}{t\_4} - \frac{-0.0001789971 \cdot t\_0}{t\_4}\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.15298196345929074}{t\_1}}{x\_m} + \frac{\mathsf{fma}\left({t\_2}^{3}, 11.259630434457211, t\_2 \cdot 0.2514179000665374\right)}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (pow (* x_m x_m) 5.0))
        (t_1 (pow (* x_m x_m) 2.0))
        (t_2 (pow (* x_m x_m) -1.0))
        (t_3 (pow (* x_m x_m) 3.0))
        (t_4
         (fma
          (* 0.0003579942 t_0)
          (* x_m x_m)
          (fma
           (* 0.0008327945 (pow t_1 2.0))
           (* x_m x_m)
           (fma
            (* 0.0140005442 t_3)
            (* x_m x_m)
            (fma
             (* 0.0694555761 t_1)
             (* x_m x_m)
             (fma
              (+ t_2 0.7715471019)
              (* x_m x_m)
              (* (* (* x_m x_m) 0.2909738639) (* x_m x_m)))))))))
   (*
    x_s
    (if (<= x_m 100.0)
      (*
       (-
        (/
         (fma
          (* 0.0005064034 t_3)
          (* x_m x_m)
          (fma
           (* 0.0072644182 t_1)
           (* x_m x_m)
           (fma (+ t_2 0.1049934947) (* x_m x_m) (* t_1 0.0424060604))))
         t_4)
        (/ (* -0.0001789971 t_0) t_4))
       x_m)
      (+
       (/ (+ 0.5 (/ 0.15298196345929074 t_1)) x_m)
       (/
        (fma (pow t_2 3.0) 11.259630434457211 (* t_2 0.2514179000665374))
        x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = pow((x_m * x_m), 5.0);
	double t_1 = pow((x_m * x_m), 2.0);
	double t_2 = pow((x_m * x_m), -1.0);
	double t_3 = pow((x_m * x_m), 3.0);
	double t_4 = fma((0.0003579942 * t_0), (x_m * x_m), fma((0.0008327945 * pow(t_1, 2.0)), (x_m * x_m), fma((0.0140005442 * t_3), (x_m * x_m), fma((0.0694555761 * t_1), (x_m * x_m), fma((t_2 + 0.7715471019), (x_m * x_m), (((x_m * x_m) * 0.2909738639) * (x_m * x_m)))))));
	double tmp;
	if (x_m <= 100.0) {
		tmp = ((fma((0.0005064034 * t_3), (x_m * x_m), fma((0.0072644182 * t_1), (x_m * x_m), fma((t_2 + 0.1049934947), (x_m * x_m), (t_1 * 0.0424060604)))) / t_4) - ((-0.0001789971 * t_0) / t_4)) * x_m;
	} else {
		tmp = ((0.5 + (0.15298196345929074 / t_1)) / x_m) + (fma(pow(t_2, 3.0), 11.259630434457211, (t_2 * 0.2514179000665374)) / x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(x_m * x_m) ^ 5.0
	t_1 = Float64(x_m * x_m) ^ 2.0
	t_2 = Float64(x_m * x_m) ^ -1.0
	t_3 = Float64(x_m * x_m) ^ 3.0
	t_4 = fma(Float64(0.0003579942 * t_0), Float64(x_m * x_m), fma(Float64(0.0008327945 * (t_1 ^ 2.0)), Float64(x_m * x_m), fma(Float64(0.0140005442 * t_3), Float64(x_m * x_m), fma(Float64(0.0694555761 * t_1), Float64(x_m * x_m), fma(Float64(t_2 + 0.7715471019), Float64(x_m * x_m), Float64(Float64(Float64(x_m * x_m) * 0.2909738639) * Float64(x_m * x_m)))))))
	tmp = 0.0
	if (x_m <= 100.0)
		tmp = Float64(Float64(Float64(fma(Float64(0.0005064034 * t_3), Float64(x_m * x_m), fma(Float64(0.0072644182 * t_1), Float64(x_m * x_m), fma(Float64(t_2 + 0.1049934947), Float64(x_m * x_m), Float64(t_1 * 0.0424060604)))) / t_4) - Float64(Float64(-0.0001789971 * t_0) / t_4)) * x_m);
	else
		tmp = Float64(Float64(Float64(0.5 + Float64(0.15298196345929074 / t_1)) / x_m) + Float64(fma((t_2 ^ 3.0), 11.259630434457211, Float64(t_2 * 0.2514179000665374)) / x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.0003579942 * t$95$0), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(0.0008327945 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(0.0140005442 * t$95$3), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(0.0694555761 * t$95$1), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(t$95$2 + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.2909738639), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 100.0], N[(N[(N[(N[(N[(0.0005064034 * t$95$3), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(0.0072644182 * t$95$1), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(t$95$2 + 0.1049934947), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(t$95$1 * 0.0424060604), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] - N[(N[(-0.0001789971 * t$95$0), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(0.5 + N[(0.15298196345929074 / t$95$1), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] + N[(N[(N[Power[t$95$2, 3.0], $MachinePrecision] * 11.259630434457211 + N[(t$95$2 * 0.2514179000665374), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 100

   1. Initial program 68.5%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   4. Applied rewrites 37.4%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0005064034 \cdot {\left(x \cdot x\right)}^{3}, x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot {\left(x \cdot x\right)}^{2}, x \cdot x, \mathsf{fma}\left({\left(x \cdot x\right)}^{-1} + 0.1049934947, x \cdot x, {\left(x \cdot x\right)}^{2} \cdot 0.0424060604\right)\right)\right)}{\mathsf{fma}\left(0.0003579942 \cdot {\left(x \cdot x\right)}^{5}, x \cdot x, \mathsf{fma}\left(0.0008327945 \cdot {\left({\left(x \cdot x\right)}^{2}\right)}^{2}, x \cdot x, \mathsf{fma}\left(0.0140005442 \cdot {\left(x \cdot x\right)}^{3}, x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot {\left(x \cdot x\right)}^{2}, x \cdot x, \mathsf{fma}\left({\left(x \cdot x\right)}^{-1} + 0.7715471019, x \cdot x, \left(\left(x \cdot x\right) \cdot 0.2909738639\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)} - \frac{-0.0001789971 \cdot {\left(x \cdot x\right)}^{5}}{\mathsf{fma}\left(0.0003579942 \cdot {\left(x \cdot x\right)}^{5}, x \cdot x, \mathsf{fma}\left(0.0008327945 \cdot {\left({\left(x \cdot x\right)}^{2}\right)}^{2}, x \cdot x, \mathsf{fma}\left(0.0140005442 \cdot {\left(x \cdot x\right)}^{3}, x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot {\left(x \cdot x\right)}^{2}, x \cdot x, \mathsf{fma}\left({\left(x \cdot x\right)}^{-1} + 0.7715471019, x \cdot x, \left(\left(x \cdot x\right) \cdot 0.2909738639\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}\right)} \cdot x \]

   if 100 < x

   1. Initial program 10.1%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \left(\frac{\frac{1307076337763}{8543989815576}}{{x}^{4}} + \left(\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.15298196345929074}{{\left(x \cdot x\right)}^{2}}}{x} + \frac{\mathsf{fma}\left({\left({\left(x \cdot x\right)}^{-1}\right)}^{3}, 11.259630434457211, {\left(x \cdot x\right)}^{-1} \cdot 0.2514179000665374\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.2% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := {\left(x\_m \cdot x\_m\right)}^{5}\\ t_1 := {\left(x\_m \cdot x\_m\right)}^{2}\\ t_2 := 0.0008327945 \cdot {t\_1}^{2}\\ t_3 := {\left(x\_m \cdot x\_m\right)}^{-1}\\ t_4 := 0.0694555761 \cdot t\_1\\ t_5 := \left(\left(x\_m \cdot x\_m\right) \cdot 0.2909738639\right) \cdot \left(x\_m \cdot x\_m\right)\\ t_6 := {\left(x\_m \cdot x\_m\right)}^{3}\\ t_7 := 0.0140005442 \cdot t\_6\\ t_8 := 0.0003579942 \cdot t\_0\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 100:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(0.0005064034 \cdot t\_6, x\_m \cdot x\_m, \mathsf{fma}\left(0.0072644182 \cdot t\_1, x\_m \cdot x\_m, \mathsf{fma}\left(t\_3 + 0.1049934947, x\_m \cdot x\_m, t\_1 \cdot 0.0424060604\right)\right)\right)}{\mathsf{fma}\left(t\_8, x\_m \cdot x\_m, \mathsf{fma}\left(t\_2, x\_m \cdot x\_m, \mathsf{fma}\left(t\_7, x\_m \cdot x\_m, \mathsf{fma}\left(t\_4, x\_m \cdot x\_m, \mathsf{fma}\left(t\_3 + 0.7715471019, x\_m \cdot x\_m, t\_5\right)\right)\right)\right)\right)} - \frac{-0.0001789971 \cdot t\_0}{\mathsf{fma}\left(t\_8, x\_m \cdot x\_m, \mathsf{fma}\left(t\_2, x\_m \cdot x\_m, \mathsf{fma}\left(t\_7, x\_m \cdot x\_m, \mathsf{fma}\left(t\_4, x\_m \cdot x\_m, \mathsf{fma}\left(0.7715471019, x\_m \cdot x\_m, t\_5\right)\right)\right)\right)\right)}\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.15298196345929074}{t\_1}}{x\_m} + \frac{\mathsf{fma}\left({t\_3}^{3}, 11.259630434457211, t\_3 \cdot 0.2514179000665374\right)}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (pow (* x_m x_m) 5.0))
        (t_1 (pow (* x_m x_m) 2.0))
        (t_2 (* 0.0008327945 (pow t_1 2.0)))
        (t_3 (pow (* x_m x_m) -1.0))
        (t_4 (* 0.0694555761 t_1))
        (t_5 (* (* (* x_m x_m) 0.2909738639) (* x_m x_m)))
        (t_6 (pow (* x_m x_m) 3.0))
        (t_7 (* 0.0140005442 t_6))
        (t_8 (* 0.0003579942 t_0)))
   (*
    x_s
    (if (<= x_m 100.0)
      (*
       (-
        (/
         (fma
          (* 0.0005064034 t_6)
          (* x_m x_m)
          (fma
           (* 0.0072644182 t_1)
           (* x_m x_m)
           (fma (+ t_3 0.1049934947) (* x_m x_m) (* t_1 0.0424060604))))
         (fma
          t_8
          (* x_m x_m)
          (fma
           t_2
           (* x_m x_m)
           (fma
            t_7
            (* x_m x_m)
            (fma
             t_4
             (* x_m x_m)
             (fma (+ t_3 0.7715471019) (* x_m x_m) t_5))))))
        (/
         (* -0.0001789971 t_0)
         (fma
          t_8
          (* x_m x_m)
          (fma
           t_2
           (* x_m x_m)
           (fma
            t_7
            (* x_m x_m)
            (fma t_4 (* x_m x_m) (fma 0.7715471019 (* x_m x_m) t_5)))))))
       x_m)
      (+
       (/ (+ 0.5 (/ 0.15298196345929074 t_1)) x_m)
       (/
        (fma (pow t_3 3.0) 11.259630434457211 (* t_3 0.2514179000665374))
        x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = pow((x_m * x_m), 5.0);
	double t_1 = pow((x_m * x_m), 2.0);
	double t_2 = 0.0008327945 * pow(t_1, 2.0);
	double t_3 = pow((x_m * x_m), -1.0);
	double t_4 = 0.0694555761 * t_1;
	double t_5 = ((x_m * x_m) * 0.2909738639) * (x_m * x_m);
	double t_6 = pow((x_m * x_m), 3.0);
	double t_7 = 0.0140005442 * t_6;
	double t_8 = 0.0003579942 * t_0;
	double tmp;
	if (x_m <= 100.0) {
		tmp = ((fma((0.0005064034 * t_6), (x_m * x_m), fma((0.0072644182 * t_1), (x_m * x_m), fma((t_3 + 0.1049934947), (x_m * x_m), (t_1 * 0.0424060604)))) / fma(t_8, (x_m * x_m), fma(t_2, (x_m * x_m), fma(t_7, (x_m * x_m), fma(t_4, (x_m * x_m), fma((t_3 + 0.7715471019), (x_m * x_m), t_5)))))) - ((-0.0001789971 * t_0) / fma(t_8, (x_m * x_m), fma(t_2, (x_m * x_m), fma(t_7, (x_m * x_m), fma(t_4, (x_m * x_m), fma(0.7715471019, (x_m * x_m), t_5))))))) * x_m;
	} else {
		tmp = ((0.5 + (0.15298196345929074 / t_1)) / x_m) + (fma(pow(t_3, 3.0), 11.259630434457211, (t_3 * 0.2514179000665374)) / x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(x_m * x_m) ^ 5.0
	t_1 = Float64(x_m * x_m) ^ 2.0
	t_2 = Float64(0.0008327945 * (t_1 ^ 2.0))
	t_3 = Float64(x_m * x_m) ^ -1.0
	t_4 = Float64(0.0694555761 * t_1)
	t_5 = Float64(Float64(Float64(x_m * x_m) * 0.2909738639) * Float64(x_m * x_m))
	t_6 = Float64(x_m * x_m) ^ 3.0
	t_7 = Float64(0.0140005442 * t_6)
	t_8 = Float64(0.0003579942 * t_0)
	tmp = 0.0
	if (x_m <= 100.0)
		tmp = Float64(Float64(Float64(fma(Float64(0.0005064034 * t_6), Float64(x_m * x_m), fma(Float64(0.0072644182 * t_1), Float64(x_m * x_m), fma(Float64(t_3 + 0.1049934947), Float64(x_m * x_m), Float64(t_1 * 0.0424060604)))) / fma(t_8, Float64(x_m * x_m), fma(t_2, Float64(x_m * x_m), fma(t_7, Float64(x_m * x_m), fma(t_4, Float64(x_m * x_m), fma(Float64(t_3 + 0.7715471019), Float64(x_m * x_m), t_5)))))) - Float64(Float64(-0.0001789971 * t_0) / fma(t_8, Float64(x_m * x_m), fma(t_2, Float64(x_m * x_m), fma(t_7, Float64(x_m * x_m), fma(t_4, Float64(x_m * x_m), fma(0.7715471019, Float64(x_m * x_m), t_5))))))) * x_m);
	else
		tmp = Float64(Float64(Float64(0.5 + Float64(0.15298196345929074 / t_1)) / x_m) + Float64(fma((t_3 ^ 3.0), 11.259630434457211, Float64(t_3 * 0.2514179000665374)) / x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(0.0008327945 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$4 = N[(0.0694555761 * t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.2909738639), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$7 = N[(0.0140005442 * t$95$6), $MachinePrecision]}, Block[{t$95$8 = N[(0.0003579942 * t$95$0), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 100.0], N[(N[(N[(N[(N[(0.0005064034 * t$95$6), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(0.0072644182 * t$95$1), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(t$95$3 + 0.1049934947), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(t$95$1 * 0.0424060604), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$8 * N[(x$95$m * x$95$m), $MachinePrecision] + N[(t$95$2 * N[(x$95$m * x$95$m), $MachinePrecision] + N[(t$95$7 * N[(x$95$m * x$95$m), $MachinePrecision] + N[(t$95$4 * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(t$95$3 + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.0001789971 * t$95$0), $MachinePrecision] / N[(t$95$8 * N[(x$95$m * x$95$m), $MachinePrecision] + N[(t$95$2 * N[(x$95$m * x$95$m), $MachinePrecision] + N[(t$95$7 * N[(x$95$m * x$95$m), $MachinePrecision] + N[(t$95$4 * N[(x$95$m * x$95$m), $MachinePrecision] + N[(0.7715471019 * N[(x$95$m * x$95$m), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(0.5 + N[(0.15298196345929074 / t$95$1), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] + N[(N[(N[Power[t$95$3, 3.0], $MachinePrecision] * 11.259630434457211 + N[(t$95$3 * 0.2514179000665374), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 100

   1. Initial program 68.5%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   4. Applied rewrites 37.4%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0005064034 \cdot {\left(x \cdot x\right)}^{3}, x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot {\left(x \cdot x\right)}^{2}, x \cdot x, \mathsf{fma}\left({\left(x \cdot x\right)}^{-1} + 0.1049934947, x \cdot x, {\left(x \cdot x\right)}^{2} \cdot 0.0424060604\right)\right)\right)}{\mathsf{fma}\left(0.0003579942 \cdot {\left(x \cdot x\right)}^{5}, x \cdot x, \mathsf{fma}\left(0.0008327945 \cdot {\left({\left(x \cdot x\right)}^{2}\right)}^{2}, x \cdot x, \mathsf{fma}\left(0.0140005442 \cdot {\left(x \cdot x\right)}^{3}, x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot {\left(x \cdot x\right)}^{2}, x \cdot x, \mathsf{fma}\left({\left(x \cdot x\right)}^{-1} + 0.7715471019, x \cdot x, \left(\left(x \cdot x\right) \cdot 0.2909738639\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)} - \frac{-0.0001789971 \cdot {\left(x \cdot x\right)}^{5}}{\mathsf{fma}\left(0.0003579942 \cdot {\left(x \cdot x\right)}^{5}, x \cdot x, \mathsf{fma}\left(0.0008327945 \cdot {\left({\left(x \cdot x\right)}^{2}\right)}^{2}, x \cdot x, \mathsf{fma}\left(0.0140005442 \cdot {\left(x \cdot x\right)}^{3}, x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot {\left(x \cdot x\right)}^{2}, x \cdot x, \mathsf{fma}\left({\left(x \cdot x\right)}^{-1} + 0.7715471019, x \cdot x, \left(\left(x \cdot x\right) \cdot 0.2909738639\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}\right)} \cdot x \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{2532017}{5000000000} \cdot {\left(x \cdot x\right)}^{3}, x \cdot x, \mathsf{fma}\left(\frac{36322091}{5000000000} \cdot {\left(x \cdot x\right)}^{2}, x \cdot x, \mathsf{fma}\left({\left(x \cdot x\right)}^{-1} + \frac{1049934947}{10000000000}, x \cdot x, {\left(x \cdot x\right)}^{2} \cdot \frac{106015151}{2500000000}\right)\right)\right)}{\mathsf{fma}\left(\frac{1789971}{5000000000} \cdot {\left(x \cdot x\right)}^{5}, x \cdot x, \mathsf{fma}\left(\frac{1665589}{2000000000} \cdot {\left({\left(x \cdot x\right)}^{2}\right)}^{2}, x \cdot x, \mathsf{fma}\left(\frac{70002721}{5000000000} \cdot {\left(x \cdot x\right)}^{3}, x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot {\left(x \cdot x\right)}^{2}, x \cdot x, \mathsf{fma}\left({\left(x \cdot x\right)}^{-1} + \frac{7715471019}{10000000000}, x \cdot x, \left(\left(x \cdot x\right) \cdot \frac{2909738639}{10000000000}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)} - \frac{\frac{-1789971}{10000000000} \cdot {\left(x \cdot x\right)}^{5}}{\mathsf{fma}\left(\frac{1789971}{5000000000} \cdot {\left(x \cdot x\right)}^{5}, x \cdot x, \mathsf{fma}\left(\frac{1665589}{2000000000} \cdot {\left({\left(x \cdot x\right)}^{2}\right)}^{2}, x \cdot x, \mathsf{fma}\left(\frac{70002721}{5000000000} \cdot {\left(x \cdot x\right)}^{3}, x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot {\left(x \cdot x\right)}^{2}, x \cdot x, \mathsf{fma}\left(\color{blue}{\frac{7715471019}{10000000000}}, x \cdot x, \left(\left(x \cdot x\right) \cdot \frac{2909738639}{10000000000}\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}\right) \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 37.4%

      \[\leadsto \left(\frac{\mathsf{fma}\left(0.0005064034 \cdot {\left(x \cdot x\right)}^{3}, x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot {\left(x \cdot x\right)}^{2}, x \cdot x, \mathsf{fma}\left({\left(x \cdot x\right)}^{-1} + 0.1049934947, x \cdot x, {\left(x \cdot x\right)}^{2} \cdot 0.0424060604\right)\right)\right)}{\mathsf{fma}\left(0.0003579942 \cdot {\left(x \cdot x\right)}^{5}, x \cdot x, \mathsf{fma}\left(0.0008327945 \cdot {\left({\left(x \cdot x\right)}^{2}\right)}^{2}, x \cdot x, \mathsf{fma}\left(0.0140005442 \cdot {\left(x \cdot x\right)}^{3}, x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot {\left(x \cdot x\right)}^{2}, x \cdot x, \mathsf{fma}\left({\left(x \cdot x\right)}^{-1} + 0.7715471019, x \cdot x, \left(\left(x \cdot x\right) \cdot 0.2909738639\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)} - \frac{-0.0001789971 \cdot {\left(x \cdot x\right)}^{5}}{\mathsf{fma}\left(0.0003579942 \cdot {\left(x \cdot x\right)}^{5}, x \cdot x, \mathsf{fma}\left(0.0008327945 \cdot {\left({\left(x \cdot x\right)}^{2}\right)}^{2}, x \cdot x, \mathsf{fma}\left(0.0140005442 \cdot {\left(x \cdot x\right)}^{3}, x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot {\left(x \cdot x\right)}^{2}, x \cdot x, \mathsf{fma}\left(\color{blue}{0.7715471019}, x \cdot x, \left(\left(x \cdot x\right) \cdot 0.2909738639\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}\right) \cdot x \]

   if 100 < x

   1. Initial program 10.1%

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \left(\frac{\frac{1307076337763}{8543989815576}}{{x}^{4}} + \left(\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.15298196345929074}{{\left(x \cdot x\right)}^{2}}}{x} + \frac{\mathsf{fma}\left({\left({\left(x \cdot x\right)}^{-1}\right)}^{3}, 11.259630434457211, {\left(x \cdot x\right)}^{-1} \cdot 0.2514179000665374\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 51.4% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := {\left(x\_m \cdot x\_m\right)}^{-1}\\ x\_s \cdot \left(\frac{0.5 + \frac{0.15298196345929074}{{\left(x\_m \cdot x\_m\right)}^{2}}}{x\_m} + \frac{\mathsf{fma}\left({t\_0}^{3}, 11.259630434457211, t\_0 \cdot 0.2514179000665374\right)}{x\_m}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (pow (* x_m x_m) -1.0)))
   (*
    x_s
    (+
     (/ (+ 0.5 (/ 0.15298196345929074 (pow (* x_m x_m) 2.0))) x_m)
     (/
      (fma (pow t_0 3.0) 11.259630434457211 (* t_0 0.2514179000665374))
      x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = pow((x_m * x_m), -1.0);
	return x_s * (((0.5 + (0.15298196345929074 / pow((x_m * x_m), 2.0))) / x_m) + (fma(pow(t_0, 3.0), 11.259630434457211, (t_0 * 0.2514179000665374)) / x_m));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(x_m * x_m) ^ -1.0
	return Float64(x_s * Float64(Float64(Float64(0.5 + Float64(0.15298196345929074 / (Float64(x_m * x_m) ^ 2.0))) / x_m) + Float64(fma((t_0 ^ 3.0), 11.259630434457211, Float64(t_0 * 0.2514179000665374)) / x_m)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], -1.0], $MachinePrecision]}, N[(x$95$s * N[(N[(N[(0.5 + N[(0.15298196345929074 / N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] + N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] * 11.259630434457211 + N[(t$95$0 * 0.2514179000665374), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 52.7%

   \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2} + \left(\frac{\frac{1307076337763}{8543989815576}}{{x}^{4}} + \left(\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
4. Step-by-step derivation

5. Applied rewrites 52.9%

   \[\leadsto \color{blue}{\frac{0.5 + \frac{0.15298196345929074}{{\left(x \cdot x\right)}^{2}}}{x} + \frac{\mathsf{fma}\left({\left({\left(x \cdot x\right)}^{-1}\right)}^{3}, 11.259630434457211, {\left(x \cdot x\right)}^{-1} \cdot 0.2514179000665374\right)}{x}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x)))) (* 0.0072644182 (* (* (* x x) (* x x)) (* x x)))) (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1.0 (* 0.7715471019 (* x x))) (* 0.2909738639 (* (* x x) (* x x)))) (* 0.0694555761 (* (* (* x x) (* x x)) (* x x)))) (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2.0 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))