Jmat.Real.dawson

Percentage Accurate: 53.8% → 100.0%

Time: 9.1s

Alternatives: 9

Speedup: 31.0×

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: 53.8% 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, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 20000000:\\ \;\;\;\;\frac{x\_m}{\frac{\mathsf{fma}\left({x\_m}^{11} \cdot 0.0003579942, x\_m, \mathsf{fma}\left(0.0008327945, {x\_m}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.0694555761, x\_m, 0.0140005442 \cdot t\_0\right), 0.2909738639 \cdot x\_m\right), 0.7715471019\right) \cdot x\_m, x\_m, 1\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, {x\_m}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.0072644182, x\_m, t\_0 \cdot 0.0005064034\right), 0.0424060604 \cdot x\_m\right), 0.1049934947\right) \cdot x\_m, x\_m, 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<= x_m 20000000.0)
      (/
       x_m
       (/
        (fma
         (* (pow x_m 11.0) 0.0003579942)
         x_m
         (fma
          0.0008327945
          (pow x_m 10.0)
          (fma
           (*
            (fma
             x_m
             (fma
              (* x_m x_m)
              (fma 0.0694555761 x_m (* 0.0140005442 t_0))
              (* 0.2909738639 x_m))
             0.7715471019)
            x_m)
           x_m
           1.0)))
        (fma
         0.0001789971
         (pow x_m 10.0)
         (fma
          (*
           (fma
            x_m
            (fma
             (* x_m x_m)
             (fma 0.0072644182 x_m (* t_0 0.0005064034))
             (* 0.0424060604 x_m))
            0.1049934947)
           x_m)
          x_m
          1.0))))
      (/ 0.5 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;
	double tmp;
	if (x_m <= 20000000.0) {
		tmp = x_m / (fma((pow(x_m, 11.0) * 0.0003579942), x_m, fma(0.0008327945, pow(x_m, 10.0), fma((fma(x_m, fma((x_m * x_m), fma(0.0694555761, x_m, (0.0140005442 * t_0)), (0.2909738639 * x_m)), 0.7715471019) * x_m), x_m, 1.0))) / fma(0.0001789971, pow(x_m, 10.0), fma((fma(x_m, fma((x_m * x_m), fma(0.0072644182, x_m, (t_0 * 0.0005064034)), (0.0424060604 * x_m)), 0.1049934947) * x_m), x_m, 1.0)));
	} else {
		tmp = 0.5 / 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) * x_m)
	tmp = 0.0
	if (x_m <= 20000000.0)
		tmp = Float64(x_m / Float64(fma(Float64((x_m ^ 11.0) * 0.0003579942), x_m, fma(0.0008327945, (x_m ^ 10.0), fma(Float64(fma(x_m, fma(Float64(x_m * x_m), fma(0.0694555761, x_m, Float64(0.0140005442 * t_0)), Float64(0.2909738639 * x_m)), 0.7715471019) * x_m), x_m, 1.0))) / fma(0.0001789971, (x_m ^ 10.0), fma(Float64(fma(x_m, fma(Float64(x_m * x_m), fma(0.0072644182, x_m, Float64(t_0 * 0.0005064034)), Float64(0.0424060604 * x_m)), 0.1049934947) * x_m), x_m, 1.0))));
	else
		tmp = Float64(0.5 / 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] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 20000000.0], N[(x$95$m / N[(N[(N[(N[Power[x$95$m, 11.0], $MachinePrecision] * 0.0003579942), $MachinePrecision] * x$95$m + N[(0.0008327945 * N[Power[x$95$m, 10.0], $MachinePrecision] + N[(N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0694555761 * x$95$m + N[(0.0140005442 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.7715471019), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0001789971 * N[Power[x$95$m, 10.0], $MachinePrecision] + N[(N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0072644182 * x$95$m + N[(t$95$0 * 0.0005064034), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.1049934947), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e7

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   5. Applied rewrites 54.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{10} \cdot 0.0003579942, \frac{x \cdot x}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}, \frac{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}\right)}} \]

   7. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left({x}^{11} \cdot 0.0003579942, x, \mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}}} \]

   if 2e7 < x

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   5. Applied rewrites 54.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{10} \cdot 0.0003579942, \frac{x \cdot x}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}, \frac{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}\right)}} \]

   7. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}{\mathsf{fma}\left({x}^{11}, 0.0003579942, \frac{\mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)}{x}\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} \]
   9. Step-by-step derivation

      1. lower-/.f64 51.6

         \[\leadsto \frac{0.5}{\color{blue}{x}} \]

   10. Applied rewrites 51.6%

       \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 20000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0001789971, {x\_m}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.0072644182, x\_m, t\_0 \cdot 0.0005064034\right), 0.0424060604 \cdot x\_m\right), 0.1049934947\right) \cdot x\_m, x\_m, 1\right)\right)}{\mathsf{fma}\left({x\_m}^{11} \cdot 0.0003579942, x\_m, \mathsf{fma}\left(0.0008327945, {x\_m}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.0694555761, x\_m, 0.0140005442 \cdot t\_0\right), 0.2909738639 \cdot x\_m\right), 0.7715471019\right) \cdot x\_m, x\_m, 1\right)\right)\right)} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<= x_m 20000000.0)
      (*
       (/
        (fma
         0.0001789971
         (pow x_m 10.0)
         (fma
          (*
           (fma
            x_m
            (fma
             (* x_m x_m)
             (fma 0.0072644182 x_m (* t_0 0.0005064034))
             (* 0.0424060604 x_m))
            0.1049934947)
           x_m)
          x_m
          1.0))
        (fma
         (* (pow x_m 11.0) 0.0003579942)
         x_m
         (fma
          0.0008327945
          (pow x_m 10.0)
          (fma
           (*
            (fma
             x_m
             (fma
              (* x_m x_m)
              (fma 0.0694555761 x_m (* 0.0140005442 t_0))
              (* 0.2909738639 x_m))
             0.7715471019)
            x_m)
           x_m
           1.0))))
       x_m)
      (/ 0.5 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;
	double tmp;
	if (x_m <= 20000000.0) {
		tmp = (fma(0.0001789971, pow(x_m, 10.0), fma((fma(x_m, fma((x_m * x_m), fma(0.0072644182, x_m, (t_0 * 0.0005064034)), (0.0424060604 * x_m)), 0.1049934947) * x_m), x_m, 1.0)) / fma((pow(x_m, 11.0) * 0.0003579942), x_m, fma(0.0008327945, pow(x_m, 10.0), fma((fma(x_m, fma((x_m * x_m), fma(0.0694555761, x_m, (0.0140005442 * t_0)), (0.2909738639 * x_m)), 0.7715471019) * x_m), x_m, 1.0)))) * x_m;
	} else {
		tmp = 0.5 / 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) * x_m)
	tmp = 0.0
	if (x_m <= 20000000.0)
		tmp = Float64(Float64(fma(0.0001789971, (x_m ^ 10.0), fma(Float64(fma(x_m, fma(Float64(x_m * x_m), fma(0.0072644182, x_m, Float64(t_0 * 0.0005064034)), Float64(0.0424060604 * x_m)), 0.1049934947) * x_m), x_m, 1.0)) / fma(Float64((x_m ^ 11.0) * 0.0003579942), x_m, fma(0.0008327945, (x_m ^ 10.0), fma(Float64(fma(x_m, fma(Float64(x_m * x_m), fma(0.0694555761, x_m, Float64(0.0140005442 * t_0)), Float64(0.2909738639 * x_m)), 0.7715471019) * x_m), x_m, 1.0)))) * x_m);
	else
		tmp = Float64(0.5 / 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] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 20000000.0], N[(N[(N[(0.0001789971 * N[Power[x$95$m, 10.0], $MachinePrecision] + N[(N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0072644182 * x$95$m + N[(t$95$0 * 0.0005064034), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.1049934947), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[x$95$m, 11.0], $MachinePrecision] * 0.0003579942), $MachinePrecision] * x$95$m + N[(0.0008327945 * N[Power[x$95$m, 10.0], $MachinePrecision] + N[(N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0694555761 * x$95$m + N[(0.0140005442 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.7715471019), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e7

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   5. Applied rewrites 54.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{10} \cdot 0.0003579942, \frac{x \cdot x}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}, \frac{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}\right)}} \]

   7. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}{\mathsf{fma}\left({x}^{11} \cdot 0.0003579942, x, \mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)} \cdot x} \]

   if 2e7 < x

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   5. Applied rewrites 54.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{10} \cdot 0.0003579942, \frac{x \cdot x}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}, \frac{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}\right)}} \]

   7. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}{\mathsf{fma}\left({x}^{11}, 0.0003579942, \frac{\mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)}{x}\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} \]
   9. Step-by-step derivation

      1. lower-/.f64 51.6

         \[\leadsto \frac{0.5}{\color{blue}{x}} \]

   10. Applied rewrites 51.6%

       \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 100000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0001789971, {x\_m}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.0072644182, x\_m, t\_0 \cdot 0.0005064034\right), 0.0424060604 \cdot x\_m\right), 0.1049934947\right) \cdot x\_m, x\_m, 1\right)\right) \cdot x\_m}{\mathsf{fma}\left({x\_m}^{11} \cdot 0.0003579942, x\_m, \mathsf{fma}\left(0.0008327945, {x\_m}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.0694555761, x\_m, 0.0140005442 \cdot t\_0\right), 0.2909738639 \cdot x\_m\right), 0.7715471019\right) \cdot x\_m, x\_m, 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<= x_m 100000000.0)
      (/
       (*
        (fma
         0.0001789971
         (pow x_m 10.0)
         (fma
          (*
           (fma
            x_m
            (fma
             (* x_m x_m)
             (fma 0.0072644182 x_m (* t_0 0.0005064034))
             (* 0.0424060604 x_m))
            0.1049934947)
           x_m)
          x_m
          1.0))
        x_m)
       (fma
        (* (pow x_m 11.0) 0.0003579942)
        x_m
        (fma
         0.0008327945
         (pow x_m 10.0)
         (fma
          (*
           (fma
            x_m
            (fma
             (* x_m x_m)
             (fma 0.0694555761 x_m (* 0.0140005442 t_0))
             (* 0.2909738639 x_m))
            0.7715471019)
           x_m)
          x_m
          1.0))))
      (/ 0.5 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;
	double tmp;
	if (x_m <= 100000000.0) {
		tmp = (fma(0.0001789971, pow(x_m, 10.0), fma((fma(x_m, fma((x_m * x_m), fma(0.0072644182, x_m, (t_0 * 0.0005064034)), (0.0424060604 * x_m)), 0.1049934947) * x_m), x_m, 1.0)) * x_m) / fma((pow(x_m, 11.0) * 0.0003579942), x_m, fma(0.0008327945, pow(x_m, 10.0), fma((fma(x_m, fma((x_m * x_m), fma(0.0694555761, x_m, (0.0140005442 * t_0)), (0.2909738639 * x_m)), 0.7715471019) * x_m), x_m, 1.0)));
	} else {
		tmp = 0.5 / 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) * x_m)
	tmp = 0.0
	if (x_m <= 100000000.0)
		tmp = Float64(Float64(fma(0.0001789971, (x_m ^ 10.0), fma(Float64(fma(x_m, fma(Float64(x_m * x_m), fma(0.0072644182, x_m, Float64(t_0 * 0.0005064034)), Float64(0.0424060604 * x_m)), 0.1049934947) * x_m), x_m, 1.0)) * x_m) / fma(Float64((x_m ^ 11.0) * 0.0003579942), x_m, fma(0.0008327945, (x_m ^ 10.0), fma(Float64(fma(x_m, fma(Float64(x_m * x_m), fma(0.0694555761, x_m, Float64(0.0140005442 * t_0)), Float64(0.2909738639 * x_m)), 0.7715471019) * x_m), x_m, 1.0))));
	else
		tmp = Float64(0.5 / 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] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 100000000.0], N[(N[(N[(0.0001789971 * N[Power[x$95$m, 10.0], $MachinePrecision] + N[(N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0072644182 * x$95$m + N[(t$95$0 * 0.0005064034), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.1049934947), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(N[(N[Power[x$95$m, 11.0], $MachinePrecision] * 0.0003579942), $MachinePrecision] * x$95$m + N[(0.0008327945 * N[Power[x$95$m, 10.0], $MachinePrecision] + N[(N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0694555761 * x$95$m + N[(0.0140005442 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.7715471019), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e8

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   5. Applied rewrites 54.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{10} \cdot 0.0003579942, \frac{x \cdot x}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}, \frac{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}\right)}} \]

   7. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right) \cdot x}{\mathsf{fma}\left({x}^{11} \cdot 0.0003579942, x, \mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}} \]

   if 1e8 < x

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   5. Applied rewrites 54.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{10} \cdot 0.0003579942, \frac{x \cdot x}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}, \frac{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}\right)}} \]

   7. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}{\mathsf{fma}\left({x}^{11}, 0.0003579942, \frac{\mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)}{x}\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} \]
   9. Step-by-step derivation

      1. lower-/.f64 51.6

         \[\leadsto \frac{0.5}{\color{blue}{x}} \]

   10. Applied rewrites 51.6%

       \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, -0.6665536072, 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;-\frac{\left(\left(-\frac{\frac{11.259630434457211}{x\_m \cdot x\_m} + 0.15298196345929074}{\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)}\right) - 0.5\right) - \frac{0.2514179000665374}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.2)
    (* (fma (* x_m x_m) -0.6665536072 1.0) x_m)
    (-
     (/
      (-
       (-
        (-
         (/
          (+ (/ 11.259630434457211 (* x_m x_m)) 0.15298196345929074)
          (* (* x_m x_m) (* x_m x_m))))
        0.5)
       (/ 0.2514179000665374 (* x_m x_m)))
      x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.2) {
		tmp = fma((x_m * x_m), -0.6665536072, 1.0) * x_m;
	} else {
		tmp = -(((-(((11.259630434457211 / (x_m * x_m)) + 0.15298196345929074) / ((x_m * x_m) * (x_m * x_m))) - 0.5) - (0.2514179000665374 / (x_m * x_m))) / x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.2)
		tmp = Float64(fma(Float64(x_m * x_m), -0.6665536072, 1.0) * x_m);
	else
		tmp = Float64(-Float64(Float64(Float64(Float64(-Float64(Float64(Float64(11.259630434457211 / Float64(x_m * x_m)) + 0.15298196345929074) / Float64(Float64(x_m * x_m) * Float64(x_m * x_m)))) - 0.5) - Float64(0.2514179000665374 / Float64(x_m * x_m))) / 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_] := N[(x$95$s * If[LessEqual[x$95$m, 1.2], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.6665536072 + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], (-N[(N[(N[((-N[(N[(N[(11.259630434457211 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.15298196345929074), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) - 0.5), $MachinePrecision] - N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision])]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 53.8%

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

   3. Applied rewrites 54.6%

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

   5. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}{\mathsf{fma}\left(\left({x}^{10} \cdot 0.0003579942\right) \cdot x, x, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)} \cdot x} \]

   7. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left({x}^{11} \cdot 0.0003579942, x, \mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}}} \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{-833192009}{1250000000} \cdot {x}^{2}\right)} \cdot x \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{-833192009}{1250000000} \cdot {x}^{2} + \color{blue}{1}\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \frac{-833192009}{1250000000} + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-833192009}{1250000000}}, 1\right) \cdot x \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-833192009}{1250000000}, 1\right) \cdot x \]

      5. lift-*.f64 50.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, -0.6665536072, 1\right) \cdot x \]

   10. Applied rewrites 50.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.6665536072, 1\right)} \cdot x \]

   if 1.19999999999999996 < x

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   5. Applied rewrites 54.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{10} \cdot 0.0003579942, \frac{x \cdot x}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}, \frac{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}\right)}} \]

   7. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}{\mathsf{fma}\left({x}^{11}, 0.0003579942, \frac{\mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)}{x}\right)}} \]

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x} \]

   10. Applied rewrites 51.4%

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

Alternative 5: 99.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, -0.6665536072, 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;-\frac{\left(-\frac{\frac{0.15298196345929074}{x\_m \cdot x\_m} + 0.2514179000665374}{x\_m \cdot x\_m}\right) - 0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.0)
    (* (fma (* x_m x_m) -0.6665536072 1.0) x_m)
    (-
     (/
      (-
       (-
        (/
         (+ (/ 0.15298196345929074 (* x_m x_m)) 0.2514179000665374)
         (* x_m x_m)))
       0.5)
      x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = fma((x_m * x_m), -0.6665536072, 1.0) * x_m;
	} else {
		tmp = -((-(((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) - 0.5) / x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(fma(Float64(x_m * x_m), -0.6665536072, 1.0) * x_m);
	else
		tmp = Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(0.15298196345929074 / Float64(x_m * x_m)) + 0.2514179000665374) / Float64(x_m * x_m))) - 0.5) / 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_] := N[(x$95$s * If[LessEqual[x$95$m, 1.0], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.6665536072 + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], (-N[(N[((-N[(N[(N[(0.15298196345929074 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.2514179000665374), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]) - 0.5), $MachinePrecision] / x$95$m), $MachinePrecision])]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 53.8%

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

   3. Applied rewrites 54.6%

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

   5. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}{\mathsf{fma}\left(\left({x}^{10} \cdot 0.0003579942\right) \cdot x, x, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)} \cdot x} \]

   7. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left({x}^{11} \cdot 0.0003579942, x, \mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}}} \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{-833192009}{1250000000} \cdot {x}^{2}\right)} \cdot x \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{-833192009}{1250000000} \cdot {x}^{2} + \color{blue}{1}\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \frac{-833192009}{1250000000} + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-833192009}{1250000000}}, 1\right) \cdot x \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-833192009}{1250000000}, 1\right) \cdot x \]

      5. lift-*.f64 50.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, -0.6665536072, 1\right) \cdot x \]

   10. Applied rewrites 50.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.6665536072, 1\right)} \cdot x \]

   if 1 < x

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   5. Applied rewrites 54.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{10} \cdot 0.0003579942, \frac{x \cdot x}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}, \frac{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}\right)}} \]

   7. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}{\mathsf{fma}\left({x}^{11}, 0.0003579942, \frac{\mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)}{x}\right)}} \]

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}}{x}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}}{x}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}}{x} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}}{x} \]

   10. Applied rewrites 51.4%

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

Alternative 6: 99.5% accurate, 14.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.95:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, -0.6665536072, 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{x\_m \cdot x\_m} + 0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.95)
    (* (fma (* x_m x_m) -0.6665536072 1.0) x_m)
    (/ (+ (/ 0.2514179000665374 (* x_m x_m)) 0.5) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.95) {
		tmp = fma((x_m * x_m), -0.6665536072, 1.0) * x_m;
	} else {
		tmp = ((0.2514179000665374 / (x_m * x_m)) + 0.5) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.95)
		tmp = Float64(fma(Float64(x_m * x_m), -0.6665536072, 1.0) * x_m);
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / Float64(x_m * x_m)) + 0.5) / 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_] := N[(x$95$s * If[LessEqual[x$95$m, 0.95], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.6665536072 + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 53.8%

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

   3. Applied rewrites 54.6%

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

   5. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}{\mathsf{fma}\left(\left({x}^{10} \cdot 0.0003579942\right) \cdot x, x, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)} \cdot x} \]

   7. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left({x}^{11} \cdot 0.0003579942, x, \mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}}} \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{-833192009}{1250000000} \cdot {x}^{2}\right)} \cdot x \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{-833192009}{1250000000} \cdot {x}^{2} + \color{blue}{1}\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \frac{-833192009}{1250000000} + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-833192009}{1250000000}}, 1\right) \cdot x \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-833192009}{1250000000}, 1\right) \cdot x \]

      5. lift-*.f64 50.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, -0.6665536072, 1\right) \cdot x \]

   10. Applied rewrites 50.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.6665536072, 1\right)} \cdot x \]

   if 0.94999999999999996 < x

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   5. Applied rewrites 54.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{10} \cdot 0.0003579942, \frac{x \cdot x}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}, \frac{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}\right)}} \]

   7. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}{\mathsf{fma}\left({x}^{11}, 0.0003579942, \frac{\mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)}{x}\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{\color{blue}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}}{x} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}}{x} \]

      4. mult-flip-rev N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628}}{{x}^{2}} + \frac{1}{2}}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628}}{{x}^{2}} + \frac{1}{2}}{x} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{\frac{600041}{2386628}}{x \cdot x} + \frac{1}{2}}{x} \]

      7. lift-*.f64 51.4

         \[\leadsto \frac{\frac{0.2514179000665374}{x \cdot x} + 0.5}{x} \]

   10. Applied rewrites 51.4%

       \[\leadsto \color{blue}{\frac{\frac{0.2514179000665374}{x \cdot x} + 0.5}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.2% accurate, 16.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, -0.6665536072, 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.8) (* (fma (* x_m x_m) -0.6665536072 1.0) x_m) (/ 0.5 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.8) {
		tmp = fma((x_m * x_m), -0.6665536072, 1.0) * x_m;
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.8)
		tmp = Float64(fma(Float64(x_m * x_m), -0.6665536072, 1.0) * x_m);
	else
		tmp = Float64(0.5 / 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_] := N[(x$95$s * If[LessEqual[x$95$m, 0.8], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.6665536072 + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 53.8%

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

   3. Applied rewrites 54.6%

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

   5. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}{\mathsf{fma}\left(\left({x}^{10} \cdot 0.0003579942\right) \cdot x, x, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)} \cdot x} \]

   7. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left({x}^{11} \cdot 0.0003579942, x, \mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}}} \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{-833192009}{1250000000} \cdot {x}^{2}\right)} \cdot x \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{-833192009}{1250000000} \cdot {x}^{2} + \color{blue}{1}\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \frac{-833192009}{1250000000} + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-833192009}{1250000000}}, 1\right) \cdot x \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{-833192009}{1250000000}, 1\right) \cdot x \]

      5. lift-*.f64 50.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, -0.6665536072, 1\right) \cdot x \]

   10. Applied rewrites 50.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.6665536072, 1\right)} \cdot x \]

   if 0.80000000000000004 < x

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   5. Applied rewrites 54.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{10} \cdot 0.0003579942, \frac{x \cdot x}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}, \frac{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}\right)}} \]

   7. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}{\mathsf{fma}\left({x}^{11}, 0.0003579942, \frac{\mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)}{x}\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} \]
   9. Step-by-step derivation

      1. lower-/.f64 51.6

         \[\leadsto \frac{0.5}{\color{blue}{x}} \]

   10. Applied rewrites 51.6%

       \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.9% accurate, 31.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.7:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 0.7) x_m (/ 0.5 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.7) {
		tmp = x_m;
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.7d0) then
        tmp = x_m
    else
        tmp = 0.5d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.7) {
		tmp = x_m;
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.7:
		tmp = x_m
	else:
		tmp = 0.5 / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.7)
		tmp = x_m;
	else
		tmp = Float64(0.5 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.7)
		tmp = x_m;
	else
		tmp = 0.5 / x_m;
	end
	tmp_2 = 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_] := N[(x$95$s * If[LessEqual[x$95$m, 0.7], x$95$m, N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   5. Applied rewrites 54.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{10} \cdot 0.0003579942, \frac{x \cdot x}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}, \frac{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}\right)}} \]

   7. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}{\mathsf{fma}\left({x}^{11}, 0.0003579942, \frac{\mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)}{x}\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.1%

       \[\leadsto \color{blue}{x} \]

   if 0.69999999999999996 < x

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   5. Applied rewrites 54.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{10} \cdot 0.0003579942, \frac{x \cdot x}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}, \frac{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}\right)}} \]

   7. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}{\mathsf{fma}\left({x}^{11}, 0.0003579942, \frac{\mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)}{x}\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} \]
   9. Step-by-step derivation

      1. lower-/.f64 51.6

         \[\leadsto \frac{0.5}{\color{blue}{x}} \]

   10. Applied rewrites 51.6%

       \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.1% accurate, 253.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * x_m;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 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_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

3. Applied rewrites 53.9%

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

5. Applied rewrites 54.6%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{10} \cdot 0.0003579942, \frac{x \cdot x}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}, \frac{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x \cdot x, \left(\left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x \cdot x, \left(\left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}\right)}} \]

7. Applied rewrites 54.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0005064034\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}{\mathsf{fma}\left({x}^{11}, 0.0003579942, \frac{\mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761, x, 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)}{x}\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
9. Step-by-step derivation

10. Applied rewrites 51.1%

    \[\leadsto \color{blue}{x} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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))