Jmat.Real.dawson

Percentage Accurate: 54.3% → 100.0%

Time: 8.2s

Alternatives: 14

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: 54.3% 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, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot x\_m\\ t_1 := \mathsf{fma}\left(x\_m \cdot x\_m, 0.1049934947, 1\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 100000000:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\left({x\_m}^{10} \cdot x\_m\right) \cdot x\_m, 0.0003579942, \mathsf{fma}\left(0.0008327945, {x\_m}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(t\_0 \cdot 0.0140005442, t\_0, \left(t\_0 \cdot 0.0694555761\right) \cdot x\_m\right), x\_m \cdot x\_m, \mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.2909738639, 0.7715471019\right), x\_m \cdot x\_m, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x\_m}^{10}, 0.0001789971, \mathsf{fma}\left(\mathsf{fma}\left(t\_0 \cdot 0.0005064034, t\_0, \left(t\_0 \cdot 0.0072644182\right) \cdot x\_m\right), x\_m \cdot x\_m, \left(1 + \frac{\left(0.0424060604 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(x\_m \cdot x\_m\right)}{t\_1}\right) \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

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)) (t_1 (fma (* x_m x_m) 0.1049934947 1.0)))
   (*
    x_s
    (if (<= x_m 100000000.0)
      (*
       (/
        x_m
        (fma
         (* (* (pow x_m 10.0) x_m) x_m)
         0.0003579942
         (fma
          0.0008327945
          (pow x_m 10.0)
          (fma
           (fma (* t_0 0.0140005442) t_0 (* (* t_0 0.0694555761) x_m))
           (* x_m x_m)
           (fma
            (fma (* x_m x_m) 0.2909738639 0.7715471019)
            (* x_m x_m)
            1.0)))))
       (fma
        (pow x_m 10.0)
        0.0001789971
        (fma
         (fma (* t_0 0.0005064034) t_0 (* (* t_0 0.0072644182) x_m))
         (* x_m x_m)
         (*
          (+ 1.0 (/ (* (* 0.0424060604 (* x_m x_m)) (* x_m x_m)) t_1))
          t_1))))
      (/ 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 t_1 = fma((x_m * x_m), 0.1049934947, 1.0);
	double tmp;
	if (x_m <= 100000000.0) {
		tmp = (x_m / fma(((pow(x_m, 10.0) * x_m) * x_m), 0.0003579942, fma(0.0008327945, pow(x_m, 10.0), fma(fma((t_0 * 0.0140005442), t_0, ((t_0 * 0.0694555761) * x_m)), (x_m * x_m), fma(fma((x_m * x_m), 0.2909738639, 0.7715471019), (x_m * x_m), 1.0))))) * fma(pow(x_m, 10.0), 0.0001789971, fma(fma((t_0 * 0.0005064034), t_0, ((t_0 * 0.0072644182) * x_m)), (x_m * x_m), ((1.0 + (((0.0424060604 * (x_m * x_m)) * (x_m * x_m)) / t_1)) * t_1)));
	} 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)
	t_1 = fma(Float64(x_m * x_m), 0.1049934947, 1.0)
	tmp = 0.0
	if (x_m <= 100000000.0)
		tmp = Float64(Float64(x_m / fma(Float64(Float64((x_m ^ 10.0) * x_m) * x_m), 0.0003579942, fma(0.0008327945, (x_m ^ 10.0), fma(fma(Float64(t_0 * 0.0140005442), t_0, Float64(Float64(t_0 * 0.0694555761) * x_m)), Float64(x_m * x_m), fma(fma(Float64(x_m * x_m), 0.2909738639, 0.7715471019), Float64(x_m * x_m), 1.0))))) * fma((x_m ^ 10.0), 0.0001789971, fma(fma(Float64(t_0 * 0.0005064034), t_0, Float64(Float64(t_0 * 0.0072644182) * x_m)), Float64(x_m * x_m), Float64(Float64(1.0 + Float64(Float64(Float64(0.0424060604 * Float64(x_m * x_m)) * Float64(x_m * x_m)) / t_1)) * t_1))));
	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]}, Block[{t$95$1 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.1049934947 + 1.0), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 100000000.0], N[(N[(x$95$m / N[(N[(N[(N[Power[x$95$m, 10.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.0003579942 + N[(0.0008327945 * N[Power[x$95$m, 10.0], $MachinePrecision] + N[(N[(N[(t$95$0 * 0.0140005442), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * 0.0694555761), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.2909738639 + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[x$95$m, 10.0], $MachinePrecision] * 0.0001789971 + N[(N[(N[(t$95$0 * 0.0005064034), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * 0.0072644182), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(1.0 + N[(N[(N[(0.0424060604 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$1), $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 54.3%

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

   3. Applied rewrites 54.4%

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

   5. Applied rewrites 54.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2532017}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{36322091}{5000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{106015151}{2500000000}, \frac{1049934947}{10000000000}\right), \color{blue}{x \cdot x}, 1\right)\right)\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2532017}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{36322091}{5000000000}\right) \cdot x\right), x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{106015151}{2500000000}, \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2532017}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{36322091}{5000000000}\right) \cdot x\right), x \cdot x, \color{blue}{1 + \mathsf{fma}\left(x \cdot x, \frac{106015151}{2500000000}, \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right)}\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2532017}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{36322091}{5000000000}\right) \cdot x\right), x \cdot x, 1 + \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{106015151}{2500000000}, \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right)\right)\right) \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2532017}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{36322091}{5000000000}\right) \cdot x\right), x \cdot x, 1 + \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{106015151}{2500000000} + \frac{1049934947}{10000000000}\right)} \cdot \left(x \cdot x\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2532017}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{36322091}{5000000000}\right) \cdot x\right), x \cdot x, 1 + \left(\color{blue}{\frac{106015151}{2500000000} \cdot \left(x \cdot x\right)} + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2532017}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{36322091}{5000000000}\right) \cdot x\right), x \cdot x, 1 + \color{blue}{\left(\frac{1049934947}{10000000000} + \frac{106015151}{2500000000} \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2532017}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{36322091}{5000000000}\right) \cdot x\right), x \cdot x, 1 + \color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1049934947}{10000000000} + \frac{106015151}{2500000000} \cdot \left(x \cdot x\right)\right)}\right)\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2532017}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{36322091}{5000000000}\right) \cdot x\right), x \cdot x, 1 + \color{blue}{\left(\frac{1049934947}{10000000000} \cdot \left(x \cdot x\right) + \left(\frac{106015151}{2500000000} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2532017}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{36322091}{5000000000}\right) \cdot x\right), x \cdot x, 1 + \left(\frac{1049934947}{10000000000} \cdot \left(x \cdot x\right) + \color{blue}{\frac{106015151}{2500000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

      11. associate-+l+ N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2532017}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{36322091}{5000000000}\right) \cdot x\right), x \cdot x, \color{blue}{\left(1 + \frac{1049934947}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{106015151}{2500000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right) \]

      12. sum-to-mult N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2532017}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{36322091}{5000000000}\right) \cdot x\right), x \cdot x, \color{blue}{\left(1 + \frac{\frac{106015151}{2500000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{1 + \frac{1049934947}{10000000000} \cdot \left(x \cdot x\right)}\right) \cdot \left(1 + \frac{1049934947}{10000000000} \cdot \left(x \cdot x\right)\right)}\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2532017}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{36322091}{5000000000}\right) \cdot x\right), x \cdot x, \color{blue}{\left(1 + \frac{\frac{106015151}{2500000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{1 + \frac{1049934947}{10000000000} \cdot \left(x \cdot x\right)}\right) \cdot \left(1 + \frac{1049934947}{10000000000} \cdot \left(x \cdot x\right)\right)}\right)\right) \]

   7. Applied rewrites 54.3%

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

   if 1e8 < x

   1. Initial program 54.3%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 51.0

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

   4. Applied rewrites 51.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left({x\_m}^{10}, 0.0001789971, \mathsf{fma}\left(\mathsf{fma}\left(t\_0 \cdot 0.0072644182, x\_m, t\_0 \cdot \left(\left(0.0005064034 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot x\_m\right)\right) \cdot x\_m, x\_m, \mathsf{fma}\left(\mathsf{fma}\left(0.0424060604, x\_m \cdot x\_m, 0.1049934947\right), x\_m \cdot x\_m, 1\right)\right)\right) \cdot \frac{x\_m}{\mathsf{fma}\left({x\_m}^{12}, 0.0003579942, \mathsf{fma}\left({x\_m}^{10}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(t\_0 \cdot 0.0694555761, x\_m, t\_0 \cdot \left(\left(0.0140005442 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot x\_m\right)\right) \cdot x\_m, x\_m, \mathsf{fma}\left(\mathsf{fma}\left(0.2909738639, x\_m \cdot x\_m, 0.7715471019\right), x\_m \cdot x\_m, 1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

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
        (pow x_m 10.0)
        0.0001789971
        (fma
         (*
          (fma
           (* t_0 0.0072644182)
           x_m
           (* t_0 (* (* 0.0005064034 (* x_m x_m)) x_m)))
          x_m)
         x_m
         (fma (fma 0.0424060604 (* x_m x_m) 0.1049934947) (* x_m x_m) 1.0)))
       (/
        x_m
        (fma
         (pow x_m 12.0)
         0.0003579942
         (fma
          (pow x_m 10.0)
          0.0008327945
          (fma
           (*
            (fma
             (* t_0 0.0694555761)
             x_m
             (* t_0 (* (* 0.0140005442 (* x_m x_m)) x_m)))
            x_m)
           x_m
           (fma
            (fma 0.2909738639 (* x_m 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(pow(x_m, 10.0), 0.0001789971, fma((fma((t_0 * 0.0072644182), x_m, (t_0 * ((0.0005064034 * (x_m * x_m)) * x_m))) * x_m), x_m, fma(fma(0.0424060604, (x_m * x_m), 0.1049934947), (x_m * x_m), 1.0))) * (x_m / fma(pow(x_m, 12.0), 0.0003579942, fma(pow(x_m, 10.0), 0.0008327945, fma((fma((t_0 * 0.0694555761), x_m, (t_0 * ((0.0140005442 * (x_m * x_m)) * x_m))) * x_m), x_m, fma(fma(0.2909738639, (x_m * 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(fma((x_m ^ 10.0), 0.0001789971, fma(Float64(fma(Float64(t_0 * 0.0072644182), x_m, Float64(t_0 * Float64(Float64(0.0005064034 * Float64(x_m * x_m)) * x_m))) * x_m), x_m, fma(fma(0.0424060604, Float64(x_m * x_m), 0.1049934947), Float64(x_m * x_m), 1.0))) * Float64(x_m / fma((x_m ^ 12.0), 0.0003579942, fma((x_m ^ 10.0), 0.0008327945, fma(Float64(fma(Float64(t_0 * 0.0694555761), x_m, Float64(t_0 * Float64(Float64(0.0140005442 * Float64(x_m * x_m)) * x_m))) * x_m), x_m, fma(fma(0.2909738639, Float64(x_m * x_m), 0.7715471019), Float64(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[Power[x$95$m, 10.0], $MachinePrecision] * 0.0001789971 + N[(N[(N[(N[(t$95$0 * 0.0072644182), $MachinePrecision] * x$95$m + N[(t$95$0 * N[(N[(0.0005064034 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + N[(N[(0.0424060604 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.1049934947), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / N[(N[Power[x$95$m, 12.0], $MachinePrecision] * 0.0003579942 + N[(N[Power[x$95$m, 10.0], $MachinePrecision] * 0.0008327945 + N[(N[(N[(N[(t$95$0 * 0.0694555761), $MachinePrecision] * x$95$m + N[(t$95$0 * N[(N[(0.0140005442 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + N[(N[(0.2909738639 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $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 54.3%

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

   3. Applied rewrites 54.4%

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 54.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0072644182, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(0.0005064034 \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right), x \cdot x, 1\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left({x}^{12}, 0.0003579942, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0694555761, x, \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(0.0140005442 \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot x, x, \mathsf{fma}\left(\mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right), x \cdot x, 1\right)\right)\right)\right)}} \]

   if 1e8 < x

   1. Initial program 54.3%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 51.0

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

   4. Applied rewrites 51.0%

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

Alternative 3: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.3:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\left({x\_m}^{10} \cdot x\_m\right) \cdot x\_m, 0.0003579942, \mathsf{fma}\left(0.0008327945, {x\_m}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(t\_0 \cdot 0.0140005442, t\_0, \left(t\_0 \cdot 0.0694555761\right) \cdot x\_m\right), x\_m \cdot x\_m, \mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.2909738639, 0.7715471019\right), x\_m \cdot x\_m, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x\_m}^{10}, 0.0001789971, \mathsf{fma}\left(0.0072644182 \cdot {x\_m}^{4}, x\_m \cdot x\_m, \mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.0424060604, 0.1049934947\right), x\_m \cdot x\_m, 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\left(\left(-\frac{\frac{11.259630434457211}{x\_m \cdot x\_m} + 0.15298196345929074}{t\_0 \cdot x\_m}\right) - 0.5\right) - \frac{0.2514179000665374}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \end{array} \]

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 2.3)
      (*
       (/
        x_m
        (fma
         (* (* (pow x_m 10.0) x_m) x_m)
         0.0003579942
         (fma
          0.0008327945
          (pow x_m 10.0)
          (fma
           (fma (* t_0 0.0140005442) t_0 (* (* t_0 0.0694555761) x_m))
           (* x_m x_m)
           (fma
            (fma (* x_m x_m) 0.2909738639 0.7715471019)
            (* x_m x_m)
            1.0)))))
       (fma
        (pow x_m 10.0)
        0.0001789971
        (fma
         (* 0.0072644182 (pow x_m 4.0))
         (* x_m x_m)
         (fma (fma (* x_m x_m) 0.0424060604 0.1049934947) (* x_m x_m) 1.0))))
      (-
       (/
        (-
         (-
          (-
           (/
            (+ (/ 11.259630434457211 (* x_m x_m)) 0.15298196345929074)
            (* t_0 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 t_0 = (x_m * x_m) * x_m;
	double tmp;
	if (x_m <= 2.3) {
		tmp = (x_m / fma(((pow(x_m, 10.0) * x_m) * x_m), 0.0003579942, fma(0.0008327945, pow(x_m, 10.0), fma(fma((t_0 * 0.0140005442), t_0, ((t_0 * 0.0694555761) * x_m)), (x_m * x_m), fma(fma((x_m * x_m), 0.2909738639, 0.7715471019), (x_m * x_m), 1.0))))) * fma(pow(x_m, 10.0), 0.0001789971, fma((0.0072644182 * pow(x_m, 4.0)), (x_m * x_m), fma(fma((x_m * x_m), 0.0424060604, 0.1049934947), (x_m * x_m), 1.0)));
	} else {
		tmp = -(((-(((11.259630434457211 / (x_m * x_m)) + 0.15298196345929074) / (t_0 * 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)
	t_0 = Float64(Float64(x_m * x_m) * x_m)
	tmp = 0.0
	if (x_m <= 2.3)
		tmp = Float64(Float64(x_m / fma(Float64(Float64((x_m ^ 10.0) * x_m) * x_m), 0.0003579942, fma(0.0008327945, (x_m ^ 10.0), fma(fma(Float64(t_0 * 0.0140005442), t_0, Float64(Float64(t_0 * 0.0694555761) * x_m)), Float64(x_m * x_m), fma(fma(Float64(x_m * x_m), 0.2909738639, 0.7715471019), Float64(x_m * x_m), 1.0))))) * fma((x_m ^ 10.0), 0.0001789971, fma(Float64(0.0072644182 * (x_m ^ 4.0)), Float64(x_m * x_m), fma(fma(Float64(x_m * x_m), 0.0424060604, 0.1049934947), Float64(x_m * x_m), 1.0))));
	else
		tmp = Float64(-Float64(Float64(Float64(Float64(-Float64(Float64(Float64(11.259630434457211 / Float64(x_m * x_m)) + 0.15298196345929074) / Float64(t_0 * 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_] := 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, 2.3], N[(N[(x$95$m / N[(N[(N[(N[Power[x$95$m, 10.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.0003579942 + N[(0.0008327945 * N[Power[x$95$m, 10.0], $MachinePrecision] + N[(N[(N[(t$95$0 * 0.0140005442), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * 0.0694555761), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.2909738639 + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[x$95$m, 10.0], $MachinePrecision] * 0.0001789971 + N[(N[(0.0072644182 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0424060604 + 0.1049934947), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(N[((-N[(N[(N[(11.259630434457211 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.15298196345929074), $MachinePrecision] / N[(t$95$0 * x$95$m), $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 < 2.2999999999999998

   1. Initial program 54.3%

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

   3. Applied rewrites 54.4%

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\color{blue}{\frac{36322091}{5000000000} \cdot {x}^{4}}, x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{106015151}{2500000000}, \frac{1049934947}{10000000000}\right), x \cdot x, 1\right)\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{70002721}{5000000000}, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{694555761}{10000000000}\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{2909738639}{10000000000}, \frac{7715471019}{10000000000}\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \mathsf{fma}\left(\frac{36322091}{5000000000} \cdot \color{blue}{{x}^{4}}, x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{106015151}{2500000000}, \frac{1049934947}{10000000000}\right), x \cdot x, 1\right)\right)\right) \]

      2. lower-pow.f64 53.7

         \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, 0.0003579942, \mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0140005442, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0694555761\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2909738639, 0.7715471019\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(0.0072644182 \cdot {x}^{\color{blue}{4}}, x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0424060604, 0.1049934947\right), x \cdot x, 1\right)\right)\right) \]

   8. Applied rewrites 53.7%

      \[\leadsto \frac{x}{\mathsf{fma}\left(\left({x}^{10} \cdot x\right) \cdot x, 0.0003579942, \mathsf{fma}\left(0.0008327945, {x}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0140005442, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0694555761\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2909738639, 0.7715471019\right), x \cdot x, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{10}, 0.0001789971, \mathsf{fma}\left(\color{blue}{0.0072644182 \cdot {x}^{4}}, x \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0424060604, 0.1049934947\right), x \cdot x, 1\right)\right)\right) \]

   if 2.2999999999999998 < x

   1. Initial program 54.3%

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

   2. Taylor expanded in x around -inf

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.15298196345929074 + 11.259630434457211 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}\right)}{x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 50.8

         \[\leadsto -\frac{-1 \cdot \frac{0.15298196345929074 + 11.259630434457211 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}\right)}{x} \]

   6. Applied rewrites 50.8%

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

Alternative 4: 99.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.015175085973910875, x\_m \cdot x\_m, 0.17858401087518092\right), x\_m \cdot x\_m, 0.6665536072\right), x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\left(\left(-\frac{\frac{11.259630434457211}{x\_m \cdot x\_m} + 0.15298196345929074}{\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m}\right) - 0.5\right) - \frac{0.2514179000665374}{x\_m \cdot x\_m}}{x\_m}\\ \end{array} \end{array} \]

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 2.1)
    (/
     x_m
     (fma
      (fma
       (fma 0.015175085973910875 (* x_m x_m) 0.17858401087518092)
       (* x_m x_m)
       0.6665536072)
      (* x_m x_m)
      1.0))
    (-
     (/
      (-
       (-
        (-
         (/
          (+ (/ 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 <= 2.1) {
		tmp = x_m / fma(fma(fma(0.015175085973910875, (x_m * x_m), 0.17858401087518092), (x_m * x_m), 0.6665536072), (x_m * x_m), 1.0);
	} 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 <= 2.1)
		tmp = Float64(x_m / fma(fma(fma(0.015175085973910875, Float64(x_m * x_m), 0.17858401087518092), Float64(x_m * x_m), 0.6665536072), Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(-Float64(Float64(Float64(Float64(-Float64(Float64(Float64(11.259630434457211 / Float64(x_m * x_m)) + 0.15298196345929074) / Float64(Float64(Float64(x_m * x_m) * 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, 2.1], N[(x$95$m / N[(N[(N[(0.015175085973910875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.17858401087518092), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], (-N[(N[(N[((-N[(N[(N[(11.259630434457211 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.15298196345929074), $MachinePrecision] / N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $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 < 2.10000000000000009

   1. Initial program 54.3%

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

   3. Applied rewrites 54.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + \color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \color{blue}{\left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\color{blue}{\frac{833192009}{1250000000}} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \color{blue}{\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\color{blue}{\frac{2232300135939761477}{12500000000000000000}} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \color{blue}{\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot \color{blue}{{x}^{2}}\right)\right)} \]

   6. Applied rewrites 53.2%

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(0.6665536072 + {x}^{2} \cdot \left(0.17858401087518092 + 0.015175085973910875 \cdot {x}^{2}\right)\right)}} \]

   8. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.015175085973910875, x \cdot x, 0.17858401087518092\right), x \cdot x, 0.6665536072\right), x \cdot x, 1\right)}} \]

   if 2.10000000000000009 < x

   1. Initial program 54.3%

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

   2. Taylor expanded in x around -inf

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.15298196345929074 + 11.259630434457211 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}\right)}{x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 50.8

         \[\leadsto -\frac{-1 \cdot \frac{0.15298196345929074 + 11.259630434457211 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}\right)}{x} \]

   6. Applied rewrites 50.8%

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

Alternative 5: 99.7% accurate, 7.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.015175085973910875, x\_m \cdot x\_m, 0.17858401087518092\right), x\_m \cdot x\_m, 0.6665536072\right), x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\left(0.5 - \frac{0.2514179000665374}{x\_m \cdot x\_m}\right) \cdot x\_m}\\ \end{array} \end{array} \]

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 1.5)
    (/
     x_m
     (fma
      (fma
       (fma 0.015175085973910875 (* x_m x_m) 0.17858401087518092)
       (* x_m x_m)
       0.6665536072)
      (* x_m x_m)
      1.0))
    (/ 0.25 (* (- 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 <= 1.5) {
		tmp = x_m / fma(fma(fma(0.015175085973910875, (x_m * x_m), 0.17858401087518092), (x_m * x_m), 0.6665536072), (x_m * x_m), 1.0);
	} else {
		tmp = 0.25 / ((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 <= 1.5)
		tmp = Float64(x_m / fma(fma(fma(0.015175085973910875, Float64(x_m * x_m), 0.17858401087518092), Float64(x_m * x_m), 0.6665536072), Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(0.25 / Float64(Float64(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, 1.5], N[(x$95$m / N[(N[(N[(0.015175085973910875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.17858401087518092), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.25 / N[(N[(0.5 - N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 54.3%

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

   3. Applied rewrites 54.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + \color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \color{blue}{\left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\color{blue}{\frac{833192009}{1250000000}} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \color{blue}{\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)}\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\color{blue}{\frac{2232300135939761477}{12500000000000000000}} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \color{blue}{\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot \color{blue}{{x}^{2}}\right)\right)} \]

   6. Applied rewrites 53.2%

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(0.6665536072 + {x}^{2} \cdot \left(0.17858401087518092 + 0.015175085973910875 \cdot {x}^{2}\right)\right)}} \]

   8. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.015175085973910875, x \cdot x, 0.17858401087518092\right), x \cdot x, 0.6665536072\right), x \cdot x, 1\right)}} \]

   if 1.5 < x

   1. Initial program 54.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.15298196345929074}{{x}^{4}} + \mathsf{fma}\left(0.2514179000665374, \frac{1}{{x}^{2}}, 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 50.8

         \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]

   7. Applied rewrites 50.8%

      \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. frac-times N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\frac{600041}{2386628}}{{x}^{2}} \cdot \frac{\frac{600041}{2386628}}{{x}^{2}}\right) \cdot 1}{\color{blue}{\left(\frac{1}{2} - \frac{\frac{600041}{2386628}}{{x}^{2}}\right) \cdot x}} \]

   9. Applied rewrites 50.4%

      \[\leadsto \frac{\left(0.25 - \frac{0.2514179000665374}{x \cdot x} \cdot \frac{0.2514179000665374}{x \cdot x}\right) \cdot 1}{\color{blue}{\left(0.5 - \frac{0.2514179000665374}{x \cdot x}\right) \cdot x}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\frac{1}{2} - \frac{\frac{600041}{2386628}}{x \cdot x}\right)} \cdot x} \]
   11. Step-by-step derivation

   12. Applied rewrites 51.7%

       \[\leadsto \frac{0.25}{\color{blue}{\left(0.5 - \frac{0.2514179000665374}{x \cdot x}\right)} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.3:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.17858401087518092, x\_m \cdot x\_m, 0.6665536072\right), x\_m \cdot x\_m, 1\right)} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\left(0.5 - \frac{0.2514179000665374}{x\_m \cdot x\_m}\right) \cdot x\_m}\\ \end{array} \end{array} \]

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 2.3)
    (*
     (/
      1.0
      (fma (fma 0.17858401087518092 (* x_m x_m) 0.6665536072) (* x_m x_m) 1.0))
     x_m)
    (/ 0.25 (* (- 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 <= 2.3) {
		tmp = (1.0 / fma(fma(0.17858401087518092, (x_m * x_m), 0.6665536072), (x_m * x_m), 1.0)) * x_m;
	} else {
		tmp = 0.25 / ((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 <= 2.3)
		tmp = Float64(Float64(1.0 / fma(fma(0.17858401087518092, Float64(x_m * x_m), 0.6665536072), Float64(x_m * x_m), 1.0)) * x_m);
	else
		tmp = Float64(0.25 / Float64(Float64(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, 2.3], N[(N[(1.0 / N[(N[(0.17858401087518092 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(0.25 / N[(N[(0.5 - N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   1. Initial program 54.3%

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

   3. Applied rewrites 54.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + \color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \color{blue}{\left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\color{blue}{\frac{833192009}{1250000000}} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \color{blue}{\frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot \color{blue}{{x}^{2}}\right)} \]

      6. lower-pow.f64 53.4

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(0.6665536072 + 0.17858401087518092 \cdot {x}^{\color{blue}{2}}\right)} \]

   6. Applied rewrites 53.4%

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(0.6665536072 + 0.17858401087518092 \cdot {x}^{2}\right)}} \]

   8. Applied rewrites 53.4%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.17858401087518092, x \cdot x, 0.6665536072\right), x \cdot x, 1\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.4%

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.17858401087518092, x \cdot x, 0.6665536072\right), x \cdot x, 1\right)} \cdot x} \]

   if 2.2999999999999998 < x

   1. Initial program 54.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.15298196345929074}{{x}^{4}} + \mathsf{fma}\left(0.2514179000665374, \frac{1}{{x}^{2}}, 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 50.8

         \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]

   7. Applied rewrites 50.8%

      \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. frac-times N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\frac{600041}{2386628}}{{x}^{2}} \cdot \frac{\frac{600041}{2386628}}{{x}^{2}}\right) \cdot 1}{\color{blue}{\left(\frac{1}{2} - \frac{\frac{600041}{2386628}}{{x}^{2}}\right) \cdot x}} \]

   9. Applied rewrites 50.4%

      \[\leadsto \frac{\left(0.25 - \frac{0.2514179000665374}{x \cdot x} \cdot \frac{0.2514179000665374}{x \cdot x}\right) \cdot 1}{\color{blue}{\left(0.5 - \frac{0.2514179000665374}{x \cdot x}\right) \cdot x}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\frac{1}{2} - \frac{\frac{600041}{2386628}}{x \cdot x}\right)} \cdot x} \]
   11. Step-by-step derivation

   12. Applied rewrites 51.7%

       \[\leadsto \frac{0.25}{\color{blue}{\left(0.5 - \frac{0.2514179000665374}{x \cdot x}\right)} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.6% accurate, 10.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.3:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\mathsf{fma}\left(0.17858401087518092, x\_m \cdot x\_m, 0.6665536072\right), x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\left(0.5 - \frac{0.2514179000665374}{x\_m \cdot x\_m}\right) \cdot x\_m}\\ \end{array} \end{array} \]

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 2.3)
    (/
     x_m
     (fma (fma 0.17858401087518092 (* x_m x_m) 0.6665536072) (* x_m x_m) 1.0))
    (/ 0.25 (* (- 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 <= 2.3) {
		tmp = x_m / fma(fma(0.17858401087518092, (x_m * x_m), 0.6665536072), (x_m * x_m), 1.0);
	} else {
		tmp = 0.25 / ((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 <= 2.3)
		tmp = Float64(x_m / fma(fma(0.17858401087518092, Float64(x_m * x_m), 0.6665536072), Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(0.25 / Float64(Float64(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, 2.3], N[(x$95$m / N[(N[(0.17858401087518092 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.25 / N[(N[(0.5 - N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   1. Initial program 54.3%

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

   3. Applied rewrites 54.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + \color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \color{blue}{\left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\color{blue}{\frac{833192009}{1250000000}} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \color{blue}{\frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot \color{blue}{{x}^{2}}\right)} \]

      6. lower-pow.f64 53.4

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(0.6665536072 + 0.17858401087518092 \cdot {x}^{\color{blue}{2}}\right)} \]

   6. Applied rewrites 53.4%

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(0.6665536072 + 0.17858401087518092 \cdot {x}^{2}\right)}} \]

   8. Applied rewrites 53.4%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.17858401087518092, x \cdot x, 0.6665536072\right), x \cdot x, 1\right)}} \]

   if 2.2999999999999998 < x

   1. Initial program 54.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.15298196345929074}{{x}^{4}} + \mathsf{fma}\left(0.2514179000665374, \frac{1}{{x}^{2}}, 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 50.8

         \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]

   7. Applied rewrites 50.8%

      \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. frac-times N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\frac{600041}{2386628}}{{x}^{2}} \cdot \frac{\frac{600041}{2386628}}{{x}^{2}}\right) \cdot 1}{\color{blue}{\left(\frac{1}{2} - \frac{\frac{600041}{2386628}}{{x}^{2}}\right) \cdot x}} \]

   9. Applied rewrites 50.4%

      \[\leadsto \frac{\left(0.25 - \frac{0.2514179000665374}{x \cdot x} \cdot \frac{0.2514179000665374}{x \cdot x}\right) \cdot 1}{\color{blue}{\left(0.5 - \frac{0.2514179000665374}{x \cdot x}\right) \cdot x}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\frac{1}{2} - \frac{\frac{600041}{2386628}}{x \cdot x}\right)} \cdot x} \]
   11. Step-by-step derivation

   12. Applied rewrites 51.7%

       \[\leadsto \frac{0.25}{\color{blue}{\left(0.5 - \frac{0.2514179000665374}{x \cdot x}\right)} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.265709700396151, -0.6665536072\right), x\_m \cdot x\_m, 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\left(0.5 - \frac{0.2514179000665374}{x\_m \cdot x\_m}\right) \cdot x\_m}\\ \end{array} \end{array} \]

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 1.15)
    (*
     (fma (fma (* x_m x_m) 0.265709700396151 -0.6665536072) (* x_m x_m) 1.0)
     x_m)
    (/ 0.25 (* (- 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 <= 1.15) {
		tmp = fma(fma((x_m * x_m), 0.265709700396151, -0.6665536072), (x_m * x_m), 1.0) * x_m;
	} else {
		tmp = 0.25 / ((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 <= 1.15)
		tmp = Float64(fma(fma(Float64(x_m * x_m), 0.265709700396151, -0.6665536072), Float64(x_m * x_m), 1.0) * x_m);
	else
		tmp = Float64(0.25 / Float64(Float64(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, 1.15], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.265709700396151 + -0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(0.25 / N[(N[(0.5 - N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 54.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)} \cdot x \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)}\right) \cdot x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \color{blue}{\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)}\right) \cdot x \]

   4. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(0.265709700396151 \cdot {x}^{2} - 0.6665536072\right)\right)} \cdot x \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)}\right) \cdot x \]

      2. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + \color{blue}{1}\right) \cdot x \]

      3. lift-*.f64 N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right) \cdot x \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right) \cdot x \]

      5. pow2 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right) \cdot x \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

      8. lower-fma.f64 51.5

         \[\leadsto \mathsf{fma}\left(0.265709700396151 \cdot {x}^{2} - 0.6665536072, \color{blue}{x \cdot x}, 1\right) \cdot x \]

      9. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}, \color{blue}{x} \cdot x, 1\right) \cdot x \]

      10. sub-flip N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), \color{blue}{x} \cdot x, 1\right) \cdot x \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      12. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{3321371254951887171}{12500000000000000000} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{3321371254951887171}{12500000000000000000}, \mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), \color{blue}{x} \cdot x, 1\right) \cdot x \]

      17. metadata-eval 51.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.265709700396151, -0.6665536072\right), x \cdot x, 1\right) \cdot x \]

   6. Applied rewrites 51.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.265709700396151, -0.6665536072\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   if 1.1499999999999999 < x

   1. Initial program 54.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.15298196345929074}{{x}^{4}} + \mathsf{fma}\left(0.2514179000665374, \frac{1}{{x}^{2}}, 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 50.8

         \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]

   7. Applied rewrites 50.8%

      \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. frac-times N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\frac{600041}{2386628}}{{x}^{2}} \cdot \frac{\frac{600041}{2386628}}{{x}^{2}}\right) \cdot 1}{\color{blue}{\left(\frac{1}{2} - \frac{\frac{600041}{2386628}}{{x}^{2}}\right) \cdot x}} \]

   9. Applied rewrites 50.4%

      \[\leadsto \frac{\left(0.25 - \frac{0.2514179000665374}{x \cdot x} \cdot \frac{0.2514179000665374}{x \cdot x}\right) \cdot 1}{\color{blue}{\left(0.5 - \frac{0.2514179000665374}{x \cdot x}\right) \cdot x}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\frac{1}{2} - \frac{\frac{600041}{2386628}}{x \cdot x}\right)} \cdot x} \]
   11. Step-by-step derivation

   12. Applied rewrites 51.7%

       \[\leadsto \frac{0.25}{\color{blue}{\left(0.5 - \frac{0.2514179000665374}{x \cdot x}\right)} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.5% accurate, 12.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.25:\\ \;\;\;\;x\_m \cdot \frac{1}{\mathsf{fma}\left(0.6665536072, x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\left(0.5 - \frac{0.2514179000665374}{x\_m \cdot x\_m}\right) \cdot x\_m}\\ \end{array} \end{array} \]

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 1.25)
    (* x_m (/ 1.0 (fma 0.6665536072 (* x_m x_m) 1.0)))
    (/ 0.25 (* (- 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 <= 1.25) {
		tmp = x_m * (1.0 / fma(0.6665536072, (x_m * x_m), 1.0));
	} else {
		tmp = 0.25 / ((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 <= 1.25)
		tmp = Float64(x_m * Float64(1.0 / fma(0.6665536072, Float64(x_m * x_m), 1.0)));
	else
		tmp = Float64(0.25 / Float64(Float64(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, 1.25], N[(x$95$m * N[(1.0 / N[(0.6665536072 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 / N[(N[(0.5 - N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 54.3%

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

   3. Applied rewrites 54.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + \color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \color{blue}{\left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\color{blue}{\frac{833192009}{1250000000}} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \color{blue}{\frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot \color{blue}{{x}^{2}}\right)} \]

      6. lower-pow.f64 53.4

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(0.6665536072 + 0.17858401087518092 \cdot {x}^{\color{blue}{2}}\right)} \]

   6. Applied rewrites 53.4%

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(0.6665536072 + 0.17858401087518092 \cdot {x}^{2}\right)}} \]

   8. Applied rewrites 53.4%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.17858401087518092, x \cdot x, 0.6665536072\right), x \cdot x, 1\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto x \cdot \frac{1}{\mathsf{fma}\left(\frac{833192009}{1250000000}, \color{blue}{x} \cdot x, 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 56.3%

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

   if 1.25 < x

   1. Initial program 54.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.15298196345929074}{{x}^{4}} + \mathsf{fma}\left(0.2514179000665374, \frac{1}{{x}^{2}}, 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 50.8

         \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]

   7. Applied rewrites 50.8%

      \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. frac-times N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \frac{\frac{600041}{2386628}}{{x}^{2}} \cdot \frac{\frac{600041}{2386628}}{{x}^{2}}\right) \cdot 1}{\color{blue}{\left(\frac{1}{2} - \frac{\frac{600041}{2386628}}{{x}^{2}}\right) \cdot x}} \]

   9. Applied rewrites 50.4%

      \[\leadsto \frac{\left(0.25 - \frac{0.2514179000665374}{x \cdot x} \cdot \frac{0.2514179000665374}{x \cdot x}\right) \cdot 1}{\color{blue}{\left(0.5 - \frac{0.2514179000665374}{x \cdot x}\right) \cdot x}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(\frac{1}{2} - \frac{\frac{600041}{2386628}}{x \cdot x}\right)} \cdot x} \]
   11. Step-by-step derivation

   12. Applied rewrites 51.7%

       \[\leadsto \frac{0.25}{\color{blue}{\left(0.5 - \frac{0.2514179000665374}{x \cdot x}\right)} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 99.5% accurate, 13.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;x\_m \cdot \frac{1}{\mathsf{fma}\left(0.6665536072, x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{x\_m \cdot x\_m} + 0.5}{x\_m}\\ \end{array} \end{array} \]

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 1.15)
    (* x_m (/ 1.0 (fma 0.6665536072 (* x_m x_m) 1.0)))
    (/ (+ (/ 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 <= 1.15) {
		tmp = x_m * (1.0 / fma(0.6665536072, (x_m * x_m), 1.0));
	} 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 <= 1.15)
		tmp = Float64(x_m * Float64(1.0 / fma(0.6665536072, Float64(x_m * x_m), 1.0)));
	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, 1.15], N[(x$95$m * N[(1.0 / N[(0.6665536072 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $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 < 1.1499999999999999

   1. Initial program 54.3%

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

   3. Applied rewrites 54.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + \color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \color{blue}{\left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\color{blue}{\frac{833192009}{1250000000}} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \color{blue}{\frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot \color{blue}{{x}^{2}}\right)} \]

      6. lower-pow.f64 53.4

         \[\leadsto \frac{x}{1 + {x}^{2} \cdot \left(0.6665536072 + 0.17858401087518092 \cdot {x}^{\color{blue}{2}}\right)} \]

   6. Applied rewrites 53.4%

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(0.6665536072 + 0.17858401087518092 \cdot {x}^{2}\right)}} \]

   8. Applied rewrites 53.4%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.17858401087518092, x \cdot x, 0.6665536072\right), x \cdot x, 1\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto x \cdot \frac{1}{\mathsf{fma}\left(\frac{833192009}{1250000000}, \color{blue}{x} \cdot x, 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 56.3%

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

   if 1.1499999999999999 < x

   1. Initial program 54.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.15298196345929074}{{x}^{4}} + \mathsf{fma}\left(0.2514179000665374, \frac{1}{{x}^{2}}, 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 50.8

         \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]

   7. Applied rewrites 50.8%

      \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 50.8

         \[\leadsto \frac{\frac{0.2514179000665374}{{x}^{2}} + 0.5}{x} \]

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 50.8

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

   9. Applied rewrites 50.8%

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

Alternative 11: 99.5% accurate, 14.7× speedup

Math

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

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 -0.6665536072 (* x_m x_m) 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(-0.6665536072, (x_m * x_m), 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(-0.6665536072, Float64(x_m * x_m), 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[(-0.6665536072 * N[(x$95$m * x$95$m), $MachinePrecision] + 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 54.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)} \cdot x \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)}\right) \cdot x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \color{blue}{\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)}\right) \cdot x \]

   4. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(0.265709700396151 \cdot {x}^{2} - 0.6665536072\right)\right)} \cdot x \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)}\right) \cdot x \]

      2. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + \color{blue}{1}\right) \cdot x \]

      3. lift-*.f64 N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right) \cdot x \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right) \cdot x \]

      5. pow2 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right) \cdot x \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

      8. lower-fma.f64 51.5

         \[\leadsto \mathsf{fma}\left(0.265709700396151 \cdot {x}^{2} - 0.6665536072, \color{blue}{x \cdot x}, 1\right) \cdot x \]

      9. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}, \color{blue}{x} \cdot x, 1\right) \cdot x \]

      10. sub-flip N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), \color{blue}{x} \cdot x, 1\right) \cdot x \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      12. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{3321371254951887171}{12500000000000000000} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{3321371254951887171}{12500000000000000000}, \mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), \color{blue}{x} \cdot x, 1\right) \cdot x \]

      17. metadata-eval 51.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.265709700396151, -0.6665536072\right), x \cdot x, 1\right) \cdot x \]

   6. Applied rewrites 51.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.265709700396151, -0.6665536072\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{-833192009}{1250000000}, \color{blue}{x} \cdot x, 1\right) \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 50.7%

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

   if 0.94999999999999996 < x

   1. Initial program 54.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.15298196345929074}{{x}^{4}} + \mathsf{fma}\left(0.2514179000665374, \frac{1}{{x}^{2}}, 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 50.8

         \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]

   7. Applied rewrites 50.8%

      \[\leadsto \frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 50.8

         \[\leadsto \frac{\frac{0.2514179000665374}{{x}^{2}} + 0.5}{x} \]

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 50.8

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

   9. Applied rewrites 50.8%

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

Alternative 12: 99.3% accurate, 16.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.78:\\ \;\;\;\;\mathsf{fma}\left(-0.6665536072, x\_m \cdot x\_m, 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

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 0.78) (* (fma -0.6665536072 (* 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 tmp;
	if (x_m <= 0.78) {
		tmp = fma(-0.6665536072, (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)
	tmp = 0.0
	if (x_m <= 0.78)
		tmp = Float64(fma(-0.6665536072, Float64(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_] := N[(x$95$s * If[LessEqual[x$95$m, 0.78], N[(N[(-0.6665536072 * N[(x$95$m * x$95$m), $MachinePrecision] + 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 < 0.78000000000000003

   1. Initial program 54.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)} \cdot x \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)}\right) \cdot x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \color{blue}{\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)}\right) \cdot x \]

   4. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(0.265709700396151 \cdot {x}^{2} - 0.6665536072\right)\right)} \cdot x \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)}\right) \cdot x \]

      2. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + \color{blue}{1}\right) \cdot x \]

      3. lift-*.f64 N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right) \cdot x \]

      4. lift-pow.f64 N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right) \cdot x \]

      5. pow2 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right) \cdot x \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x \]

      8. lower-fma.f64 51.5

         \[\leadsto \mathsf{fma}\left(0.265709700396151 \cdot {x}^{2} - 0.6665536072, \color{blue}{x \cdot x}, 1\right) \cdot x \]

      9. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}, \color{blue}{x} \cdot x, 1\right) \cdot x \]

      10. sub-flip N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), \color{blue}{x} \cdot x, 1\right) \cdot x \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      12. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{3321371254951887171}{12500000000000000000} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), x \cdot x, 1\right) \cdot x \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{3321371254951887171}{12500000000000000000}, \mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), \color{blue}{x} \cdot x, 1\right) \cdot x \]

      17. metadata-eval 51.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.265709700396151, -0.6665536072\right), x \cdot x, 1\right) \cdot x \]

   6. Applied rewrites 51.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.265709700396151, -0.6665536072\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{-833192009}{1250000000}, \color{blue}{x} \cdot x, 1\right) \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 50.7%

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

   if 0.78000000000000003 < x

   1. Initial program 54.3%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 51.0

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

   4. Applied rewrites 51.0%

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

Alternative 13: 99.0% accurate, 31.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.72:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

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 0.72) 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 <= 0.72) {
		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 <= 0.72d0) 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 <= 0.72) {
		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 <= 0.72:
		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 <= 0.72)
		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 <= 0.72)
		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, 0.72], x$95$m, N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 54.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.8%

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

   if 0.71999999999999997 < x

   1. Initial program 54.3%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 51.0

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

   4. Applied rewrites 51.0%

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

Alternative 14: 51.8% 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 54.3%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 51.8%

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

Reproduce

herbie shell --seed 2025156 
(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))