Jmat.Real.dawson

Percentage Accurate: 54.7% → 100.0%

Time: 5.4s

Alternatives: 8

Speedup: 15.4×

Specification

Math

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

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: 54.7% accurate, 1.0× speedup

Math

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

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, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 \cdot t\_0\\ t_3 := t\_2 \cdot t\_0\\ t_4 := t\_3 \cdot t\_0\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 40000:\\ \;\;\;\;\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot t\_0\right) + 0.0424060604 \cdot t\_1\right) + 0.0072644182 \cdot t\_2\right) + 0.0005064034 \cdot t\_3\right) + 0.0001789971 \cdot t\_4}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot t\_0\right) + 0.2909738639 \cdot t\_1\right) + 0.0694555761 \cdot t\_2\right) + 0.0140005442 \cdot t\_3\right) + 0.0008327945 \cdot t\_4\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(t\_4 \cdot t\_0\right)} \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{1.0056716002661497}{t\_0}\right) \cdot \left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x)))
        (t_1 (* t_0 t_0))
        (t_2 (* t_1 t_0))
        (t_3 (* t_2 t_0))
        (t_4 (* t_3 t_0)))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 40000.0)
      (*
       (/
        (+
         (+
          (+
           (+ (+ 1.0 (* 0.1049934947 t_0)) (* 0.0424060604 t_1))
           (* 0.0072644182 t_2))
          (* 0.0005064034 t_3))
         (* 0.0001789971 t_4))
        (+
         (+
          (+
           (+
            (+ (+ 1.0 (* 0.7715471019 t_0)) (* 0.2909738639 t_1))
            (* 0.0694555761 t_2))
           (* 0.0140005442 t_3))
          (* 0.0008327945 t_4))
         (* (* 2.0 0.0001789971) (* t_4 t_0))))
       (fabs x))
      (/ 1.0 (* (- 2.0 (/ 1.0056716002661497 t_0)) (fabs x)))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 * t_0;
	double t_3 = t_2 * t_0;
	double t_4 = t_3 * t_0;
	double tmp;
	if (fabs(x) <= 40000.0) {
		tmp = ((((((1.0 + (0.1049934947 * t_0)) + (0.0424060604 * t_1)) + (0.0072644182 * t_2)) + (0.0005064034 * t_3)) + (0.0001789971 * t_4)) / ((((((1.0 + (0.7715471019 * t_0)) + (0.2909738639 * t_1)) + (0.0694555761 * t_2)) + (0.0140005442 * t_3)) + (0.0008327945 * t_4)) + ((2.0 * 0.0001789971) * (t_4 * t_0)))) * fabs(x);
	} else {
		tmp = 1.0 / ((2.0 - (1.0056716002661497 / t_0)) * fabs(x));
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * Math.abs(x);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 * t_0;
	double t_3 = t_2 * t_0;
	double t_4 = t_3 * t_0;
	double tmp;
	if (Math.abs(x) <= 40000.0) {
		tmp = ((((((1.0 + (0.1049934947 * t_0)) + (0.0424060604 * t_1)) + (0.0072644182 * t_2)) + (0.0005064034 * t_3)) + (0.0001789971 * t_4)) / ((((((1.0 + (0.7715471019 * t_0)) + (0.2909738639 * t_1)) + (0.0694555761 * t_2)) + (0.0140005442 * t_3)) + (0.0008327945 * t_4)) + ((2.0 * 0.0001789971) * (t_4 * t_0)))) * Math.abs(x);
	} else {
		tmp = 1.0 / ((2.0 - (1.0056716002661497 / t_0)) * Math.abs(x));
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) * math.fabs(x)
	t_1 = t_0 * t_0
	t_2 = t_1 * t_0
	t_3 = t_2 * t_0
	t_4 = t_3 * t_0
	tmp = 0
	if math.fabs(x) <= 40000.0:
		tmp = ((((((1.0 + (0.1049934947 * t_0)) + (0.0424060604 * t_1)) + (0.0072644182 * t_2)) + (0.0005064034 * t_3)) + (0.0001789971 * t_4)) / ((((((1.0 + (0.7715471019 * t_0)) + (0.2909738639 * t_1)) + (0.0694555761 * t_2)) + (0.0140005442 * t_3)) + (0.0008327945 * t_4)) + ((2.0 * 0.0001789971) * (t_4 * t_0)))) * math.fabs(x)
	else:
		tmp = 1.0 / ((2.0 - (1.0056716002661497 / t_0)) * math.fabs(x))
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(t_1 * t_0)
	t_3 = Float64(t_2 * t_0)
	t_4 = Float64(t_3 * t_0)
	tmp = 0.0
	if (abs(x) <= 40000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * t_0)) + Float64(0.0424060604 * t_1)) + Float64(0.0072644182 * t_2)) + Float64(0.0005064034 * t_3)) + Float64(0.0001789971 * t_4)) / Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.7715471019 * t_0)) + Float64(0.2909738639 * t_1)) + Float64(0.0694555761 * t_2)) + Float64(0.0140005442 * t_3)) + Float64(0.0008327945 * t_4)) + Float64(Float64(2.0 * 0.0001789971) * Float64(t_4 * t_0)))) * abs(x));
	else
		tmp = Float64(1.0 / Float64(Float64(2.0 - Float64(1.0056716002661497 / t_0)) * abs(x)));
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) * abs(x);
	t_1 = t_0 * t_0;
	t_2 = t_1 * t_0;
	t_3 = t_2 * t_0;
	t_4 = t_3 * t_0;
	tmp = 0.0;
	if (abs(x) <= 40000.0)
		tmp = ((((((1.0 + (0.1049934947 * t_0)) + (0.0424060604 * t_1)) + (0.0072644182 * t_2)) + (0.0005064034 * t_3)) + (0.0001789971 * t_4)) / ((((((1.0 + (0.7715471019 * t_0)) + (0.2909738639 * t_1)) + (0.0694555761 * t_2)) + (0.0140005442 * t_3)) + (0.0008327945 * t_4)) + ((2.0 * 0.0001789971) * (t_4 * t_0)))) * abs(x);
	else
		tmp = 1.0 / ((2.0 - (1.0056716002661497 / t_0)) * abs(x));
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 40000.0], N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$4), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(2.0 - N[(1.0056716002661497 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4e4

   1. Initial program 54.7%

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

   if 4e4 < x

   1. Initial program 54.7%

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

   2. 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 51.3%

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

   6. Applied rewrites 51.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\left(\left(\frac{0.15298196345929074}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \frac{-0.2514179000665374}{x \cdot x}\right) - \frac{-11.259630434457211}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right) - -0.5}}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(2 - \color{blue}{\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(2 - \frac{600041}{596657} \cdot \color{blue}{\frac{1}{{x}^{2}}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(2 - \frac{600041}{596657} \cdot \frac{1}{\color{blue}{{x}^{2}}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{\color{blue}{2}}}\right)} \]

      5. lower-pow.f64 52.2%

         \[\leadsto \frac{1}{x \cdot \left(2 - 1.0056716002661497 \cdot \frac{1}{{x}^{2}}\right)} \]

   9. Applied rewrites 52.2%

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 - 1.0056716002661497 \cdot \frac{1}{{x}^{2}}\right)}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(2 - \color{blue}{\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{\left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

       3. lower-*.f64 52.2%

          \[\leadsto \frac{1}{\left(2 - 1.0056716002661497 \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{1}{\left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

       6. lift-pow.f64 N/A

          \[\leadsto \frac{1}{\left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

       7. pow2 N/A

          \[\leadsto \frac{1}{\left(2 - \frac{600041}{596657} \cdot \frac{1}{x \cdot x}\right) \cdot x} \]

       8. mult-flip-rev N/A

          \[\leadsto \frac{1}{\left(2 - \frac{\frac{600041}{596657}}{x \cdot x}\right) \cdot x} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{1}{\left(2 - \frac{\frac{600041}{596657}}{x \cdot x}\right) \cdot x} \]

       10. lift-*.f64 52.2%

           \[\leadsto \frac{1}{\left(2 - \frac{1.0056716002661497}{x \cdot x}\right) \cdot x} \]

   11. Applied rewrites 52.2%

       \[\leadsto \frac{1}{\left(2 - \frac{1.0056716002661497}{x \cdot x}\right) \cdot \color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -0.6665536072, 1\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left|x\right|}{\frac{\frac{\frac{0.15298196345929074}{t\_0} - -0.2514179000665374}{\left|x\right|} - \frac{-11.259630434457211}{\left(\left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|}}{\left|x\right|} - -0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 1.2)
      (* (fma t_0 -0.6665536072 1.0) (fabs x))
      (/
       1.0
       (/
        (fabs x)
        (-
         (/
          (-
           (/ (- (/ 0.15298196345929074 t_0) -0.2514179000665374) (fabs x))
           (/ -11.259630434457211 (* (* (* t_0 (fabs x)) (fabs x)) (fabs x))))
          (fabs x))
         -0.5)))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 1.2) {
		tmp = fma(t_0, -0.6665536072, 1.0) * fabs(x);
	} else {
		tmp = 1.0 / (fabs(x) / ((((((0.15298196345929074 / t_0) - -0.2514179000665374) / fabs(x)) - (-11.259630434457211 / (((t_0 * fabs(x)) * fabs(x)) * fabs(x)))) / fabs(x)) - -0.5));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 1.2)
		tmp = Float64(fma(t_0, -0.6665536072, 1.0) * abs(x));
	else
		tmp = Float64(1.0 / Float64(abs(x) / Float64(Float64(Float64(Float64(Float64(Float64(0.15298196345929074 / t_0) - -0.2514179000665374) / abs(x)) - Float64(-11.259630434457211 / Float64(Float64(Float64(t_0 * abs(x)) * abs(x)) * abs(x)))) / abs(x)) - -0.5)));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.2], N[(N[(t$95$0 * -0.6665536072 + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Abs[x], $MachinePrecision] / N[(N[(N[(N[(N[(N[(0.15298196345929074 / t$95$0), $MachinePrecision] - -0.2514179000665374), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] - N[(-11.259630434457211 / N[(N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 54.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.2%

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

   4. Applied rewrites 50.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

   if 1.19999999999999996 < x

   1. Initial program 54.7%

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

   2. 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 51.3%

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

   6. Applied rewrites 51.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lower-/.f64 51.3%

         \[\leadsto \frac{\left(\left(\frac{0.15298196345929074}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \frac{-0.2514179000665374}{x \cdot x}\right) - \frac{-11.259630434457211}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right) - -0.5}{\color{blue}{x}} \]

   8. Applied rewrites 51.3%

      \[\leadsto \frac{\left(\frac{\frac{0.15298196345929074}{\left(x \cdot x\right) \cdot x} - \frac{-0.2514179000665374}{x}}{x} - \frac{-11.259630434457211}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}\right) - -0.5}{\color{blue}{x}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 51.2%

         \[\leadsto \frac{1}{\frac{x}{\color{blue}{\left(\frac{\frac{0.15298196345929074}{\left(x \cdot x\right) \cdot x} - \frac{-0.2514179000665374}{x}}{x} - \frac{-11.259630434457211}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}\right) - -0.5}}} \]

   10. Applied rewrites 51.2%

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

Alternative 3: 99.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -0.6665536072, 1\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.15298196345929074}{t\_0} - -0.2514179000665374}{\left|x\right|} - \frac{-11.259630434457211}{\left(\left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|}}{\left|x\right|} - -0.5}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 1.2)
      (* (fma t_0 -0.6665536072 1.0) (fabs x))
      (/
       (-
        (/
         (-
          (/ (- (/ 0.15298196345929074 t_0) -0.2514179000665374) (fabs x))
          (/ -11.259630434457211 (* (* (* t_0 (fabs x)) (fabs x)) (fabs x))))
         (fabs x))
        -0.5)
       (fabs x))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 1.2) {
		tmp = fma(t_0, -0.6665536072, 1.0) * fabs(x);
	} else {
		tmp = ((((((0.15298196345929074 / t_0) - -0.2514179000665374) / fabs(x)) - (-11.259630434457211 / (((t_0 * fabs(x)) * fabs(x)) * fabs(x)))) / fabs(x)) - -0.5) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 1.2)
		tmp = Float64(fma(t_0, -0.6665536072, 1.0) * abs(x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.15298196345929074 / t_0) - -0.2514179000665374) / abs(x)) - Float64(-11.259630434457211 / Float64(Float64(Float64(t_0 * abs(x)) * abs(x)) * abs(x)))) / abs(x)) - -0.5) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.2], N[(N[(t$95$0 * -0.6665536072 + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.15298196345929074 / t$95$0), $MachinePrecision] - -0.2514179000665374), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] - N[(-11.259630434457211 / N[(N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 54.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.2%

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

   4. Applied rewrites 50.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

   if 1.19999999999999996 < x

   1. Initial program 54.7%

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

   2. 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 51.3%

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

   6. Applied rewrites 51.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lower-/.f64 51.3%

         \[\leadsto \frac{\left(\left(\frac{0.15298196345929074}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \frac{-0.2514179000665374}{x \cdot x}\right) - \frac{-11.259630434457211}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right) - -0.5}{\color{blue}{x}} \]

   8. Applied rewrites 51.3%

      \[\leadsto \frac{\left(\frac{\frac{0.15298196345929074}{\left(x \cdot x\right) \cdot x} - \frac{-0.2514179000665374}{x}}{x} - \frac{-11.259630434457211}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}\right) - -0.5}{\color{blue}{x}} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.3%

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

Alternative 4: 99.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -0.6665536072, 1\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left|x\right|, 2, \left|x\right| \cdot \frac{-1.0056716002661497}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 1.2)
      (* (fma t_0 -0.6665536072 1.0) (fabs x))
      (/ 1.0 (fma (fabs x) 2.0 (* (fabs x) (/ -1.0056716002661497 t_0))))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 1.2) {
		tmp = fma(t_0, -0.6665536072, 1.0) * fabs(x);
	} else {
		tmp = 1.0 / fma(fabs(x), 2.0, (fabs(x) * (-1.0056716002661497 / t_0)));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 1.2)
		tmp = Float64(fma(t_0, -0.6665536072, 1.0) * abs(x));
	else
		tmp = Float64(1.0 / fma(abs(x), 2.0, Float64(abs(x) * Float64(-1.0056716002661497 / t_0))));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.2], N[(N[(t$95$0 * -0.6665536072 + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Abs[x], $MachinePrecision] * 2.0 + N[(N[Abs[x], $MachinePrecision] * N[(-1.0056716002661497 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 54.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.2%

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

   4. Applied rewrites 50.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

   if 1.19999999999999996 < x

   1. Initial program 54.7%

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

   2. 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 51.3%

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

   6. Applied rewrites 51.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\left(\left(\frac{0.15298196345929074}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \frac{-0.2514179000665374}{x \cdot x}\right) - \frac{-11.259630434457211}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right) - -0.5}}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(2 - \color{blue}{\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(2 - \frac{600041}{596657} \cdot \color{blue}{\frac{1}{{x}^{2}}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(2 - \frac{600041}{596657} \cdot \frac{1}{\color{blue}{{x}^{2}}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{\color{blue}{2}}}\right)} \]

      5. lower-pow.f64 52.2%

         \[\leadsto \frac{1}{x \cdot \left(2 - 1.0056716002661497 \cdot \frac{1}{{x}^{2}}\right)} \]

   9. Applied rewrites 52.2%

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 - 1.0056716002661497 \cdot \frac{1}{{x}^{2}}\right)}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(2 - \color{blue}{\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}}\right)} \]

       2. lift--.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(2 - \frac{600041}{596657} \cdot \color{blue}{\frac{1}{{x}^{2}}}\right)} \]

       3. sub-flip N/A

          \[\leadsto \frac{1}{x \cdot \left(2 + \left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

       4. distribute-lft-in N/A

          \[\leadsto \frac{1}{x \cdot 2 + x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)}} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, 2, x \cdot \left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, 2, x \cdot \left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, 2, x \cdot \left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

       8. lift-/.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, 2, x \cdot \left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

       9. lift-pow.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, 2, x \cdot \left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(x, 2, x \cdot \left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{x \cdot x}\right)\right)\right)} \]

       11. mult-flip-rev N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(x, 2, x \cdot \left(\mathsf{neg}\left(\frac{\frac{600041}{596657}}{x \cdot x}\right)\right)\right)} \]

       12. distribute-neg-frac N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(x, 2, x \cdot \frac{\mathsf{neg}\left(\frac{600041}{596657}\right)}{x \cdot x}\right)} \]

       13. lower-/.f64 N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(x, 2, x \cdot \frac{\mathsf{neg}\left(\frac{600041}{596657}\right)}{x \cdot x}\right)} \]

       14. metadata-eval N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(x, 2, x \cdot \frac{\frac{-600041}{596657}}{x \cdot x}\right)} \]

       15. lift-*.f64 52.2%

           \[\leadsto \frac{1}{\mathsf{fma}\left(x, 2, x \cdot \frac{-1.0056716002661497}{x \cdot x}\right)} \]

   11. Applied rewrites 52.2%

       \[\leadsto \frac{1}{\mathsf{fma}\left(x, 2, x \cdot \frac{-1.0056716002661497}{x \cdot x}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -0.6665536072, 1\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{1.0056716002661497}{t\_0}\right) \cdot \left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 1.2)
      (* (fma t_0 -0.6665536072 1.0) (fabs x))
      (/ 1.0 (* (- 2.0 (/ 1.0056716002661497 t_0)) (fabs x)))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 1.2) {
		tmp = fma(t_0, -0.6665536072, 1.0) * fabs(x);
	} else {
		tmp = 1.0 / ((2.0 - (1.0056716002661497 / t_0)) * fabs(x));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 1.2)
		tmp = Float64(fma(t_0, -0.6665536072, 1.0) * abs(x));
	else
		tmp = Float64(1.0 / Float64(Float64(2.0 - Float64(1.0056716002661497 / t_0)) * abs(x)));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.2], N[(N[(t$95$0 * -0.6665536072 + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(2.0 - N[(1.0056716002661497 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 54.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.2%

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

   4. Applied rewrites 50.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

   if 1.19999999999999996 < x

   1. Initial program 54.7%

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

   2. 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 51.3%

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

   6. Applied rewrites 51.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\left(\left(\frac{0.15298196345929074}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - \frac{-0.2514179000665374}{x \cdot x}\right) - \frac{-11.259630434457211}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right) - -0.5}}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(2 - \color{blue}{\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(2 - \frac{600041}{596657} \cdot \color{blue}{\frac{1}{{x}^{2}}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(2 - \frac{600041}{596657} \cdot \frac{1}{\color{blue}{{x}^{2}}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{\color{blue}{2}}}\right)} \]

      5. lower-pow.f64 52.2%

         \[\leadsto \frac{1}{x \cdot \left(2 - 1.0056716002661497 \cdot \frac{1}{{x}^{2}}\right)} \]

   9. Applied rewrites 52.2%

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 - 1.0056716002661497 \cdot \frac{1}{{x}^{2}}\right)}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(2 - \color{blue}{\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{\left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

       3. lower-*.f64 52.2%

          \[\leadsto \frac{1}{\left(2 - 1.0056716002661497 \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{1}{\left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

       6. lift-pow.f64 N/A

          \[\leadsto \frac{1}{\left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

       7. pow2 N/A

          \[\leadsto \frac{1}{\left(2 - \frac{600041}{596657} \cdot \frac{1}{x \cdot x}\right) \cdot x} \]

       8. mult-flip-rev N/A

          \[\leadsto \frac{1}{\left(2 - \frac{\frac{600041}{596657}}{x \cdot x}\right) \cdot x} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{1}{\left(2 - \frac{\frac{600041}{596657}}{x \cdot x}\right) \cdot x} \]

       10. lift-*.f64 52.2%

           \[\leadsto \frac{1}{\left(2 - \frac{1.0056716002661497}{x \cdot x}\right) \cdot x} \]

   11. Applied rewrites 52.2%

       \[\leadsto \frac{1}{\left(2 - \frac{1.0056716002661497}{x \cdot x}\right) \cdot \color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.2% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (copysign 1.0 x)
  (if (<= (fabs x) 1.2)
    (* (fma (* (fabs x) (fabs x)) -0.6665536072 1.0) (fabs x))
    (/ 0.5 (fabs x)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1.2) {
		tmp = fma((fabs(x) * fabs(x)), -0.6665536072, 1.0) * fabs(x);
	} else {
		tmp = 0.5 / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1.2)
		tmp = Float64(fma(Float64(abs(x) * abs(x)), -0.6665536072, 1.0) * abs(x));
	else
		tmp = Float64(0.5 / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.2], N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * -0.6665536072 + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 54.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.2%

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

   4. Applied rewrites 50.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

   if 1.19999999999999996 < x

   1. Initial program 54.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 51.5%

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

   4. Applied rewrites 51.5%

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

Alternative 7: 98.9% accurate, 15.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (copysign 1.0 x) (if (<= (fabs x) 0.7) (fabs x) (/ 0.5 (fabs x)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.7)
		tmp = abs(x);
	else
		tmp = 0.5 / abs(x);
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 0.7], N[Abs[x], $MachinePrecision], N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 54.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.2%

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

   if 0.69999999999999996 < x

   1. Initial program 54.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 51.5%

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

   4. Applied rewrites 51.5%

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

Alternative 8: 51.2% accurate, 253.1× speedup

Math

\[x \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return 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
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 54.7%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 51.2%

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

Reproduce

herbie shell --seed 2025187 
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
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