Jmat.Real.dawson

Percentage Accurate: 53.9% → 100.0%

Time: 9.1s

Alternatives: 13

Speedup: 23.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(\left(\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(x\_m \cdot x\_m\right)\\ t_1 := t\_0 \cdot \left(x\_m \cdot x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5000:\\ \;\;\;\;\frac{\left(\left(\left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot x\_m\right) \cdot x\_m + 0.1049934947\right) \cdot \left(x\_m \cdot x\_m\right) + 1\right) + 0.0005064034 \cdot t\_0\right) + 0.0001789971 \cdot t\_1}{\left(\left(\left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot 0.0694555761 + 0.2909738639\right) \cdot \left(x\_m \cdot x\_m\right) + 0.7715471019\right) \cdot \left(x\_m \cdot x\_m\right) + 1\right) + 0.0140005442 \cdot t\_0\right) + 0.0008327945 \cdot t\_1\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(t\_1 \cdot \left(x\_m \cdot x\_m\right)\right)} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.2514179000665374}{x\_m \cdot x\_m}}{-x\_m} - \frac{-0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (* (* (* x_m x_m) (* x_m x_m)) (* x_m x_m)) (* x_m x_m)))
        (t_1 (* t_0 (* x_m x_m))))
   (*
    x_s
    (if (<= x_m 5000.0)
      (*
       (/
        (+
         (+
          (+
           (*
            (+
             (* (* (+ (* (* x_m x_m) 0.0072644182) 0.0424060604) x_m) x_m)
             0.1049934947)
            (* x_m x_m))
           1.0)
          (* 0.0005064034 t_0))
         (* 0.0001789971 t_1))
        (+
         (+
          (+
           (+
            (*
             (+
              (* (+ (* (* x_m x_m) 0.0694555761) 0.2909738639) (* x_m x_m))
              0.7715471019)
             (* x_m x_m))
            1.0)
           (* 0.0140005442 t_0))
          (* 0.0008327945 t_1))
         (* (* 2.0 0.0001789971) (* t_1 (* x_m x_m)))))
       x_m)
      (- (/ (/ -0.2514179000665374 (* x_m x_m)) (- x_m)) (/ -0.5 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = (((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)) * (x_m * x_m);
	double t_1 = t_0 * (x_m * x_m);
	double tmp;
	if (x_m <= 5000.0) {
		tmp = (((((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * x_m) * x_m) + 0.1049934947) * (x_m * x_m)) + 1.0) + (0.0005064034 * t_0)) + (0.0001789971 * t_1)) / ((((((((((x_m * x_m) * 0.0694555761) + 0.2909738639) * (x_m * x_m)) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0140005442 * t_0)) + (0.0008327945 * t_1)) + ((2.0 * 0.0001789971) * (t_1 * (x_m * x_m))))) * x_m;
	} else {
		tmp = ((-0.2514179000665374 / (x_m * x_m)) / -x_m) - (-0.5 / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)) * (x_m * x_m)
    t_1 = t_0 * (x_m * x_m)
    if (x_m <= 5000.0d0) then
        tmp = (((((((((((x_m * x_m) * 0.0072644182d0) + 0.0424060604d0) * x_m) * x_m) + 0.1049934947d0) * (x_m * x_m)) + 1.0d0) + (0.0005064034d0 * t_0)) + (0.0001789971d0 * t_1)) / ((((((((((x_m * x_m) * 0.0694555761d0) + 0.2909738639d0) * (x_m * x_m)) + 0.7715471019d0) * (x_m * x_m)) + 1.0d0) + (0.0140005442d0 * t_0)) + (0.0008327945d0 * t_1)) + ((2.0d0 * 0.0001789971d0) * (t_1 * (x_m * x_m))))) * x_m
    else
        tmp = (((-0.2514179000665374d0) / (x_m * x_m)) / -x_m) - ((-0.5d0) / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = (((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)) * (x_m * x_m);
	double t_1 = t_0 * (x_m * x_m);
	double tmp;
	if (x_m <= 5000.0) {
		tmp = (((((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * x_m) * x_m) + 0.1049934947) * (x_m * x_m)) + 1.0) + (0.0005064034 * t_0)) + (0.0001789971 * t_1)) / ((((((((((x_m * x_m) * 0.0694555761) + 0.2909738639) * (x_m * x_m)) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0140005442 * t_0)) + (0.0008327945 * t_1)) + ((2.0 * 0.0001789971) * (t_1 * (x_m * x_m))))) * x_m;
	} else {
		tmp = ((-0.2514179000665374 / (x_m * x_m)) / -x_m) - (-0.5 / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = (((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)) * (x_m * x_m)
	t_1 = t_0 * (x_m * x_m)
	tmp = 0
	if x_m <= 5000.0:
		tmp = (((((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * x_m) * x_m) + 0.1049934947) * (x_m * x_m)) + 1.0) + (0.0005064034 * t_0)) + (0.0001789971 * t_1)) / ((((((((((x_m * x_m) * 0.0694555761) + 0.2909738639) * (x_m * x_m)) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0140005442 * t_0)) + (0.0008327945 * t_1)) + ((2.0 * 0.0001789971) * (t_1 * (x_m * x_m))))) * x_m
	else:
		tmp = ((-0.2514179000665374 / (x_m * x_m)) / -x_m) - (-0.5 / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) * Float64(x_m * x_m)) * Float64(x_m * x_m)) * Float64(x_m * x_m))
	t_1 = Float64(t_0 * Float64(x_m * x_m))
	tmp = 0.0
	if (x_m <= 5000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * 0.0072644182) + 0.0424060604) * x_m) * x_m) + 0.1049934947) * Float64(x_m * x_m)) + 1.0) + Float64(0.0005064034 * t_0)) + Float64(0.0001789971 * t_1)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * 0.0694555761) + 0.2909738639) * Float64(x_m * x_m)) + 0.7715471019) * Float64(x_m * x_m)) + 1.0) + Float64(0.0140005442 * t_0)) + Float64(0.0008327945 * t_1)) + Float64(Float64(2.0 * 0.0001789971) * Float64(t_1 * Float64(x_m * x_m))))) * x_m);
	else
		tmp = Float64(Float64(Float64(-0.2514179000665374 / Float64(x_m * x_m)) / Float64(-x_m)) - Float64(-0.5 / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = (((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)) * (x_m * x_m);
	t_1 = t_0 * (x_m * x_m);
	tmp = 0.0;
	if (x_m <= 5000.0)
		tmp = (((((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * x_m) * x_m) + 0.1049934947) * (x_m * x_m)) + 1.0) + (0.0005064034 * t_0)) + (0.0001789971 * t_1)) / ((((((((((x_m * x_m) * 0.0694555761) + 0.2909738639) * (x_m * x_m)) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0140005442 * t_0)) + (0.0008327945 * t_1)) + ((2.0 * 0.0001789971) * (t_1 * (x_m * x_m))))) * x_m;
	else
		tmp = ((-0.2514179000665374 / (x_m * x_m)) / -x_m) - (-0.5 / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 5000.0], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0072644182), $MachinePrecision] + 0.0424060604), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] + 0.1049934947), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(0.0005064034 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0694555761), $MachinePrecision] + 0.2909738639), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(0.0140005442 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(t$95$1 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(-0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / (-x$95$m)), $MachinePrecision] - N[(-0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 67.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\left(\left(\left(\left(1 + \frac{1049934947}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{106015151}{2500000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{36322091}{5000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{2532017}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1789971}{10000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left({x}^{2} \cdot \left(\frac{7715471019}{10000000000} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + \frac{694555761}{10000000000} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\left(\left(\left(\left(1 + \frac{1049934947}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{106015151}{2500000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{36322091}{5000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{2532017}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1789971}{10000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left({x}^{2} \cdot \left(\frac{7715471019}{10000000000} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + \frac{694555761}{10000000000} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\left(\left({x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right) + \frac{2532017}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1789971}{10000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \frac{694555761}{10000000000} + \frac{2909738639}{10000000000}\right) \cdot \left(x \cdot x\right) + \frac{7715471019}{10000000000}\right) \cdot \left(x \cdot x\right) + 1\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   8. Applied rewrites 67.4%

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

   if 5e3 < x

   1. Initial program 11.6%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 100.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. div-sub N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

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

Alternative 2: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(x\_m \cdot x\_m\right)\\ t_1 := t\_0 \cdot \left(x\_m \cdot x\_m\right)\\ t_2 := t\_1 \cdot \left(x\_m \cdot x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1:\\ \;\;\;\;\frac{\left(\left(\left(x\_m \cdot x\_m\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot \left(x\_m \cdot x\_m\right) + 0.1049934947\right) \cdot \left(x\_m \cdot x\_m\right) + 1}{\left(\left(\left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot 0.2909738639 + 0.7715471019\right) \cdot \left(x\_m \cdot x\_m\right) + 1\right) + 0.0694555761 \cdot t\_0\right) + 0.0140005442 \cdot t\_1\right) + 0.0008327945 \cdot t\_2\right) + 0.0003579942 \cdot \left(t\_2 \cdot \left(x\_m \cdot x\_m\right)\right)} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 + \frac{0.15298196345929074}{{x\_m}^{4}}\right) + \left({x\_m}^{-2} \cdot 0.2514179000665374 - -11.259630434457211 \cdot {x\_m}^{-6}\right)}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (* (* x_m x_m) (* x_m x_m)) (* x_m x_m)))
        (t_1 (* t_0 (* x_m x_m)))
        (t_2 (* t_1 (* x_m x_m))))
   (*
    x_s
    (if (<= x_m 2.1)
      (*
       (/
        (+
         (*
          (+
           (* (+ (* (* x_m x_m) 0.0072644182) 0.0424060604) (* x_m x_m))
           0.1049934947)
          (* x_m x_m))
         1.0)
        (+
         (+
          (+
           (+
            (+
             (* (+ (* (* x_m x_m) 0.2909738639) 0.7715471019) (* x_m x_m))
             1.0)
            (* 0.0694555761 t_0))
           (* 0.0140005442 t_1))
          (* 0.0008327945 t_2))
         (* 0.0003579942 (* t_2 (* x_m x_m)))))
       x_m)
      (/
       (+
        (+ 0.5 (/ 0.15298196345929074 (pow x_m 4.0)))
        (-
         (* (pow x_m -2.0) 0.2514179000665374)
         (* -11.259630434457211 (pow x_m -6.0))))
       x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = ((x_m * x_m) * (x_m * x_m)) * (x_m * x_m);
	double t_1 = t_0 * (x_m * x_m);
	double t_2 = t_1 * (x_m * x_m);
	double tmp;
	if (x_m <= 2.1) {
		tmp = ((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * (x_m * x_m)) + 0.1049934947) * (x_m * x_m)) + 1.0) / (((((((((x_m * x_m) * 0.2909738639) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0694555761 * t_0)) + (0.0140005442 * t_1)) + (0.0008327945 * t_2)) + (0.0003579942 * (t_2 * (x_m * x_m))))) * x_m;
	} else {
		tmp = ((0.5 + (0.15298196345929074 / pow(x_m, 4.0))) + ((pow(x_m, -2.0) * 0.2514179000665374) - (-11.259630434457211 * pow(x_m, -6.0)))) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)
    t_1 = t_0 * (x_m * x_m)
    t_2 = t_1 * (x_m * x_m)
    if (x_m <= 2.1d0) then
        tmp = ((((((((x_m * x_m) * 0.0072644182d0) + 0.0424060604d0) * (x_m * x_m)) + 0.1049934947d0) * (x_m * x_m)) + 1.0d0) / (((((((((x_m * x_m) * 0.2909738639d0) + 0.7715471019d0) * (x_m * x_m)) + 1.0d0) + (0.0694555761d0 * t_0)) + (0.0140005442d0 * t_1)) + (0.0008327945d0 * t_2)) + (0.0003579942d0 * (t_2 * (x_m * x_m))))) * x_m
    else
        tmp = ((0.5d0 + (0.15298196345929074d0 / (x_m ** 4.0d0))) + (((x_m ** (-2.0d0)) * 0.2514179000665374d0) - ((-11.259630434457211d0) * (x_m ** (-6.0d0))))) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = ((x_m * x_m) * (x_m * x_m)) * (x_m * x_m);
	double t_1 = t_0 * (x_m * x_m);
	double t_2 = t_1 * (x_m * x_m);
	double tmp;
	if (x_m <= 2.1) {
		tmp = ((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * (x_m * x_m)) + 0.1049934947) * (x_m * x_m)) + 1.0) / (((((((((x_m * x_m) * 0.2909738639) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0694555761 * t_0)) + (0.0140005442 * t_1)) + (0.0008327945 * t_2)) + (0.0003579942 * (t_2 * (x_m * x_m))))) * x_m;
	} else {
		tmp = ((0.5 + (0.15298196345929074 / Math.pow(x_m, 4.0))) + ((Math.pow(x_m, -2.0) * 0.2514179000665374) - (-11.259630434457211 * Math.pow(x_m, -6.0)))) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = ((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)
	t_1 = t_0 * (x_m * x_m)
	t_2 = t_1 * (x_m * x_m)
	tmp = 0
	if x_m <= 2.1:
		tmp = ((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * (x_m * x_m)) + 0.1049934947) * (x_m * x_m)) + 1.0) / (((((((((x_m * x_m) * 0.2909738639) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0694555761 * t_0)) + (0.0140005442 * t_1)) + (0.0008327945 * t_2)) + (0.0003579942 * (t_2 * (x_m * x_m))))) * x_m
	else:
		tmp = ((0.5 + (0.15298196345929074 / math.pow(x_m, 4.0))) + ((math.pow(x_m, -2.0) * 0.2514179000665374) - (-11.259630434457211 * math.pow(x_m, -6.0)))) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(Float64(x_m * x_m) * Float64(x_m * x_m)) * Float64(x_m * x_m))
	t_1 = Float64(t_0 * Float64(x_m * x_m))
	t_2 = Float64(t_1 * Float64(x_m * x_m))
	tmp = 0.0
	if (x_m <= 2.1)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * 0.0072644182) + 0.0424060604) * Float64(x_m * x_m)) + 0.1049934947) * Float64(x_m * x_m)) + 1.0) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * 0.2909738639) + 0.7715471019) * Float64(x_m * x_m)) + 1.0) + Float64(0.0694555761 * t_0)) + Float64(0.0140005442 * t_1)) + Float64(0.0008327945 * t_2)) + Float64(0.0003579942 * Float64(t_2 * Float64(x_m * x_m))))) * x_m);
	else
		tmp = Float64(Float64(Float64(0.5 + Float64(0.15298196345929074 / (x_m ^ 4.0))) + Float64(Float64((x_m ^ -2.0) * 0.2514179000665374) - Float64(-11.259630434457211 * (x_m ^ -6.0)))) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = ((x_m * x_m) * (x_m * x_m)) * (x_m * x_m);
	t_1 = t_0 * (x_m * x_m);
	t_2 = t_1 * (x_m * x_m);
	tmp = 0.0;
	if (x_m <= 2.1)
		tmp = ((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * (x_m * x_m)) + 0.1049934947) * (x_m * x_m)) + 1.0) / (((((((((x_m * x_m) * 0.2909738639) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0694555761 * t_0)) + (0.0140005442 * t_1)) + (0.0008327945 * t_2)) + (0.0003579942 * (t_2 * (x_m * x_m))))) * x_m;
	else
		tmp = ((0.5 + (0.15298196345929074 / (x_m ^ 4.0))) + (((x_m ^ -2.0) * 0.2514179000665374) - (-11.259630434457211 * (x_m ^ -6.0)))) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 2.1], N[(N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0072644182), $MachinePrecision] + 0.0424060604), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.1049934947), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.2909738639), $MachinePrecision] + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(0.0694555761 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0003579942 * N[(t$95$2 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(0.5 + N[(0.15298196345929074 / N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x$95$m, -2.0], $MachinePrecision] * 0.2514179000665374), $MachinePrecision] - N[(-11.259630434457211 * N[Power[x$95$m, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right) + \color{blue}{1}}{\left(\left(\left(\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right) + \color{blue}{1}}{\left(\left(\left(\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   5. Applied rewrites 62.7%

      \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot \left(x \cdot x\right) + 0.1049934947\right) \cdot \left(x \cdot x\right) + 1}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\color{blue}{\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)} + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(\color{blue}{\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right)} + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      3. associate-+l+ N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\color{blue}{\left(1 + \left(\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)} + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \left(\color{blue}{\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)} + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \left(\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right) + \color{blue}{\frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \left(\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right) + \frac{2909738639}{10000000000} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \left(\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right) + \color{blue}{\left(\frac{2909738639}{10000000000} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \left(\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right) + \left(\frac{2909738639}{10000000000} \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \left(\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right) + \left(\frac{2909738639}{10000000000} \cdot \color{blue}{{x}^{2}}\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      10. distribute-rgt-in N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \color{blue}{\left(x \cdot x\right) \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot {x}^{2}\right)}\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot {x}^{2}\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      12. pow2 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \color{blue}{{x}^{2}} \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot {x}^{2}\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      13. +-commutative N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot {x}^{2}\right) + 1\right)} + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot {x}^{2}\right) + 1\right)} + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   7. Applied rewrites 62.7%

      \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot \left(x \cdot x\right) + 0.1049934947\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\color{blue}{\left(\left(\left(x \cdot x\right) \cdot 0.2909738639 + 0.7715471019\right) \cdot \left(x \cdot x\right) + 1\right)} + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \frac{2909738639}{10000000000} + \frac{7715471019}{10000000000}\right) \cdot \left(x \cdot x\right) + 1\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \color{blue}{\left(2 \cdot \frac{1789971}{10000000000}\right)} \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      2. metadata-eval 62.7

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot \left(x \cdot x\right) + 0.1049934947\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot 0.2909738639 + 0.7715471019\right) \cdot \left(x \cdot x\right) + 1\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \color{blue}{0.0003579942} \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   9. Applied rewrites 62.7%

      \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot \left(x \cdot x\right) + 0.1049934947\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot 0.2909738639 + 0.7715471019\right) \cdot \left(x \cdot x\right) + 1\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \color{blue}{0.0003579942} \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   if 2.10000000000000009 < x

   1. Initial program 14.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\left(0.5 + \frac{0.15298196345929074}{{x}^{4}}\right) + \left({x}^{-2} \cdot 0.2514179000665374 - -11.259630434457211 \cdot {x}^{-6}\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(x\_m \cdot x\_m\right)\\ t_1 := t\_0 \cdot \left(x\_m \cdot x\_m\right)\\ t_2 := t\_1 \cdot \left(x\_m \cdot x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5:\\ \;\;\;\;\frac{\left(\left(\left(x\_m \cdot x\_m\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot \left(x\_m \cdot x\_m\right) + 0.1049934947\right) \cdot \left(x\_m \cdot x\_m\right) + 1}{\left(\left(\left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot 0.2909738639 + 0.7715471019\right) \cdot \left(x\_m \cdot x\_m\right) + 1\right) + 0.0694555761 \cdot t\_0\right) + 0.0140005442 \cdot t\_1\right) + 0.0008327945 \cdot t\_2\right) + 0.0003579942 \cdot \left(t\_2 \cdot \left(x\_m \cdot x\_m\right)\right)} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.15298196345929074}{x\_m \cdot x\_m} + 0.2514179000665374}{x\_m \cdot x\_m}}{x\_m} - \frac{-0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (* (* x_m x_m) (* x_m x_m)) (* x_m x_m)))
        (t_1 (* t_0 (* x_m x_m)))
        (t_2 (* t_1 (* x_m x_m))))
   (*
    x_s
    (if (<= x_m 1.5)
      (*
       (/
        (+
         (*
          (+
           (* (+ (* (* x_m x_m) 0.0072644182) 0.0424060604) (* x_m x_m))
           0.1049934947)
          (* x_m x_m))
         1.0)
        (+
         (+
          (+
           (+
            (+
             (* (+ (* (* x_m x_m) 0.2909738639) 0.7715471019) (* x_m x_m))
             1.0)
            (* 0.0694555761 t_0))
           (* 0.0140005442 t_1))
          (* 0.0008327945 t_2))
         (* 0.0003579942 (* t_2 (* x_m x_m)))))
       x_m)
      (-
       (/
        (/
         (+ (/ 0.15298196345929074 (* x_m x_m)) 0.2514179000665374)
         (* x_m x_m))
        x_m)
       (/ -0.5 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = ((x_m * x_m) * (x_m * x_m)) * (x_m * x_m);
	double t_1 = t_0 * (x_m * x_m);
	double t_2 = t_1 * (x_m * x_m);
	double tmp;
	if (x_m <= 1.5) {
		tmp = ((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * (x_m * x_m)) + 0.1049934947) * (x_m * x_m)) + 1.0) / (((((((((x_m * x_m) * 0.2909738639) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0694555761 * t_0)) + (0.0140005442 * t_1)) + (0.0008327945 * t_2)) + (0.0003579942 * (t_2 * (x_m * x_m))))) * x_m;
	} else {
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) / x_m) - (-0.5 / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)
    t_1 = t_0 * (x_m * x_m)
    t_2 = t_1 * (x_m * x_m)
    if (x_m <= 1.5d0) then
        tmp = ((((((((x_m * x_m) * 0.0072644182d0) + 0.0424060604d0) * (x_m * x_m)) + 0.1049934947d0) * (x_m * x_m)) + 1.0d0) / (((((((((x_m * x_m) * 0.2909738639d0) + 0.7715471019d0) * (x_m * x_m)) + 1.0d0) + (0.0694555761d0 * t_0)) + (0.0140005442d0 * t_1)) + (0.0008327945d0 * t_2)) + (0.0003579942d0 * (t_2 * (x_m * x_m))))) * x_m
    else
        tmp = ((((0.15298196345929074d0 / (x_m * x_m)) + 0.2514179000665374d0) / (x_m * x_m)) / x_m) - ((-0.5d0) / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = ((x_m * x_m) * (x_m * x_m)) * (x_m * x_m);
	double t_1 = t_0 * (x_m * x_m);
	double t_2 = t_1 * (x_m * x_m);
	double tmp;
	if (x_m <= 1.5) {
		tmp = ((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * (x_m * x_m)) + 0.1049934947) * (x_m * x_m)) + 1.0) / (((((((((x_m * x_m) * 0.2909738639) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0694555761 * t_0)) + (0.0140005442 * t_1)) + (0.0008327945 * t_2)) + (0.0003579942 * (t_2 * (x_m * x_m))))) * x_m;
	} else {
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) / x_m) - (-0.5 / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = ((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)
	t_1 = t_0 * (x_m * x_m)
	t_2 = t_1 * (x_m * x_m)
	tmp = 0
	if x_m <= 1.5:
		tmp = ((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * (x_m * x_m)) + 0.1049934947) * (x_m * x_m)) + 1.0) / (((((((((x_m * x_m) * 0.2909738639) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0694555761 * t_0)) + (0.0140005442 * t_1)) + (0.0008327945 * t_2)) + (0.0003579942 * (t_2 * (x_m * x_m))))) * x_m
	else:
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) / x_m) - (-0.5 / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(Float64(x_m * x_m) * Float64(x_m * x_m)) * Float64(x_m * x_m))
	t_1 = Float64(t_0 * Float64(x_m * x_m))
	t_2 = Float64(t_1 * Float64(x_m * x_m))
	tmp = 0.0
	if (x_m <= 1.5)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * 0.0072644182) + 0.0424060604) * Float64(x_m * x_m)) + 0.1049934947) * Float64(x_m * x_m)) + 1.0) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * 0.2909738639) + 0.7715471019) * Float64(x_m * x_m)) + 1.0) + Float64(0.0694555761 * t_0)) + Float64(0.0140005442 * t_1)) + Float64(0.0008327945 * t_2)) + Float64(0.0003579942 * Float64(t_2 * Float64(x_m * x_m))))) * x_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.15298196345929074 / Float64(x_m * x_m)) + 0.2514179000665374) / Float64(x_m * x_m)) / x_m) - Float64(-0.5 / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = ((x_m * x_m) * (x_m * x_m)) * (x_m * x_m);
	t_1 = t_0 * (x_m * x_m);
	t_2 = t_1 * (x_m * x_m);
	tmp = 0.0;
	if (x_m <= 1.5)
		tmp = ((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * (x_m * x_m)) + 0.1049934947) * (x_m * x_m)) + 1.0) / (((((((((x_m * x_m) * 0.2909738639) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0694555761 * t_0)) + (0.0140005442 * t_1)) + (0.0008327945 * t_2)) + (0.0003579942 * (t_2 * (x_m * x_m))))) * x_m;
	else
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) / x_m) - (-0.5 / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 1.5], N[(N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0072644182), $MachinePrecision] + 0.0424060604), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.1049934947), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.2909738639), $MachinePrecision] + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(0.0694555761 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0003579942 * N[(t$95$2 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(N[(N[(0.15298196345929074 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.2514179000665374), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - N[(-0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right) + \color{blue}{1}}{\left(\left(\left(\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right) + \color{blue}{1}}{\left(\left(\left(\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   5. Applied rewrites 62.7%

      \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot \left(x \cdot x\right) + 0.1049934947\right) \cdot \left(x \cdot x\right) + 1}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\color{blue}{\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)} + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(\color{blue}{\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right)} + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      3. associate-+l+ N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\color{blue}{\left(1 + \left(\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)} + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \left(\color{blue}{\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)} + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \left(\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right) + \color{blue}{\frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \left(\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right) + \frac{2909738639}{10000000000} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \left(\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right) + \color{blue}{\left(\frac{2909738639}{10000000000} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \left(\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right) + \left(\frac{2909738639}{10000000000} \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \left(\frac{7715471019}{10000000000} \cdot \left(x \cdot x\right) + \left(\frac{2909738639}{10000000000} \cdot \color{blue}{{x}^{2}}\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      10. distribute-rgt-in N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \color{blue}{\left(x \cdot x\right) \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot {x}^{2}\right)}\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot {x}^{2}\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      12. pow2 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(1 + \color{blue}{{x}^{2}} \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot {x}^{2}\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      13. +-commutative N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot {x}^{2}\right) + 1\right)} + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot {x}^{2}\right) + 1\right)} + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   7. Applied rewrites 62.7%

      \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot \left(x \cdot x\right) + 0.1049934947\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\color{blue}{\left(\left(\left(x \cdot x\right) \cdot 0.2909738639 + 0.7715471019\right) \cdot \left(x \cdot x\right) + 1\right)} + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \frac{2909738639}{10000000000} + \frac{7715471019}{10000000000}\right) \cdot \left(x \cdot x\right) + 1\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \color{blue}{\left(2 \cdot \frac{1789971}{10000000000}\right)} \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      2. metadata-eval 62.7

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot \left(x \cdot x\right) + 0.1049934947\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot 0.2909738639 + 0.7715471019\right) \cdot \left(x \cdot x\right) + 1\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \color{blue}{0.0003579942} \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   9. Applied rewrites 62.7%

      \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot \left(x \cdot x\right) + 0.1049934947\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot 0.2909738639 + 0.7715471019\right) \cdot \left(x \cdot x\right) + 1\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \color{blue}{0.0003579942} \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   if 1.5 < x

   1. Initial program 14.1%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 99.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. div-sub N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

   7. Applied rewrites 99.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot \left(x \cdot x\right) + 0.1049934947\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot 0.2909738639 + 0.7715471019\right) \cdot \left(x \cdot x\right) + 1\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0003579942 \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.15298196345929074}{x \cdot x} + 0.2514179000665374}{x \cdot x}}{x} - \frac{-0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(\left(\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(x\_m \cdot x\_m\right)\\ t_1 := t\_0 \cdot \left(x\_m \cdot x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5:\\ \;\;\;\;\frac{\left(\left(\left(x\_m \cdot x\_m\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot \left(x\_m \cdot x\_m\right) + 0.1049934947\right) \cdot \left(x\_m \cdot x\_m\right) + 1}{\left(\left(\left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot 0.0694555761 + 0.2909738639\right) \cdot \left(x\_m \cdot x\_m\right) + 0.7715471019\right) \cdot \left(x\_m \cdot x\_m\right) + 1\right) + 0.0140005442 \cdot t\_0\right) + 0.0008327945 \cdot t\_1\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(t\_1 \cdot \left(x\_m \cdot x\_m\right)\right)} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.15298196345929074}{x\_m \cdot x\_m} + 0.2514179000665374}{x\_m \cdot x\_m}}{x\_m} - \frac{-0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (* (* (* x_m x_m) (* x_m x_m)) (* x_m x_m)) (* x_m x_m)))
        (t_1 (* t_0 (* x_m x_m))))
   (*
    x_s
    (if (<= x_m 1.5)
      (*
       (/
        (+
         (*
          (+
           (* (+ (* (* x_m x_m) 0.0072644182) 0.0424060604) (* x_m x_m))
           0.1049934947)
          (* x_m x_m))
         1.0)
        (+
         (+
          (+
           (+
            (*
             (+
              (* (+ (* (* x_m x_m) 0.0694555761) 0.2909738639) (* x_m x_m))
              0.7715471019)
             (* x_m x_m))
            1.0)
           (* 0.0140005442 t_0))
          (* 0.0008327945 t_1))
         (* (* 2.0 0.0001789971) (* t_1 (* x_m x_m)))))
       x_m)
      (-
       (/
        (/
         (+ (/ 0.15298196345929074 (* x_m x_m)) 0.2514179000665374)
         (* x_m x_m))
        x_m)
       (/ -0.5 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = (((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)) * (x_m * x_m);
	double t_1 = t_0 * (x_m * x_m);
	double tmp;
	if (x_m <= 1.5) {
		tmp = ((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * (x_m * x_m)) + 0.1049934947) * (x_m * x_m)) + 1.0) / ((((((((((x_m * x_m) * 0.0694555761) + 0.2909738639) * (x_m * x_m)) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0140005442 * t_0)) + (0.0008327945 * t_1)) + ((2.0 * 0.0001789971) * (t_1 * (x_m * x_m))))) * x_m;
	} else {
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) / x_m) - (-0.5 / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)) * (x_m * x_m)
    t_1 = t_0 * (x_m * x_m)
    if (x_m <= 1.5d0) then
        tmp = ((((((((x_m * x_m) * 0.0072644182d0) + 0.0424060604d0) * (x_m * x_m)) + 0.1049934947d0) * (x_m * x_m)) + 1.0d0) / ((((((((((x_m * x_m) * 0.0694555761d0) + 0.2909738639d0) * (x_m * x_m)) + 0.7715471019d0) * (x_m * x_m)) + 1.0d0) + (0.0140005442d0 * t_0)) + (0.0008327945d0 * t_1)) + ((2.0d0 * 0.0001789971d0) * (t_1 * (x_m * x_m))))) * x_m
    else
        tmp = ((((0.15298196345929074d0 / (x_m * x_m)) + 0.2514179000665374d0) / (x_m * x_m)) / x_m) - ((-0.5d0) / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = (((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)) * (x_m * x_m);
	double t_1 = t_0 * (x_m * x_m);
	double tmp;
	if (x_m <= 1.5) {
		tmp = ((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * (x_m * x_m)) + 0.1049934947) * (x_m * x_m)) + 1.0) / ((((((((((x_m * x_m) * 0.0694555761) + 0.2909738639) * (x_m * x_m)) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0140005442 * t_0)) + (0.0008327945 * t_1)) + ((2.0 * 0.0001789971) * (t_1 * (x_m * x_m))))) * x_m;
	} else {
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) / x_m) - (-0.5 / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = (((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)) * (x_m * x_m)
	t_1 = t_0 * (x_m * x_m)
	tmp = 0
	if x_m <= 1.5:
		tmp = ((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * (x_m * x_m)) + 0.1049934947) * (x_m * x_m)) + 1.0) / ((((((((((x_m * x_m) * 0.0694555761) + 0.2909738639) * (x_m * x_m)) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0140005442 * t_0)) + (0.0008327945 * t_1)) + ((2.0 * 0.0001789971) * (t_1 * (x_m * x_m))))) * x_m
	else:
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) / x_m) - (-0.5 / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) * Float64(x_m * x_m)) * Float64(x_m * x_m)) * Float64(x_m * x_m))
	t_1 = Float64(t_0 * Float64(x_m * x_m))
	tmp = 0.0
	if (x_m <= 1.5)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * 0.0072644182) + 0.0424060604) * Float64(x_m * x_m)) + 0.1049934947) * Float64(x_m * x_m)) + 1.0) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x_m * x_m) * 0.0694555761) + 0.2909738639) * Float64(x_m * x_m)) + 0.7715471019) * Float64(x_m * x_m)) + 1.0) + Float64(0.0140005442 * t_0)) + Float64(0.0008327945 * t_1)) + Float64(Float64(2.0 * 0.0001789971) * Float64(t_1 * Float64(x_m * x_m))))) * x_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.15298196345929074 / Float64(x_m * x_m)) + 0.2514179000665374) / Float64(x_m * x_m)) / x_m) - Float64(-0.5 / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = (((x_m * x_m) * (x_m * x_m)) * (x_m * x_m)) * (x_m * x_m);
	t_1 = t_0 * (x_m * x_m);
	tmp = 0.0;
	if (x_m <= 1.5)
		tmp = ((((((((x_m * x_m) * 0.0072644182) + 0.0424060604) * (x_m * x_m)) + 0.1049934947) * (x_m * x_m)) + 1.0) / ((((((((((x_m * x_m) * 0.0694555761) + 0.2909738639) * (x_m * x_m)) + 0.7715471019) * (x_m * x_m)) + 1.0) + (0.0140005442 * t_0)) + (0.0008327945 * t_1)) + ((2.0 * 0.0001789971) * (t_1 * (x_m * x_m))))) * x_m;
	else
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) / x_m) - (-0.5 / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 1.5], N[(N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0072644182), $MachinePrecision] + 0.0424060604), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.1049934947), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0694555761), $MachinePrecision] + 0.2909738639), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(0.0140005442 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(t$95$1 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(N[(N[(0.15298196345929074 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.2514179000665374), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - N[(-0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right) + \color{blue}{1}}{\left(\left(\left(\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right) + \color{blue}{1}}{\left(\left(\left(\left(\left(1 + \frac{7715471019}{10000000000} \cdot \left(x \cdot x\right)\right) + \frac{2909738639}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{694555761}{10000000000} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left({x}^{2} \cdot \left(\frac{7715471019}{10000000000} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + \frac{694555761}{10000000000} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{36322091}{5000000000} + \frac{106015151}{2500000000}\right) \cdot \left(x \cdot x\right) + \frac{1049934947}{10000000000}\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left({x}^{2} \cdot \left(\frac{7715471019}{10000000000} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + \frac{694555761}{10000000000} \cdot {x}^{2}\right)\right) + \color{blue}{1}\right) + \frac{70002721}{5000000000} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1665589}{2000000000} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot \frac{1789971}{10000000000}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   8. Applied rewrites 62.7%

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

   if 1.5 < x

   1. Initial program 14.1%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 99.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. div-sub N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

   7. Applied rewrites 99.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot x\right) \cdot 0.0072644182 + 0.0424060604\right) \cdot \left(x \cdot x\right) + 0.1049934947\right) \cdot \left(x \cdot x\right) + 1}{\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot 0.0694555761 + 0.2909738639\right) \cdot \left(x \cdot x\right) + 0.7715471019\right) \cdot \left(x \cdot x\right) + 1\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.15298196345929074}{x \cdot x} + 0.2514179000665374}{x \cdot x}}{x} - \frac{-0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2:\\ \;\;\;\;\left(\left(\left(-0.0732490286039007 \cdot \left(x\_m \cdot x\_m\right) + 0.265709700396151\right) \cdot \left(x\_m \cdot x\_m\right) - 0.6665536072\right) \cdot \left(x\_m \cdot x\_m\right) + 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.15298196345929074}{x\_m \cdot x\_m} + 0.2514179000665374}{x\_m \cdot x\_m}}{x\_m} - \frac{-0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.2)
    (*
     (+
      (*
       (-
        (*
         (+ (* -0.0732490286039007 (* x_m x_m)) 0.265709700396151)
         (* x_m x_m))
        0.6665536072)
       (* x_m x_m))
      1.0)
     x_m)
    (-
     (/
      (/
       (+ (/ 0.15298196345929074 (* x_m x_m)) 0.2514179000665374)
       (* x_m x_m))
      x_m)
     (/ -0.5 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.2) {
		tmp = ((((((-0.0732490286039007 * (x_m * x_m)) + 0.265709700396151) * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	} else {
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) / x_m) - (-0.5 / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.2d0) then
        tmp = (((((((-0.0732490286039007d0) * (x_m * x_m)) + 0.265709700396151d0) * (x_m * x_m)) - 0.6665536072d0) * (x_m * x_m)) + 1.0d0) * x_m
    else
        tmp = ((((0.15298196345929074d0 / (x_m * x_m)) + 0.2514179000665374d0) / (x_m * x_m)) / x_m) - ((-0.5d0) / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.2) {
		tmp = ((((((-0.0732490286039007 * (x_m * x_m)) + 0.265709700396151) * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	} else {
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) / x_m) - (-0.5 / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.2:
		tmp = ((((((-0.0732490286039007 * (x_m * x_m)) + 0.265709700396151) * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m
	else:
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) / x_m) - (-0.5 / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.2)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0732490286039007 * Float64(x_m * x_m)) + 0.265709700396151) * Float64(x_m * x_m)) - 0.6665536072) * Float64(x_m * x_m)) + 1.0) * x_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.15298196345929074 / Float64(x_m * x_m)) + 0.2514179000665374) / Float64(x_m * x_m)) / x_m) - Float64(-0.5 / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.2)
		tmp = ((((((-0.0732490286039007 * (x_m * x_m)) + 0.265709700396151) * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	else
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) / x_m) - (-0.5 / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.2], N[(N[(N[(N[(N[(N[(N[(-0.0732490286039007 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.265709700396151), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(N[(N[(0.15298196345929074 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.2514179000665374), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - N[(-0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right)\right)} \cdot x \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right) + \color{blue}{1}\right) \cdot x \]

   5. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\left(\left(\left(-0.0732490286039007 \cdot \left(x \cdot x\right) + 0.265709700396151\right) \cdot \left(x \cdot x\right) - 0.6665536072\right) \cdot \left(x \cdot x\right) + 1\right)} \cdot x \]

   if 1.19999999999999996 < x

   1. Initial program 14.1%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 99.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. div-sub N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

   7. Applied rewrites 99.3%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;\left(\left(\left(-0.0732490286039007 \cdot \left(x \cdot x\right) + 0.265709700396151\right) \cdot \left(x \cdot x\right) - 0.6665536072\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.15298196345929074}{x \cdot x} + 0.2514179000665374}{x \cdot x}}{x} - \frac{-0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2:\\ \;\;\;\;\left(\left(\left(-0.0732490286039007 \cdot \left(x\_m \cdot x\_m\right) + 0.265709700396151\right) \cdot \left(x\_m \cdot x\_m\right) - 0.6665536072\right) \cdot \left(x\_m \cdot x\_m\right) + 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.15298196345929074}{x\_m \cdot x\_m} + 0.2514179000665374}{x\_m \cdot x\_m} + 0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.2)
    (*
     (+
      (*
       (-
        (*
         (+ (* -0.0732490286039007 (* x_m x_m)) 0.265709700396151)
         (* x_m x_m))
        0.6665536072)
       (* x_m x_m))
      1.0)
     x_m)
    (/
     (+
      (/
       (+ (/ 0.15298196345929074 (* x_m x_m)) 0.2514179000665374)
       (* x_m x_m))
      0.5)
     x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.2) {
		tmp = ((((((-0.0732490286039007 * (x_m * x_m)) + 0.265709700396151) * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	} else {
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) + 0.5) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.2d0) then
        tmp = (((((((-0.0732490286039007d0) * (x_m * x_m)) + 0.265709700396151d0) * (x_m * x_m)) - 0.6665536072d0) * (x_m * x_m)) + 1.0d0) * x_m
    else
        tmp = ((((0.15298196345929074d0 / (x_m * x_m)) + 0.2514179000665374d0) / (x_m * x_m)) + 0.5d0) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.2) {
		tmp = ((((((-0.0732490286039007 * (x_m * x_m)) + 0.265709700396151) * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	} else {
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) + 0.5) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.2:
		tmp = ((((((-0.0732490286039007 * (x_m * x_m)) + 0.265709700396151) * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m
	else:
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) + 0.5) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.2)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0732490286039007 * Float64(x_m * x_m)) + 0.265709700396151) * Float64(x_m * x_m)) - 0.6665536072) * Float64(x_m * x_m)) + 1.0) * x_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.15298196345929074 / Float64(x_m * x_m)) + 0.2514179000665374) / Float64(x_m * x_m)) + 0.5) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.2)
		tmp = ((((((-0.0732490286039007 * (x_m * x_m)) + 0.265709700396151) * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	else
		tmp = ((((0.15298196345929074 / (x_m * x_m)) + 0.2514179000665374) / (x_m * x_m)) + 0.5) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.2], N[(N[(N[(N[(N[(N[(N[(-0.0732490286039007 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.265709700396151), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(N[(N[(0.15298196345929074 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.2514179000665374), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right)\right)} \cdot x \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right) + \color{blue}{1}\right) \cdot x \]

   5. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\left(\left(\left(-0.0732490286039007 \cdot \left(x \cdot x\right) + 0.265709700396151\right) \cdot \left(x \cdot x\right) - 0.6665536072\right) \cdot \left(x \cdot x\right) + 1\right)} \cdot x \]

   if 1.19999999999999996 < x

   1. Initial program 14.1%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;\left(\left(\left(-0.0732490286039007 \cdot \left(x \cdot x\right) + 0.265709700396151\right) \cdot \left(x \cdot x\right) - 0.6665536072\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.15298196345929074}{x \cdot x} + 0.2514179000665374}{x \cdot x} + 0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;\left(\left(\left(-0.0732490286039007 \cdot \left(x\_m \cdot x\_m\right) + 0.265709700396151\right) \cdot \left(x\_m \cdot x\_m\right) - 0.6665536072\right) \cdot \left(x\_m \cdot x\_m\right) + 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.2514179000665374}{x\_m \cdot x\_m}}{-x\_m} - \frac{-0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.15)
    (*
     (+
      (*
       (-
        (*
         (+ (* -0.0732490286039007 (* x_m x_m)) 0.265709700396151)
         (* x_m x_m))
        0.6665536072)
       (* x_m x_m))
      1.0)
     x_m)
    (- (/ (/ -0.2514179000665374 (* x_m x_m)) (- x_m)) (/ -0.5 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.15) {
		tmp = ((((((-0.0732490286039007 * (x_m * x_m)) + 0.265709700396151) * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	} else {
		tmp = ((-0.2514179000665374 / (x_m * x_m)) / -x_m) - (-0.5 / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.15d0) then
        tmp = (((((((-0.0732490286039007d0) * (x_m * x_m)) + 0.265709700396151d0) * (x_m * x_m)) - 0.6665536072d0) * (x_m * x_m)) + 1.0d0) * x_m
    else
        tmp = (((-0.2514179000665374d0) / (x_m * x_m)) / -x_m) - ((-0.5d0) / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.15) {
		tmp = ((((((-0.0732490286039007 * (x_m * x_m)) + 0.265709700396151) * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	} else {
		tmp = ((-0.2514179000665374 / (x_m * x_m)) / -x_m) - (-0.5 / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.15:
		tmp = ((((((-0.0732490286039007 * (x_m * x_m)) + 0.265709700396151) * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m
	else:
		tmp = ((-0.2514179000665374 / (x_m * x_m)) / -x_m) - (-0.5 / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.15)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0732490286039007 * Float64(x_m * x_m)) + 0.265709700396151) * Float64(x_m * x_m)) - 0.6665536072) * Float64(x_m * x_m)) + 1.0) * x_m);
	else
		tmp = Float64(Float64(Float64(-0.2514179000665374 / Float64(x_m * x_m)) / Float64(-x_m)) - Float64(-0.5 / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.15)
		tmp = ((((((-0.0732490286039007 * (x_m * x_m)) + 0.265709700396151) * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	else
		tmp = ((-0.2514179000665374 / (x_m * x_m)) / -x_m) - (-0.5 / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.15], N[(N[(N[(N[(N[(N[(N[(-0.0732490286039007 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.265709700396151), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(-0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / (-x$95$m)), $MachinePrecision] - N[(-0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right)\right)} \cdot x \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right) + \color{blue}{1}\right) \cdot x \]

   5. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\left(\left(\left(-0.0732490286039007 \cdot \left(x \cdot x\right) + 0.265709700396151\right) \cdot \left(x \cdot x\right) - 0.6665536072\right) \cdot \left(x \cdot x\right) + 1\right)} \cdot x \]

   if 1.1499999999999999 < x

   1. Initial program 14.1%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 99.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. div-sub N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.0%

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

Alternative 8: 99.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\left(\left(0.265709700396151 \cdot \left(x\_m \cdot x\_m\right) - 0.6665536072\right) \cdot \left(x\_m \cdot x\_m\right) + 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.2514179000665374}{x\_m \cdot x\_m}}{-x\_m} - \frac{-0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.1)
    (*
     (+ (* (- (* 0.265709700396151 (* x_m x_m)) 0.6665536072) (* x_m x_m)) 1.0)
     x_m)
    (- (/ (/ -0.2514179000665374 (* x_m x_m)) (- x_m)) (/ -0.5 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = ((((0.265709700396151 * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	} else {
		tmp = ((-0.2514179000665374 / (x_m * x_m)) / -x_m) - (-0.5 / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.1d0) then
        tmp = ((((0.265709700396151d0 * (x_m * x_m)) - 0.6665536072d0) * (x_m * x_m)) + 1.0d0) * x_m
    else
        tmp = (((-0.2514179000665374d0) / (x_m * x_m)) / -x_m) - ((-0.5d0) / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = ((((0.265709700396151 * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	} else {
		tmp = ((-0.2514179000665374 / (x_m * x_m)) / -x_m) - (-0.5 / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.1:
		tmp = ((((0.265709700396151 * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m
	else:
		tmp = ((-0.2514179000665374 / (x_m * x_m)) / -x_m) - (-0.5 / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(Float64(Float64(Float64(Float64(0.265709700396151 * Float64(x_m * x_m)) - 0.6665536072) * Float64(x_m * x_m)) + 1.0) * x_m);
	else
		tmp = Float64(Float64(Float64(-0.2514179000665374 / Float64(x_m * x_m)) / Float64(-x_m)) - Float64(-0.5 / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.1)
		tmp = ((((0.265709700396151 * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	else
		tmp = ((-0.2514179000665374 / (x_m * x_m)) / -x_m) - (-0.5 / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.1], N[(N[(N[(N[(N[(0.265709700396151 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(-0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / (-x$95$m)), $MachinePrecision] - N[(-0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 63.1%

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

   if 1.1000000000000001 < x

   1. Initial program 14.1%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 99.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. div-sub N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.0%

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

Alternative 9: 99.5% accurate, 10.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\left(\left(0.265709700396151 \cdot \left(x\_m \cdot x\_m\right) - 0.6665536072\right) \cdot \left(x\_m \cdot x\_m\right) + 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{x\_m \cdot x\_m} + 0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.1)
    (*
     (+ (* (- (* 0.265709700396151 (* x_m x_m)) 0.6665536072) (* x_m x_m)) 1.0)
     x_m)
    (/ (+ (/ 0.2514179000665374 (* x_m x_m)) 0.5) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = ((((0.265709700396151 * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	} else {
		tmp = ((0.2514179000665374 / (x_m * x_m)) + 0.5) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.1d0) then
        tmp = ((((0.265709700396151d0 * (x_m * x_m)) - 0.6665536072d0) * (x_m * x_m)) + 1.0d0) * x_m
    else
        tmp = ((0.2514179000665374d0 / (x_m * x_m)) + 0.5d0) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = ((((0.265709700396151 * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	} else {
		tmp = ((0.2514179000665374 / (x_m * x_m)) + 0.5) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.1:
		tmp = ((((0.265709700396151 * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m
	else:
		tmp = ((0.2514179000665374 / (x_m * x_m)) + 0.5) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(Float64(Float64(Float64(Float64(0.265709700396151 * Float64(x_m * x_m)) - 0.6665536072) * Float64(x_m * x_m)) + 1.0) * x_m);
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / Float64(x_m * x_m)) + 0.5) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.1)
		tmp = ((((0.265709700396151 * (x_m * x_m)) - 0.6665536072) * (x_m * x_m)) + 1.0) * x_m;
	else
		tmp = ((0.2514179000665374 / (x_m * x_m)) + 0.5) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.1], N[(N[(N[(N[(N[(0.265709700396151 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 63.1%

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

   if 1.1000000000000001 < x

   1. Initial program 14.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{{x}^{-2} \cdot 0.2514179000665374 + 0.5}{x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-flip N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

Alternative 10: 99.4% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.95d0) then
        tmp = (((x_m * x_m) * (-0.6665536072d0)) + 1.0d0) * x_m
    else
        tmp = ((0.2514179000665374d0 / (x_m * x_m)) + 0.5d0) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.95) {
		tmp = (((x_m * x_m) * -0.6665536072) + 1.0) * x_m;
	} else {
		tmp = ((0.2514179000665374 / (x_m * x_m)) + 0.5) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.95:
		tmp = (((x_m * x_m) * -0.6665536072) + 1.0) * x_m
	else:
		tmp = ((0.2514179000665374 / (x_m * x_m)) + 0.5) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.95)
		tmp = Float64(Float64(Float64(Float64(x_m * x_m) * -0.6665536072) + 1.0) * x_m);
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / Float64(x_m * x_m)) + 0.5) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.95)
		tmp = (((x_m * x_m) * -0.6665536072) + 1.0) * x_m;
	else
		tmp = ((0.2514179000665374 / (x_m * x_m)) + 0.5) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.95], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.6665536072), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 62.4%

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

   if 0.94999999999999996 < x

   1. Initial program 14.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{{x}^{-2} \cdot 0.2514179000665374 + 0.5}{x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-flip N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

Alternative 11: 99.2% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.8d0) then
        tmp = (((x_m * x_m) * (-0.6665536072d0)) + 1.0d0) * x_m
    else
        tmp = 0.5d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.8) {
		tmp = (((x_m * x_m) * -0.6665536072) + 1.0) * x_m;
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.8:
		tmp = (((x_m * x_m) * -0.6665536072) + 1.0) * x_m
	else:
		tmp = 0.5 / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.8)
		tmp = Float64(Float64(Float64(Float64(x_m * x_m) * -0.6665536072) + 1.0) * x_m);
	else
		tmp = Float64(0.5 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.8)
		tmp = (((x_m * x_m) * -0.6665536072) + 1.0) * x_m;
	else
		tmp = 0.5 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.8], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.6665536072), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 62.4%

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

   if 0.80000000000000004 < x

   1. Initial program 14.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 12: 98.9% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.7)
		tmp = x_m;
	else
		tmp = 0.5 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.7], x$95$m, N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.2%

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

   if 0.69999999999999996 < x

   1. Initial program 14.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 13: 51.4% accurate, 415.0× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 52.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 46.8%

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

Reproduce

herbie shell --seed 2025065 
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x)))) (* 0.0072644182 (* (* (* x x) (* x x)) (* x x)))) (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1.0 (* 0.7715471019 (* x x))) (* 0.2909738639 (* (* x x) (* x x)))) (* 0.0694555761 (* (* (* x x) (* x x)) (* x x)))) (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2.0 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))