Jmat.Real.dawson

Percentage Accurate: 53.7% → 100.0%

Time: 7.8s

Alternatives: 10

Speedup: 31.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% 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, 1.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10400:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.0001789971, {\left(x\_m \cdot x\_m\right)}^{4}, \mathsf{fma}\left(x\_m \cdot \mathsf{fma}\left(x\_m \cdot \left(0.0072644182 + 0.0005064034 \cdot \left(x\_m \cdot x\_m\right)\right), x\_m, 0.0424060604\right), x\_m, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left({\left(x\_m \cdot x\_m\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x\_m \cdot x\_m\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x\_m, \mathsf{fma}\left(x\_m, \mathsf{fma}\left(0.0694555761 \cdot x\_m, x\_m, \left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\_m\right), 0.7715471019\right) \cdot x\_m, x\_m, 1\right)\right)\right)} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x\_m}^{2}}}{x\_m}\\ \end{array} \end{array} \]

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 10400.0)
    (*
     (/
      (fma
       (* x_m x_m)
       (fma
        0.0001789971
        (pow (* x_m x_m) 4.0)
        (fma
         (*
          x_m
          (fma
           (* x_m (+ 0.0072644182 (* 0.0005064034 (* x_m x_m))))
           x_m
           0.0424060604))
         x_m
         0.1049934947))
       1.0)
      (fma
       (pow (* x_m x_m) 6.0)
       0.0003579942
       (fma
        (pow (* x_m x_m) 5.0)
        0.0008327945
        (fma
         (*
          (fma
           x_m
           (fma
            x_m
            (fma
             (* 0.0694555761 x_m)
             x_m
             (* (* (* (* x_m x_m) x_m) x_m) 0.0140005442))
            (* 0.2909738639 x_m))
           0.7715471019)
          x_m)
         x_m
         1.0))))
     x_m)
    (/ (+ 0.5 (* 0.2514179000665374 (/ 1.0 (pow x_m 2.0)))) 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 <= 10400.0) {
		tmp = (fma((x_m * x_m), fma(0.0001789971, pow((x_m * x_m), 4.0), fma((x_m * fma((x_m * (0.0072644182 + (0.0005064034 * (x_m * x_m)))), x_m, 0.0424060604)), x_m, 0.1049934947)), 1.0) / fma(pow((x_m * x_m), 6.0), 0.0003579942, fma(pow((x_m * x_m), 5.0), 0.0008327945, fma((fma(x_m, fma(x_m, fma((0.0694555761 * x_m), x_m, ((((x_m * x_m) * x_m) * x_m) * 0.0140005442)), (0.2909738639 * x_m)), 0.7715471019) * x_m), x_m, 1.0)))) * x_m;
	} else {
		tmp = (0.5 + (0.2514179000665374 * (1.0 / pow(x_m, 2.0)))) / 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 <= 10400.0)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), fma(0.0001789971, (Float64(x_m * x_m) ^ 4.0), fma(Float64(x_m * fma(Float64(x_m * Float64(0.0072644182 + Float64(0.0005064034 * Float64(x_m * x_m)))), x_m, 0.0424060604)), x_m, 0.1049934947)), 1.0) / fma((Float64(x_m * x_m) ^ 6.0), 0.0003579942, fma((Float64(x_m * x_m) ^ 5.0), 0.0008327945, fma(Float64(fma(x_m, fma(x_m, fma(Float64(0.0694555761 * x_m), x_m, Float64(Float64(Float64(Float64(x_m * x_m) * x_m) * x_m) * 0.0140005442)), Float64(0.2909738639 * x_m)), 0.7715471019) * x_m), x_m, 1.0)))) * x_m);
	else
		tmp = Float64(Float64(0.5 + Float64(0.2514179000665374 * Float64(1.0 / (x_m ^ 2.0)))) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 10400.0], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0001789971 * N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 4.0], $MachinePrecision] + N[(N[(x$95$m * N[(N[(x$95$m * N[(0.0072644182 + N[(0.0005064034 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m + 0.0424060604), $MachinePrecision]), $MachinePrecision] * x$95$m + 0.1049934947), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 6.0], $MachinePrecision] * 0.0003579942 + N[(N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 5.0], $MachinePrecision] * 0.0008327945 + N[(N[(N[(x$95$m * N[(x$95$m * N[(N[(0.0694555761 * x$95$m), $MachinePrecision] * x$95$m + N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.0140005442), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.7715471019), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(0.5 + N[(0.2514179000665374 * N[(1.0 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 10400

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}} \cdot x \]

   9. Applied rewrites 53.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0001789971, {\left(x \cdot x\right)}^{4}, \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot \left(0.0072644182 + 0.0005064034 \cdot \left(x \cdot x\right)\right), x, 0.0424060604\right), x, 0.1049934947\right)\right), 1\right)}}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)} \cdot x \]

   if 10400 < x

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 51.6%

       \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := {\left(x\_m \cdot x\_m\right)}^{5}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot \mathsf{fma}\left(x\_m \cdot \left(0.0072644182 + 0.0005064034 \cdot \left(x\_m \cdot x\_m\right)\right), x\_m, 0.0424060604\right), x\_m, 0.1049934947\right) \cdot x\_m, x\_m, \mathsf{fma}\left(t\_0, 0.0001789971, 1\right)\right) \cdot x\_m}{\mathsf{fma}\left(0.0003579942, {\left(x\_m \cdot x\_m\right)}^{6}, \mathsf{fma}\left(0.0008327945, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot \left(\mathsf{fma}\left(0.0694555761 \cdot x\_m, x\_m, 0.0140005442 \cdot \left(\left(\left(x\_m \cdot x\_m\right) \cdot x\_m\right) \cdot x\_m\right)\right) - -0.2909738639\right), x\_m, 0.7715471019\right) \cdot x\_m, x\_m, 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x\_m}^{2}}}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (pow (* x_m x_m) 5.0)))
   (*
    x_s
    (if (<= x_m 5000.0)
      (/
       (*
        (fma
         (*
          (fma
           (*
            x_m
            (fma
             (* x_m (+ 0.0072644182 (* 0.0005064034 (* x_m x_m))))
             x_m
             0.0424060604))
           x_m
           0.1049934947)
          x_m)
         x_m
         (fma t_0 0.0001789971 1.0))
        x_m)
       (fma
        0.0003579942
        (pow (* x_m x_m) 6.0)
        (fma
         0.0008327945
         t_0
         (fma
          (*
           (fma
            (*
             x_m
             (-
              (fma
               (* 0.0694555761 x_m)
               x_m
               (* 0.0140005442 (* (* (* x_m x_m) x_m) x_m)))
              -0.2909738639))
            x_m
            0.7715471019)
           x_m)
          x_m
          1.0))))
      (/ (+ 0.5 (* 0.2514179000665374 (/ 1.0 (pow x_m 2.0)))) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = pow((x_m * x_m), 5.0);
	double tmp;
	if (x_m <= 5000.0) {
		tmp = (fma((fma((x_m * fma((x_m * (0.0072644182 + (0.0005064034 * (x_m * x_m)))), x_m, 0.0424060604)), x_m, 0.1049934947) * x_m), x_m, fma(t_0, 0.0001789971, 1.0)) * x_m) / fma(0.0003579942, pow((x_m * x_m), 6.0), fma(0.0008327945, t_0, fma((fma((x_m * (fma((0.0694555761 * x_m), x_m, (0.0140005442 * (((x_m * x_m) * x_m) * x_m))) - -0.2909738639)), x_m, 0.7715471019) * x_m), x_m, 1.0)));
	} else {
		tmp = (0.5 + (0.2514179000665374 * (1.0 / pow(x_m, 2.0)))) / 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(x_m * x_m) ^ 5.0
	tmp = 0.0
	if (x_m <= 5000.0)
		tmp = Float64(Float64(fma(Float64(fma(Float64(x_m * fma(Float64(x_m * Float64(0.0072644182 + Float64(0.0005064034 * Float64(x_m * x_m)))), x_m, 0.0424060604)), x_m, 0.1049934947) * x_m), x_m, fma(t_0, 0.0001789971, 1.0)) * x_m) / fma(0.0003579942, (Float64(x_m * x_m) ^ 6.0), fma(0.0008327945, t_0, fma(Float64(fma(Float64(x_m * Float64(fma(Float64(0.0694555761 * x_m), x_m, Float64(0.0140005442 * Float64(Float64(Float64(x_m * x_m) * x_m) * x_m))) - -0.2909738639)), x_m, 0.7715471019) * x_m), x_m, 1.0))));
	else
		tmp = Float64(Float64(0.5 + Float64(0.2514179000665374 * Float64(1.0 / (x_m ^ 2.0)))) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 5.0], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 5000.0], N[(N[(N[(N[(N[(N[(x$95$m * N[(N[(x$95$m * N[(0.0072644182 + N[(0.0005064034 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m + 0.0424060604), $MachinePrecision]), $MachinePrecision] * x$95$m + 0.1049934947), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + N[(t$95$0 * 0.0001789971 + 1.0), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(0.0003579942 * N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 6.0], $MachinePrecision] + N[(0.0008327945 * t$95$0 + N[(N[(N[(N[(x$95$m * N[(N[(N[(0.0694555761 * x$95$m), $MachinePrecision] * x$95$m + N[(0.0140005442 * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.2909738639), $MachinePrecision]), $MachinePrecision] * x$95$m + 0.7715471019), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(0.2514179000665374 * N[(1.0 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   9. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot \left(0.0072644182 + 0.0005064034 \cdot \left(x \cdot x\right)\right), x, 0.0424060604\right), x, 0.1049934947\right) \cdot x, x, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, 1\right)\right) \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \left(\mathsf{fma}\left(0.0694555761 \cdot x, x, 0.0140005442 \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right) - -0.2909738639\right), x, 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}} \]

   if 5e3 < x

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 51.6%

       \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{1}{{x\_m}^{2}}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.46:\\ \;\;\;\;\left(1 + {x\_m}^{2} \cdot \left(0.265709700396151 \cdot {x\_m}^{2} - 0.6665536072\right)\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{-1 \cdot \frac{0.15298196345929074 + 11.259630434457211 \cdot t\_0}{{x\_m}^{4}} - \left(0.5 + 0.2514179000665374 \cdot t\_0\right)}{x\_m}\\ \end{array} \end{array} \end{array} \]

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 (/ 1.0 (pow x_m 2.0))))
   (*
    x_s
    (if (<= x_m 1.46)
      (*
       (+
        1.0
        (* (pow x_m 2.0) (- (* 0.265709700396151 (pow x_m 2.0)) 0.6665536072)))
       x_m)
      (*
       -1.0
       (/
        (-
         (*
          -1.0
          (/ (+ 0.15298196345929074 (* 11.259630434457211 t_0)) (pow x_m 4.0)))
         (+ 0.5 (* 0.2514179000665374 t_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 = 1.0 / pow(x_m, 2.0);
	double tmp;
	if (x_m <= 1.46) {
		tmp = (1.0 + (pow(x_m, 2.0) * ((0.265709700396151 * pow(x_m, 2.0)) - 0.6665536072))) * x_m;
	} else {
		tmp = -1.0 * (((-1.0 * ((0.15298196345929074 + (11.259630434457211 * t_0)) / pow(x_m, 4.0))) - (0.5 + (0.2514179000665374 * t_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) :: tmp
    t_0 = 1.0d0 / (x_m ** 2.0d0)
    if (x_m <= 1.46d0) then
        tmp = (1.0d0 + ((x_m ** 2.0d0) * ((0.265709700396151d0 * (x_m ** 2.0d0)) - 0.6665536072d0))) * x_m
    else
        tmp = (-1.0d0) * ((((-1.0d0) * ((0.15298196345929074d0 + (11.259630434457211d0 * t_0)) / (x_m ** 4.0d0))) - (0.5d0 + (0.2514179000665374d0 * t_0))) / 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 = 1.0 / Math.pow(x_m, 2.0);
	double tmp;
	if (x_m <= 1.46) {
		tmp = (1.0 + (Math.pow(x_m, 2.0) * ((0.265709700396151 * Math.pow(x_m, 2.0)) - 0.6665536072))) * x_m;
	} else {
		tmp = -1.0 * (((-1.0 * ((0.15298196345929074 + (11.259630434457211 * t_0)) / Math.pow(x_m, 4.0))) - (0.5 + (0.2514179000665374 * t_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 = 1.0 / math.pow(x_m, 2.0)
	tmp = 0
	if x_m <= 1.46:
		tmp = (1.0 + (math.pow(x_m, 2.0) * ((0.265709700396151 * math.pow(x_m, 2.0)) - 0.6665536072))) * x_m
	else:
		tmp = -1.0 * (((-1.0 * ((0.15298196345929074 + (11.259630434457211 * t_0)) / math.pow(x_m, 4.0))) - (0.5 + (0.2514179000665374 * t_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(1.0 / (x_m ^ 2.0))
	tmp = 0.0
	if (x_m <= 1.46)
		tmp = Float64(Float64(1.0 + Float64((x_m ^ 2.0) * Float64(Float64(0.265709700396151 * (x_m ^ 2.0)) - 0.6665536072))) * x_m);
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(-1.0 * Float64(Float64(0.15298196345929074 + Float64(11.259630434457211 * t_0)) / (x_m ^ 4.0))) - Float64(0.5 + Float64(0.2514179000665374 * t_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 = 1.0 / (x_m ^ 2.0);
	tmp = 0.0;
	if (x_m <= 1.46)
		tmp = (1.0 + ((x_m ^ 2.0) * ((0.265709700396151 * (x_m ^ 2.0)) - 0.6665536072))) * x_m;
	else
		tmp = -1.0 * (((-1.0 * ((0.15298196345929074 + (11.259630434457211 * t_0)) / (x_m ^ 4.0))) - (0.5 + (0.2514179000665374 * t_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[(1.0 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 1.46], N[(N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(0.265709700396151 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(0.15298196345929074 + N[(11.259630434457211 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 + N[(0.2514179000665374 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.46

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 50.7

         \[\leadsto \left(1 + {x}^{2} \cdot \left(0.265709700396151 \cdot {x}^{2} - 0.6665536072\right)\right) \cdot x \]

   10. Applied rewrites 50.7%

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

   if 1.46 < x

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

   10. Applied rewrites 51.5%

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

Alternative 4: 99.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.46:\\ \;\;\;\;\left(1 + {x\_m}^{2} \cdot \left(0.265709700396151 \cdot {x\_m}^{2} - 0.6665536072\right)\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \left(\frac{0.15298196345929074}{{x\_m}^{4}} + \mathsf{fma}\left(0.2514179000665374, \frac{1}{{x\_m}^{2}}, 11.259630434457211 \cdot \frac{1}{{x\_m}^{6}}\right)\right)}{x\_m}\\ \end{array} \end{array} \]

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 1.46)
    (*
     (+
      1.0
      (* (pow x_m 2.0) (- (* 0.265709700396151 (pow x_m 2.0)) 0.6665536072)))
     x_m)
    (/
     (+
      0.5
      (+
       (/ 0.15298196345929074 (pow x_m 4.0))
       (fma
        0.2514179000665374
        (/ 1.0 (pow x_m 2.0))
        (* 11.259630434457211 (/ 1.0 (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 tmp;
	if (x_m <= 1.46) {
		tmp = (1.0 + (pow(x_m, 2.0) * ((0.265709700396151 * pow(x_m, 2.0)) - 0.6665536072))) * x_m;
	} else {
		tmp = (0.5 + ((0.15298196345929074 / pow(x_m, 4.0)) + fma(0.2514179000665374, (1.0 / pow(x_m, 2.0)), (11.259630434457211 * (1.0 / 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)
	tmp = 0.0
	if (x_m <= 1.46)
		tmp = Float64(Float64(1.0 + Float64((x_m ^ 2.0) * Float64(Float64(0.265709700396151 * (x_m ^ 2.0)) - 0.6665536072))) * x_m);
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.15298196345929074 / (x_m ^ 4.0)) + fma(0.2514179000665374, Float64(1.0 / (x_m ^ 2.0)), Float64(11.259630434457211 * Float64(1.0 / (x_m ^ 6.0)))))) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.46], N[(N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(0.265709700396151 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(0.5 + N[(N[(0.15298196345929074 / N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.2514179000665374 * N[(1.0 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(11.259630434457211 * N[(1.0 / N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.46

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 50.7

         \[\leadsto \left(1 + {x}^{2} \cdot \left(0.265709700396151 \cdot {x}^{2} - 0.6665536072\right)\right) \cdot x \]

   10. Applied rewrites 50.7%

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

   if 1.46 < x

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 51.5%

       \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.15298196345929074}{{x}^{4}} + \mathsf{fma}\left(0.2514179000665374, \frac{1}{{x}^{2}}, 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;\left(1 + {x\_m}^{2} \cdot \left(0.265709700396151 \cdot {x\_m}^{2} - 0.6665536072\right)\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \left(\frac{0.15298196345929074}{{x\_m}^{4}} + 0.2514179000665374 \cdot \frac{1}{{x\_m}^{2}}\right)}{x\_m}\\ \end{array} \end{array} \]

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)
    (*
     (+
      1.0
      (* (pow x_m 2.0) (- (* 0.265709700396151 (pow x_m 2.0)) 0.6665536072)))
     x_m)
    (/
     (+
      0.5
      (+
       (/ 0.15298196345929074 (pow x_m 4.0))
       (* 0.2514179000665374 (/ 1.0 (pow x_m 2.0)))))
     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 = (1.0 + (pow(x_m, 2.0) * ((0.265709700396151 * pow(x_m, 2.0)) - 0.6665536072))) * x_m;
	} else {
		tmp = (0.5 + ((0.15298196345929074 / pow(x_m, 4.0)) + (0.2514179000665374 * (1.0 / pow(x_m, 2.0))))) / 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) :: tmp
    if (x_m <= 1.15d0) then
        tmp = (1.0d0 + ((x_m ** 2.0d0) * ((0.265709700396151d0 * (x_m ** 2.0d0)) - 0.6665536072d0))) * x_m
    else
        tmp = (0.5d0 + ((0.15298196345929074d0 / (x_m ** 4.0d0)) + (0.2514179000665374d0 * (1.0d0 / (x_m ** 2.0d0))))) / 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 tmp;
	if (x_m <= 1.15) {
		tmp = (1.0 + (Math.pow(x_m, 2.0) * ((0.265709700396151 * Math.pow(x_m, 2.0)) - 0.6665536072))) * x_m;
	} else {
		tmp = (0.5 + ((0.15298196345929074 / Math.pow(x_m, 4.0)) + (0.2514179000665374 * (1.0 / Math.pow(x_m, 2.0))))) / 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):
	tmp = 0
	if x_m <= 1.15:
		tmp = (1.0 + (math.pow(x_m, 2.0) * ((0.265709700396151 * math.pow(x_m, 2.0)) - 0.6665536072))) * x_m
	else:
		tmp = (0.5 + ((0.15298196345929074 / math.pow(x_m, 4.0)) + (0.2514179000665374 * (1.0 / math.pow(x_m, 2.0))))) / 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(1.0 + Float64((x_m ^ 2.0) * Float64(Float64(0.265709700396151 * (x_m ^ 2.0)) - 0.6665536072))) * x_m);
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.15298196345929074 / (x_m ^ 4.0)) + Float64(0.2514179000665374 * Float64(1.0 / (x_m ^ 2.0))))) / 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)
	tmp = 0.0;
	if (x_m <= 1.15)
		tmp = (1.0 + ((x_m ^ 2.0) * ((0.265709700396151 * (x_m ^ 2.0)) - 0.6665536072))) * x_m;
	else
		tmp = (0.5 + ((0.15298196345929074 / (x_m ^ 4.0)) + (0.2514179000665374 * (1.0 / (x_m ^ 2.0))))) / 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_] := N[(x$95$s * If[LessEqual[x$95$m, 1.15], N[(N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(0.265709700396151 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(0.5 + N[(N[(0.15298196345929074 / N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.2514179000665374 * N[(1.0 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 50.7

         \[\leadsto \left(1 + {x}^{2} \cdot \left(0.265709700396151 \cdot {x}^{2} - 0.6665536072\right)\right) \cdot x \]

   10. Applied rewrites 50.7%

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

   if 1.1499999999999999 < x

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 51.6%

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

Alternative 6: 99.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\left(1 + {x\_m}^{2} \cdot \left(0.265709700396151 \cdot {x\_m}^{2} - 0.6665536072\right)\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x\_m}^{2}}}{x\_m}\\ \end{array} \end{array} \]

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)
    (*
     (+
      1.0
      (* (pow x_m 2.0) (- (* 0.265709700396151 (pow x_m 2.0)) 0.6665536072)))
     x_m)
    (/ (+ 0.5 (* 0.2514179000665374 (/ 1.0 (pow x_m 2.0)))) 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 = (1.0 + (pow(x_m, 2.0) * ((0.265709700396151 * pow(x_m, 2.0)) - 0.6665536072))) * x_m;
	} else {
		tmp = (0.5 + (0.2514179000665374 * (1.0 / pow(x_m, 2.0)))) / 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) :: tmp
    if (x_m <= 1.1d0) then
        tmp = (1.0d0 + ((x_m ** 2.0d0) * ((0.265709700396151d0 * (x_m ** 2.0d0)) - 0.6665536072d0))) * x_m
    else
        tmp = (0.5d0 + (0.2514179000665374d0 * (1.0d0 / (x_m ** 2.0d0)))) / 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 tmp;
	if (x_m <= 1.1) {
		tmp = (1.0 + (Math.pow(x_m, 2.0) * ((0.265709700396151 * Math.pow(x_m, 2.0)) - 0.6665536072))) * x_m;
	} else {
		tmp = (0.5 + (0.2514179000665374 * (1.0 / Math.pow(x_m, 2.0)))) / 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):
	tmp = 0
	if x_m <= 1.1:
		tmp = (1.0 + (math.pow(x_m, 2.0) * ((0.265709700396151 * math.pow(x_m, 2.0)) - 0.6665536072))) * x_m
	else:
		tmp = (0.5 + (0.2514179000665374 * (1.0 / math.pow(x_m, 2.0)))) / 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(1.0 + Float64((x_m ^ 2.0) * Float64(Float64(0.265709700396151 * (x_m ^ 2.0)) - 0.6665536072))) * x_m);
	else
		tmp = Float64(Float64(0.5 + Float64(0.2514179000665374 * Float64(1.0 / (x_m ^ 2.0)))) / 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)
	tmp = 0.0;
	if (x_m <= 1.1)
		tmp = (1.0 + ((x_m ^ 2.0) * ((0.265709700396151 * (x_m ^ 2.0)) - 0.6665536072))) * x_m;
	else
		tmp = (0.5 + (0.2514179000665374 * (1.0 / (x_m ^ 2.0)))) / 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_] := N[(x$95$s * If[LessEqual[x$95$m, 1.1], N[(N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(0.265709700396151 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(0.5 + N[(0.2514179000665374 * N[(1.0 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 50.7

         \[\leadsto \left(1 + {x}^{2} \cdot \left(0.265709700396151 \cdot {x}^{2} - 0.6665536072\right)\right) \cdot x \]

   10. Applied rewrites 50.7%

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

   if 1.1000000000000001 < x

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 51.6%

       \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.95:\\ \;\;\;\;\left(1 + -0.6665536072 \cdot {x\_m}^{2}\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x\_m}^{2}}}{x\_m}\\ \end{array} \end{array} \]

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)
    (* (+ 1.0 (* -0.6665536072 (pow x_m 2.0))) x_m)
    (/ (+ 0.5 (* 0.2514179000665374 (/ 1.0 (pow x_m 2.0)))) 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 = (1.0 + (-0.6665536072 * pow(x_m, 2.0))) * x_m;
	} else {
		tmp = (0.5 + (0.2514179000665374 * (1.0 / pow(x_m, 2.0)))) / 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) :: tmp
    if (x_m <= 0.95d0) then
        tmp = (1.0d0 + ((-0.6665536072d0) * (x_m ** 2.0d0))) * x_m
    else
        tmp = (0.5d0 + (0.2514179000665374d0 * (1.0d0 / (x_m ** 2.0d0)))) / 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 tmp;
	if (x_m <= 0.95) {
		tmp = (1.0 + (-0.6665536072 * Math.pow(x_m, 2.0))) * x_m;
	} else {
		tmp = (0.5 + (0.2514179000665374 * (1.0 / Math.pow(x_m, 2.0)))) / 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):
	tmp = 0
	if x_m <= 0.95:
		tmp = (1.0 + (-0.6665536072 * math.pow(x_m, 2.0))) * x_m
	else:
		tmp = (0.5 + (0.2514179000665374 * (1.0 / math.pow(x_m, 2.0)))) / 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(1.0 + Float64(-0.6665536072 * (x_m ^ 2.0))) * x_m);
	else
		tmp = Float64(Float64(0.5 + Float64(0.2514179000665374 * Float64(1.0 / (x_m ^ 2.0)))) / 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)
	tmp = 0.0;
	if (x_m <= 0.95)
		tmp = (1.0 + (-0.6665536072 * (x_m ^ 2.0))) * x_m;
	else
		tmp = (0.5 + (0.2514179000665374 * (1.0 / (x_m ^ 2.0)))) / 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_] := N[(x$95$s * If[LessEqual[x$95$m, 0.95], N[(N[(1.0 + N[(-0.6665536072 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(0.5 + N[(0.2514179000665374 * N[(1.0 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 49.8

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

   10. Applied rewrites 49.8%

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

   if 0.94999999999999996 < x

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 51.6%

       \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.8:\\ \;\;\;\;\left(1 + -0.6665536072 \cdot {x\_m}^{2}\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

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)
    (* (+ 1.0 (* -0.6665536072 (pow x_m 2.0))) 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 = (1.0 + (-0.6665536072 * pow(x_m, 2.0))) * 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 = (1.0d0 + ((-0.6665536072d0) * (x_m ** 2.0d0))) * 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 = (1.0 + (-0.6665536072 * Math.pow(x_m, 2.0))) * 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 = (1.0 + (-0.6665536072 * math.pow(x_m, 2.0))) * 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(1.0 + Float64(-0.6665536072 * (x_m ^ 2.0))) * 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 = (1.0 + (-0.6665536072 * (x_m ^ 2.0))) * 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[(1.0 + N[(-0.6665536072 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 49.8

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

   10. Applied rewrites 49.8%

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

   if 0.80000000000000004 < x

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 51.8

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

   10. Applied rewrites 51.8%

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

Alternative 9: 98.8% accurate, 31.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 0.72) x_m (/ 0.5 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.72) {
		tmp = x_m;
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Fortran

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

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

real(8) function code(x_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.72d0) then
        tmp = x_m
    else
        tmp = 0.5d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.72) {
		tmp = x_m;
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.72:
		tmp = x_m
	else:
		tmp = 0.5 / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.72)
		tmp = x_m;
	else
		tmp = Float64(0.5 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.72)
		tmp = x_m;
	else
		tmp = 0.5 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.72], x$95$m, N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 50.9%

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

   if 0.71999999999999997 < x

   1. Initial program 53.7%

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

   3. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

   7. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 51.8

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

   10. Applied rewrites 51.8%

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

Alternative 10: 50.9% accurate, 253.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.7%

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

3. Applied rewrites 53.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \cdot x} \]

5. Applied rewrites 53.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.0001789971 \cdot {\left(x \cdot x\right)}^{4}, \frac{x \cdot x}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}, \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot \left(x \cdot x\right), x, 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.0424060604, x \cdot x, 0.1049934947\right)\right), 1\right)}{\mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot \left(x \cdot x\right), x, 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right)\right), 1\right)\right)\right)}\right)} \cdot x \]

7. Applied rewrites 53.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 0.0001789971 \cdot \frac{x \cdot x}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0072644182 \cdot x, x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), 0.0424060604 \cdot x\right), 0.1049934947\right) \cdot x, x, 1\right)}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0008327945, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(0.0694555761 \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.0140005442\right), 0.2909738639 \cdot x\right), 0.7715471019\right) \cdot x, x, 1\right)\right)\right)}\right)} \cdot x \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 50.9%

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

Reproduce

herbie shell --seed 2025164 
(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))