Jmat.Real.dawson

Percentage Accurate: 54.1% → 100.0%

Time: 5.9s

Alternatives: 11

Speedup: 15.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 \cdot t\_0\\ t_3 := t\_2 \cdot t\_0\\ t_4 := t\_3 \cdot t\_0\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 500:\\ \;\;\;\;\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot t\_0\right) + 0.0424060604 \cdot t\_1\right) + 0.0072644182 \cdot t\_2\right) + 0.0005064034 \cdot t\_3\right) + 0.0001789971 \cdot t\_4}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot t\_0\right) + 0.2909738639 \cdot t\_1\right) + 0.0694555761 \cdot t\_2\right) + 0.0140005442 \cdot t\_3\right) + 0.0008327945 \cdot t\_4\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(t\_4 \cdot t\_0\right)} \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.15298196345929074}{t\_0} + 0.2514179000665374}{\left|x\right|}}{\left|x\right|} - -0.5}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x)))
        (t_1 (* t_0 t_0))
        (t_2 (* t_1 t_0))
        (t_3 (* t_2 t_0))
        (t_4 (* t_3 t_0)))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 500.0)
      (*
       (/
        (+
         (+
          (+
           (+ (+ 1.0 (* 0.1049934947 t_0)) (* 0.0424060604 t_1))
           (* 0.0072644182 t_2))
          (* 0.0005064034 t_3))
         (* 0.0001789971 t_4))
        (+
         (+
          (+
           (+
            (+ (+ 1.0 (* 0.7715471019 t_0)) (* 0.2909738639 t_1))
            (* 0.0694555761 t_2))
           (* 0.0140005442 t_3))
          (* 0.0008327945 t_4))
         (* (* 2.0 0.0001789971) (* t_4 t_0))))
       (fabs x))
      (/
       (-
        (/
         (/ (+ (/ 0.15298196345929074 t_0) 0.2514179000665374) (fabs x))
         (fabs x))
        -0.5)
       (fabs x))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 * t_0;
	double t_3 = t_2 * t_0;
	double t_4 = t_3 * t_0;
	double tmp;
	if (fabs(x) <= 500.0) {
		tmp = ((((((1.0 + (0.1049934947 * t_0)) + (0.0424060604 * t_1)) + (0.0072644182 * t_2)) + (0.0005064034 * t_3)) + (0.0001789971 * t_4)) / ((((((1.0 + (0.7715471019 * t_0)) + (0.2909738639 * t_1)) + (0.0694555761 * t_2)) + (0.0140005442 * t_3)) + (0.0008327945 * t_4)) + ((2.0 * 0.0001789971) * (t_4 * t_0)))) * fabs(x);
	} else {
		tmp = (((((0.15298196345929074 / t_0) + 0.2514179000665374) / fabs(x)) / fabs(x)) - -0.5) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * Math.abs(x);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 * t_0;
	double t_3 = t_2 * t_0;
	double t_4 = t_3 * t_0;
	double tmp;
	if (Math.abs(x) <= 500.0) {
		tmp = ((((((1.0 + (0.1049934947 * t_0)) + (0.0424060604 * t_1)) + (0.0072644182 * t_2)) + (0.0005064034 * t_3)) + (0.0001789971 * t_4)) / ((((((1.0 + (0.7715471019 * t_0)) + (0.2909738639 * t_1)) + (0.0694555761 * t_2)) + (0.0140005442 * t_3)) + (0.0008327945 * t_4)) + ((2.0 * 0.0001789971) * (t_4 * t_0)))) * Math.abs(x);
	} else {
		tmp = (((((0.15298196345929074 / t_0) + 0.2514179000665374) / Math.abs(x)) / Math.abs(x)) - -0.5) / Math.abs(x);
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) * math.fabs(x)
	t_1 = t_0 * t_0
	t_2 = t_1 * t_0
	t_3 = t_2 * t_0
	t_4 = t_3 * t_0
	tmp = 0
	if math.fabs(x) <= 500.0:
		tmp = ((((((1.0 + (0.1049934947 * t_0)) + (0.0424060604 * t_1)) + (0.0072644182 * t_2)) + (0.0005064034 * t_3)) + (0.0001789971 * t_4)) / ((((((1.0 + (0.7715471019 * t_0)) + (0.2909738639 * t_1)) + (0.0694555761 * t_2)) + (0.0140005442 * t_3)) + (0.0008327945 * t_4)) + ((2.0 * 0.0001789971) * (t_4 * t_0)))) * math.fabs(x)
	else:
		tmp = (((((0.15298196345929074 / t_0) + 0.2514179000665374) / math.fabs(x)) / math.fabs(x)) - -0.5) / math.fabs(x)
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(t_1 * t_0)
	t_3 = Float64(t_2 * t_0)
	t_4 = Float64(t_3 * t_0)
	tmp = 0.0
	if (abs(x) <= 500.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * t_0)) + Float64(0.0424060604 * t_1)) + Float64(0.0072644182 * t_2)) + Float64(0.0005064034 * t_3)) + Float64(0.0001789971 * t_4)) / Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.7715471019 * t_0)) + Float64(0.2909738639 * t_1)) + Float64(0.0694555761 * t_2)) + Float64(0.0140005442 * t_3)) + Float64(0.0008327945 * t_4)) + Float64(Float64(2.0 * 0.0001789971) * Float64(t_4 * t_0)))) * abs(x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.15298196345929074 / t_0) + 0.2514179000665374) / abs(x)) / abs(x)) - -0.5) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) * abs(x);
	t_1 = t_0 * t_0;
	t_2 = t_1 * t_0;
	t_3 = t_2 * t_0;
	t_4 = t_3 * t_0;
	tmp = 0.0;
	if (abs(x) <= 500.0)
		tmp = ((((((1.0 + (0.1049934947 * t_0)) + (0.0424060604 * t_1)) + (0.0072644182 * t_2)) + (0.0005064034 * t_3)) + (0.0001789971 * t_4)) / ((((((1.0 + (0.7715471019 * t_0)) + (0.2909738639 * t_1)) + (0.0694555761 * t_2)) + (0.0140005442 * t_3)) + (0.0008327945 * t_4)) + ((2.0 * 0.0001789971) * (t_4 * t_0)))) * abs(x);
	else
		tmp = (((((0.15298196345929074 / t_0) + 0.2514179000665374) / abs(x)) / abs(x)) - -0.5) / abs(x);
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 500.0], N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$4), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.15298196345929074 / t$95$0), $MachinePrecision] + 0.2514179000665374), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 54.1%

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

   if 500 < x

   1. Initial program 54.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 51.3%

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 51.3%

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

   6. Applied rewrites 51.3%

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

   8. Applied rewrites 51.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 51.3%

          \[\leadsto \frac{\frac{\frac{\frac{0.15298196345929074}{x \cdot x} + 0.2514179000665374}{x}}{x} - -0.5}{x} \]

   10. Applied rewrites 51.3%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := t\_0 \cdot \left|x\right|\\ t_2 := {t\_0}^{5}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, 0.0001789971, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(0.0072644182 \cdot t\_1, \left|x\right|, \left(0.0005064034 \cdot t\_1\right) \cdot t\_1\right), \mathsf{fma}\left(t\_0, 0.1049934947 + 0.0424060604 \cdot t\_0, 1\right)\right)\right) \cdot \left|x\right|}{\mathsf{fma}\left({t\_0}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, t\_2, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(0.0694555761 \cdot t\_1, \left|x\right|, \left(0.0140005442 \cdot t\_1\right) \cdot t\_1\right), \mathsf{fma}\left(t\_0, 0.7715471019 + 0.2909738639 \cdot t\_0, 1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{t\_0} - -0.5}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))) (t_1 (* t_0 (fabs x))) (t_2 (pow t_0 5.0)))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 5000.0)
      (/
       (*
        (fma
         t_2
         0.0001789971
         (fma
          t_0
          (fma (* 0.0072644182 t_1) (fabs x) (* (* 0.0005064034 t_1) t_1))
          (fma t_0 (+ 0.1049934947 (* 0.0424060604 t_0)) 1.0)))
        (fabs x))
       (fma
        (pow t_0 6.0)
        0.0003579942
        (fma
         0.0008327945
         t_2
         (fma
          t_0
          (fma (* 0.0694555761 t_1) (fabs x) (* (* 0.0140005442 t_1) t_1))
          (fma t_0 (+ 0.7715471019 (* 0.2909738639 t_0)) 1.0)))))
      (/ (- (/ 0.2514179000665374 t_0) -0.5) (fabs x))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = t_0 * fabs(x);
	double t_2 = pow(t_0, 5.0);
	double tmp;
	if (fabs(x) <= 5000.0) {
		tmp = (fma(t_2, 0.0001789971, fma(t_0, fma((0.0072644182 * t_1), fabs(x), ((0.0005064034 * t_1) * t_1)), fma(t_0, (0.1049934947 + (0.0424060604 * t_0)), 1.0))) * fabs(x)) / fma(pow(t_0, 6.0), 0.0003579942, fma(0.0008327945, t_2, fma(t_0, fma((0.0694555761 * t_1), fabs(x), ((0.0140005442 * t_1) * t_1)), fma(t_0, (0.7715471019 + (0.2909738639 * t_0)), 1.0))));
	} else {
		tmp = ((0.2514179000665374 / t_0) - -0.5) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = Float64(t_0 * abs(x))
	t_2 = t_0 ^ 5.0
	tmp = 0.0
	if (abs(x) <= 5000.0)
		tmp = Float64(Float64(fma(t_2, 0.0001789971, fma(t_0, fma(Float64(0.0072644182 * t_1), abs(x), Float64(Float64(0.0005064034 * t_1) * t_1)), fma(t_0, Float64(0.1049934947 + Float64(0.0424060604 * t_0)), 1.0))) * abs(x)) / fma((t_0 ^ 6.0), 0.0003579942, fma(0.0008327945, t_2, fma(t_0, fma(Float64(0.0694555761 * t_1), abs(x), Float64(Float64(0.0140005442 * t_1) * t_1)), fma(t_0, Float64(0.7715471019 + Float64(0.2909738639 * t_0)), 1.0)))));
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / t_0) - -0.5) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$0, 5.0], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 5000.0], N[(N[(N[(t$95$2 * 0.0001789971 + N[(t$95$0 * N[(N[(0.0072644182 * t$95$1), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(N[(0.0005064034 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(0.1049934947 + N[(0.0424060604 * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$0, 6.0], $MachinePrecision] * 0.0003579942 + N[(0.0008327945 * t$95$2 + N[(t$95$0 * N[(N[(0.0694555761 * t$95$1), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(N[(0.0140005442 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(0.7715471019 + N[(0.2909738639 * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.2514179000665374 / t$95$0), $MachinePrecision] - -0.5), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 54.1%

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

   3. Applied rewrites 54.1%

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

   if 5e3 < x

   1. Initial program 54.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 51.4%

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

   6. Applied rewrites 51.4%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|x\right|\right)}^{2}\\ t_1 := \left|x\right| \cdot \left|x\right|\\ t_2 := t\_1 \cdot \left|x\right|\\ t_3 := t\_1 \cdot t\_1\\ t_4 := t\_3 \cdot t\_1\\ t_5 := t\_4 \cdot t\_1\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.9:\\ \;\;\;\;\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot t\_1\right) + 0.0424060604 \cdot t\_3\right) + 0.0072644182 \cdot t\_4\right) + 0.0005064034 \cdot t\_5\right) + 0.0001789971 \cdot \left(t\_5 \cdot t\_1\right)}{1 + t\_0 \cdot \left(0.7715471019 + t\_0 \cdot \left(0.2909738639 + 0.0694555761 \cdot t\_0\right)\right)} \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{0.2514179000665374}{\left|x\right|} + \frac{0.15298196345929074}{t\_2}}{\left|x\right|} - -0.5\right) - \frac{-11.259630434457211}{t\_2 \cdot t\_2}}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (fabs x) 2.0))
        (t_1 (* (fabs x) (fabs x)))
        (t_2 (* t_1 (fabs x)))
        (t_3 (* t_1 t_1))
        (t_4 (* t_3 t_1))
        (t_5 (* t_4 t_1)))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 1.9)
      (*
       (/
        (+
         (+
          (+
           (+ (+ 1.0 (* 0.1049934947 t_1)) (* 0.0424060604 t_3))
           (* 0.0072644182 t_4))
          (* 0.0005064034 t_5))
         (* 0.0001789971 (* t_5 t_1)))
        (+
         1.0
         (*
          t_0
          (+ 0.7715471019 (* t_0 (+ 0.2909738639 (* 0.0694555761 t_0)))))))
       (fabs x))
      (/
       (-
        (-
         (/
          (+ (/ 0.2514179000665374 (fabs x)) (/ 0.15298196345929074 t_2))
          (fabs x))
         -0.5)
        (/ -11.259630434457211 (* t_2 t_2)))
       (fabs x))))))

C

double code(double x) {
	double t_0 = pow(fabs(x), 2.0);
	double t_1 = fabs(x) * fabs(x);
	double t_2 = t_1 * fabs(x);
	double t_3 = t_1 * t_1;
	double t_4 = t_3 * t_1;
	double t_5 = t_4 * t_1;
	double tmp;
	if (fabs(x) <= 1.9) {
		tmp = ((((((1.0 + (0.1049934947 * t_1)) + (0.0424060604 * t_3)) + (0.0072644182 * t_4)) + (0.0005064034 * t_5)) + (0.0001789971 * (t_5 * t_1))) / (1.0 + (t_0 * (0.7715471019 + (t_0 * (0.2909738639 + (0.0694555761 * t_0))))))) * fabs(x);
	} else {
		tmp = (((((0.2514179000665374 / fabs(x)) + (0.15298196345929074 / t_2)) / fabs(x)) - -0.5) - (-11.259630434457211 / (t_2 * t_2))) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.abs(x), 2.0);
	double t_1 = Math.abs(x) * Math.abs(x);
	double t_2 = t_1 * Math.abs(x);
	double t_3 = t_1 * t_1;
	double t_4 = t_3 * t_1;
	double t_5 = t_4 * t_1;
	double tmp;
	if (Math.abs(x) <= 1.9) {
		tmp = ((((((1.0 + (0.1049934947 * t_1)) + (0.0424060604 * t_3)) + (0.0072644182 * t_4)) + (0.0005064034 * t_5)) + (0.0001789971 * (t_5 * t_1))) / (1.0 + (t_0 * (0.7715471019 + (t_0 * (0.2909738639 + (0.0694555761 * t_0))))))) * Math.abs(x);
	} else {
		tmp = (((((0.2514179000665374 / Math.abs(x)) + (0.15298196345929074 / t_2)) / Math.abs(x)) - -0.5) - (-11.259630434457211 / (t_2 * t_2))) / Math.abs(x);
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	t_0 = math.pow(math.fabs(x), 2.0)
	t_1 = math.fabs(x) * math.fabs(x)
	t_2 = t_1 * math.fabs(x)
	t_3 = t_1 * t_1
	t_4 = t_3 * t_1
	t_5 = t_4 * t_1
	tmp = 0
	if math.fabs(x) <= 1.9:
		tmp = ((((((1.0 + (0.1049934947 * t_1)) + (0.0424060604 * t_3)) + (0.0072644182 * t_4)) + (0.0005064034 * t_5)) + (0.0001789971 * (t_5 * t_1))) / (1.0 + (t_0 * (0.7715471019 + (t_0 * (0.2909738639 + (0.0694555761 * t_0))))))) * math.fabs(x)
	else:
		tmp = (((((0.2514179000665374 / math.fabs(x)) + (0.15298196345929074 / t_2)) / math.fabs(x)) - -0.5) - (-11.259630434457211 / (t_2 * t_2))) / math.fabs(x)
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	t_0 = abs(x) ^ 2.0
	t_1 = Float64(abs(x) * abs(x))
	t_2 = Float64(t_1 * abs(x))
	t_3 = Float64(t_1 * t_1)
	t_4 = Float64(t_3 * t_1)
	t_5 = Float64(t_4 * t_1)
	tmp = 0.0
	if (abs(x) <= 1.9)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * t_1)) + Float64(0.0424060604 * t_3)) + Float64(0.0072644182 * t_4)) + Float64(0.0005064034 * t_5)) + Float64(0.0001789971 * Float64(t_5 * t_1))) / Float64(1.0 + Float64(t_0 * Float64(0.7715471019 + Float64(t_0 * Float64(0.2909738639 + Float64(0.0694555761 * t_0))))))) * abs(x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.2514179000665374 / abs(x)) + Float64(0.15298196345929074 / t_2)) / abs(x)) - -0.5) - Float64(-11.259630434457211 / Float64(t_2 * t_2))) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) ^ 2.0;
	t_1 = abs(x) * abs(x);
	t_2 = t_1 * abs(x);
	t_3 = t_1 * t_1;
	t_4 = t_3 * t_1;
	t_5 = t_4 * t_1;
	tmp = 0.0;
	if (abs(x) <= 1.9)
		tmp = ((((((1.0 + (0.1049934947 * t_1)) + (0.0424060604 * t_3)) + (0.0072644182 * t_4)) + (0.0005064034 * t_5)) + (0.0001789971 * (t_5 * t_1))) / (1.0 + (t_0 * (0.7715471019 + (t_0 * (0.2909738639 + (0.0694555761 * t_0))))))) * abs(x);
	else
		tmp = (((((0.2514179000665374 / abs(x)) + (0.15298196345929074 / t_2)) / abs(x)) - -0.5) - (-11.259630434457211 / (t_2 * t_2))) / abs(x);
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * t$95$1), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.9], N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$5), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * N[(t$95$5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 * N[(0.7715471019 + N[(t$95$0 * N[(0.2909738639 + N[(0.0694555761 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.2514179000665374 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.15298196345929074 / t$95$2), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] - N[(-11.259630434457211 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 54.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{7715471019}{10000000000} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + \frac{694555761}{10000000000} \cdot {x}^{2}\right)\right)}} \cdot x \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 50.2%

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

   4. Applied rewrites 50.2%

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

   if 1.8999999999999999 < x

   1. Initial program 54.1%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 51.3%

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

   6. Applied rewrites 51.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. add-flip N/A

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

      5. associate-+r- N/A

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

      6. lift-+.f64 N/A

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

      7. lower--.f64 N/A

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

   8. Applied rewrites 51.3%

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

Alternative 4: 99.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|x\right|\right)}^{2}\\ t_1 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.5:\\ \;\;\;\;\left(1 + t\_0 \cdot \left(t\_0 \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot t\_0\right) - 0.6665536072\right)\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{0.2514179000665374}{\left|x\right|} + \frac{0.15298196345929074}{t\_1}}{\left|x\right|} - -0.5\right) - \frac{-11.259630434457211}{t\_1 \cdot t\_1}}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (fabs x) 2.0)) (t_1 (* (* (fabs x) (fabs x)) (fabs x))))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 1.5)
      (*
       (+
        1.0
        (*
         t_0
         (-
          (* t_0 (+ 0.265709700396151 (* -0.0732490286039007 t_0)))
          0.6665536072)))
       (fabs x))
      (/
       (-
        (-
         (/
          (+ (/ 0.2514179000665374 (fabs x)) (/ 0.15298196345929074 t_1))
          (fabs x))
         -0.5)
        (/ -11.259630434457211 (* t_1 t_1)))
       (fabs x))))))

C

double code(double x) {
	double t_0 = pow(fabs(x), 2.0);
	double t_1 = (fabs(x) * fabs(x)) * fabs(x);
	double tmp;
	if (fabs(x) <= 1.5) {
		tmp = (1.0 + (t_0 * ((t_0 * (0.265709700396151 + (-0.0732490286039007 * t_0))) - 0.6665536072))) * fabs(x);
	} else {
		tmp = (((((0.2514179000665374 / fabs(x)) + (0.15298196345929074 / t_1)) / fabs(x)) - -0.5) - (-11.259630434457211 / (t_1 * t_1))) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.abs(x), 2.0);
	double t_1 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 1.5) {
		tmp = (1.0 + (t_0 * ((t_0 * (0.265709700396151 + (-0.0732490286039007 * t_0))) - 0.6665536072))) * Math.abs(x);
	} else {
		tmp = (((((0.2514179000665374 / Math.abs(x)) + (0.15298196345929074 / t_1)) / Math.abs(x)) - -0.5) - (-11.259630434457211 / (t_1 * t_1))) / Math.abs(x);
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	t_0 = math.pow(math.fabs(x), 2.0)
	t_1 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 1.5:
		tmp = (1.0 + (t_0 * ((t_0 * (0.265709700396151 + (-0.0732490286039007 * t_0))) - 0.6665536072))) * math.fabs(x)
	else:
		tmp = (((((0.2514179000665374 / math.fabs(x)) + (0.15298196345929074 / t_1)) / math.fabs(x)) - -0.5) - (-11.259630434457211 / (t_1 * t_1))) / math.fabs(x)
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	t_0 = abs(x) ^ 2.0
	t_1 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(x) <= 1.5)
		tmp = Float64(Float64(1.0 + Float64(t_0 * Float64(Float64(t_0 * Float64(0.265709700396151 + Float64(-0.0732490286039007 * t_0))) - 0.6665536072))) * abs(x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.2514179000665374 / abs(x)) + Float64(0.15298196345929074 / t_1)) / abs(x)) - -0.5) - Float64(-11.259630434457211 / Float64(t_1 * t_1))) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) ^ 2.0;
	t_1 = (abs(x) * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs(x) <= 1.5)
		tmp = (1.0 + (t_0 * ((t_0 * (0.265709700396151 + (-0.0732490286039007 * t_0))) - 0.6665536072))) * abs(x);
	else
		tmp = (((((0.2514179000665374 / abs(x)) + (0.15298196345929074 / t_1)) / abs(x)) - -0.5) - (-11.259630434457211 / (t_1 * t_1))) / abs(x);
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.5], N[(N[(1.0 + N[(t$95$0 * N[(N[(t$95$0 * N[(0.265709700396151 + N[(-0.0732490286039007 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.2514179000665374 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.15298196345929074 / t$95$1), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] - N[(-11.259630434457211 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 54.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right) - 0.6665536072\right)\right)} \cdot x \]

   if 1.5 < x

   1. Initial program 54.1%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 51.3%

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

   6. Applied rewrites 51.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. add-flip N/A

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

      5. associate-+r- N/A

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

      6. lift-+.f64 N/A

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

      7. lower--.f64 N/A

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

   8. Applied rewrites 51.3%

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

Alternative 5: 99.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := {\left(\left|x\right|\right)}^{2}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.45:\\ \;\;\;\;\left(1 + t\_1 \cdot \left(0.265709700396151 \cdot t\_1 - 0.6665536072\right)\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{0.2514179000665374}{\left|x\right|} + \frac{0.15298196345929074}{t\_0}}{\left|x\right|} - -0.5\right) - \frac{-11.259630434457211}{t\_0 \cdot t\_0}}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x))) (t_1 (pow (fabs x) 2.0)))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 1.45)
      (* (+ 1.0 (* t_1 (- (* 0.265709700396151 t_1) 0.6665536072))) (fabs x))
      (/
       (-
        (-
         (/
          (+ (/ 0.2514179000665374 (fabs x)) (/ 0.15298196345929074 t_0))
          (fabs x))
         -0.5)
        (/ -11.259630434457211 (* t_0 t_0)))
       (fabs x))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = pow(fabs(x), 2.0);
	double tmp;
	if (fabs(x) <= 1.45) {
		tmp = (1.0 + (t_1 * ((0.265709700396151 * t_1) - 0.6665536072))) * fabs(x);
	} else {
		tmp = (((((0.2514179000665374 / fabs(x)) + (0.15298196345929074 / t_0)) / fabs(x)) - -0.5) - (-11.259630434457211 / (t_0 * t_0))) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = Math.pow(Math.abs(x), 2.0);
	double tmp;
	if (Math.abs(x) <= 1.45) {
		tmp = (1.0 + (t_1 * ((0.265709700396151 * t_1) - 0.6665536072))) * Math.abs(x);
	} else {
		tmp = (((((0.2514179000665374 / Math.abs(x)) + (0.15298196345929074 / t_0)) / Math.abs(x)) - -0.5) - (-11.259630434457211 / (t_0 * t_0))) / Math.abs(x);
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = math.pow(math.fabs(x), 2.0)
	tmp = 0
	if math.fabs(x) <= 1.45:
		tmp = (1.0 + (t_1 * ((0.265709700396151 * t_1) - 0.6665536072))) * math.fabs(x)
	else:
		tmp = (((((0.2514179000665374 / math.fabs(x)) + (0.15298196345929074 / t_0)) / math.fabs(x)) - -0.5) - (-11.259630434457211 / (t_0 * t_0))) / math.fabs(x)
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = abs(x) ^ 2.0
	tmp = 0.0
	if (abs(x) <= 1.45)
		tmp = Float64(Float64(1.0 + Float64(t_1 * Float64(Float64(0.265709700396151 * t_1) - 0.6665536072))) * abs(x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.2514179000665374 / abs(x)) + Float64(0.15298196345929074 / t_0)) / abs(x)) - -0.5) - Float64(-11.259630434457211 / Float64(t_0 * t_0))) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = abs(x) ^ 2.0;
	tmp = 0.0;
	if (abs(x) <= 1.45)
		tmp = (1.0 + (t_1 * ((0.265709700396151 * t_1) - 0.6665536072))) * abs(x);
	else
		tmp = (((((0.2514179000665374 / abs(x)) + (0.15298196345929074 / t_0)) / abs(x)) - -0.5) - (-11.259630434457211 / (t_0 * t_0))) / abs(x);
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.45], N[(N[(1.0 + N[(t$95$1 * N[(N[(0.265709700396151 * t$95$1), $MachinePrecision] - 0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.2514179000665374 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.15298196345929074 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] - N[(-11.259630434457211 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45

   1. Initial program 54.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 51.1%

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

   if 1.45 < x

   1. Initial program 54.1%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 51.3%

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

   6. Applied rewrites 51.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. add-flip N/A

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

      5. associate-+r- N/A

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

      6. lift-+.f64 N/A

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

      7. lower--.f64 N/A

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

   8. Applied rewrites 51.3%

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

Alternative 6: 99.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := t\_0 \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -0.6665536072, 1\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{0.2514179000665374}{\left|x\right|} + \frac{0.15298196345929074}{t\_1}}{\left|x\right|} - -0.5\right) - \frac{-11.259630434457211}{t\_1 \cdot t\_1}}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))) (t_1 (* t_0 (fabs x))))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 1.25)
      (* (fma t_0 -0.6665536072 1.0) (fabs x))
      (/
       (-
        (-
         (/
          (+ (/ 0.2514179000665374 (fabs x)) (/ 0.15298196345929074 t_1))
          (fabs x))
         -0.5)
        (/ -11.259630434457211 (* t_1 t_1)))
       (fabs x))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = t_0 * fabs(x);
	double tmp;
	if (fabs(x) <= 1.25) {
		tmp = fma(t_0, -0.6665536072, 1.0) * fabs(x);
	} else {
		tmp = (((((0.2514179000665374 / fabs(x)) + (0.15298196345929074 / t_1)) / fabs(x)) - -0.5) - (-11.259630434457211 / (t_1 * t_1))) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = Float64(t_0 * abs(x))
	tmp = 0.0
	if (abs(x) <= 1.25)
		tmp = Float64(fma(t_0, -0.6665536072, 1.0) * abs(x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.2514179000665374 / abs(x)) + Float64(0.15298196345929074 / t_1)) / abs(x)) - -0.5) - Float64(-11.259630434457211 / Float64(t_1 * t_1))) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.25], N[(N[(t$95$0 * -0.6665536072 + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.2514179000665374 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.15298196345929074 / t$95$1), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] - N[(-11.259630434457211 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 54.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.2%

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

   4. Applied rewrites 50.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

   if 1.25 < x

   1. Initial program 54.1%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 51.3%

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

   6. Applied rewrites 51.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. add-flip N/A

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

      5. associate-+r- N/A

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

      6. lift-+.f64 N/A

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

      7. lower--.f64 N/A

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

   8. Applied rewrites 51.3%

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

Alternative 7: 99.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -0.6665536072, 1\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.15298196345929074}{t\_0} + 0.2514179000665374}{\left|x\right|}}{\left|x\right|} - -0.5}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 1.25)
      (* (fma t_0 -0.6665536072 1.0) (fabs x))
      (/
       (-
        (/
         (/ (+ (/ 0.15298196345929074 t_0) 0.2514179000665374) (fabs x))
         (fabs x))
        -0.5)
       (fabs x))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 1.25) {
		tmp = fma(t_0, -0.6665536072, 1.0) * fabs(x);
	} else {
		tmp = (((((0.15298196345929074 / t_0) + 0.2514179000665374) / fabs(x)) / fabs(x)) - -0.5) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 1.25)
		tmp = Float64(fma(t_0, -0.6665536072, 1.0) * abs(x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.15298196345929074 / t_0) + 0.2514179000665374) / abs(x)) / abs(x)) - -0.5) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.25], N[(N[(t$95$0 * -0.6665536072 + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.15298196345929074 / t$95$0), $MachinePrecision] + 0.2514179000665374), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 54.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.2%

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

   4. Applied rewrites 50.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

   if 1.25 < x

   1. Initial program 54.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 51.3%

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. mult-flip-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 51.3%

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

   6. Applied rewrites 51.3%

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

   8. Applied rewrites 51.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 51.3%

          \[\leadsto \frac{\frac{\frac{\frac{0.15298196345929074}{x \cdot x} + 0.2514179000665374}{x}}{x} - -0.5}{x} \]

   10. Applied rewrites 51.3%

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

Alternative 8: 99.5% accurate, 9.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -0.6665536072, 1\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{t\_0} - -0.5}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 1.25)
      (* (fma t_0 -0.6665536072 1.0) (fabs x))
      (/ (- (/ 0.2514179000665374 t_0) -0.5) (fabs x))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 1.25) {
		tmp = fma(t_0, -0.6665536072, 1.0) * fabs(x);
	} else {
		tmp = ((0.2514179000665374 / t_0) - -0.5) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 1.25)
		tmp = Float64(fma(t_0, -0.6665536072, 1.0) * abs(x));
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / t_0) - -0.5) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.25], N[(N[(t$95$0 * -0.6665536072 + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.2514179000665374 / t$95$0), $MachinePrecision] - -0.5), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 54.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.2%

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

   4. Applied rewrites 50.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

   if 1.25 < x

   1. Initial program 54.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 51.4%

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

   6. Applied rewrites 51.4%

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

Alternative 9: 99.3% accurate, 10.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (copysign 1.0 x)
  (if (<= (fabs x) 1.25)
    (* (fma (* (fabs x) (fabs x)) -0.6665536072 1.0) (fabs x))
    (/ 0.5 (fabs x)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1.25) {
		tmp = fma((fabs(x) * fabs(x)), -0.6665536072, 1.0) * fabs(x);
	} else {
		tmp = 0.5 / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1.25)
		tmp = Float64(fma(Float64(abs(x) * abs(x)), -0.6665536072, 1.0) * abs(x));
	else
		tmp = Float64(0.5 / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.25], N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * -0.6665536072 + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 54.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.2%

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

   4. Applied rewrites 50.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

   if 1.25 < x

   1. Initial program 54.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 51.6%

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

   4. Applied rewrites 51.6%

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

Alternative 10: 99.0% accurate, 15.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (copysign 1.0 x) (if (<= (fabs x) 0.72) (fabs x) (/ 0.5 (fabs x)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.72)
		tmp = abs(x);
	else
		tmp = 0.5 / abs(x);
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 0.72], N[Abs[x], $MachinePrecision], N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 54.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.3%

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

   if 0.71999999999999997 < x

   1. Initial program 54.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 51.6%

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

   4. Applied rewrites 51.6%

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

Alternative 11: 51.3% accurate, 255.4× speedup

Math

\[x \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = x
end function

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 51.3%

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

Reproduce

herbie shell --seed 2025201 
(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))