Jmat.Real.dawson

Percentage Accurate: 54.6% → 100.0%

Time: 5.2s

Alternatives: 13

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: 54.6% 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

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 \cdot t\_0\\ t_3 := t\_2 \cdot t\_0\\ t_4 := t\_3 \cdot t\_0\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 20000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(0.0424060604 \cdot \left|x\right|, \left|x\right|, \left(0.0072644182 \cdot \left(t\_0 \cdot \left|x\right|\right)\right) \cdot \left|x\right|\right), \mathsf{fma}\left(t\_0, 0.1049934947, 1\right)\right) - \mathsf{fma}\left(-0.0005064034, {\left(\left|x\right|\right)}^{8}, -0.0001789971 \cdot {t\_0}^{5}\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot t\_0\right) + 0.2909738639 \cdot t\_1\right) + 0.0694555761 \cdot t\_2\right) + 0.0140005442 \cdot t\_3\right) + 0.0008327945 \cdot t\_4\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(t\_4 \cdot t\_0\right)} \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|x\right|} \cdot \left(\frac{0.2514179000665374}{t\_0} - -0.5\right)\\ \end{array} \end{array} \]

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) 20000000.0)
     (*
      (/
       (-
        (fma
         t_0
         (fma
          (* 0.0424060604 (fabs x))
          (fabs x)
          (* (* 0.0072644182 (* t_0 (fabs x))) (fabs x)))
         (fma t_0 0.1049934947 1.0))
        (fma
         -0.0005064034
         (pow (fabs x) 8.0)
         (* -0.0001789971 (pow t_0 5.0))))
       (+
        (+
         (+
          (+
           (+ (+ 1.0 (* 0.7715471019 t_0)) (* 0.2909738639 t_1))
           (* 0.0694555761 t_2))
          (* 0.0140005442 t_3))
         (* 0.0008327945 t_4))
        (* (* 2.0 0.0001789971) (* t_4 t_0))))
      (fabs x))
     (* (/ 1.0 (fabs x)) (- (/ 0.2514179000665374 t_0) -0.5))))))

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) <= 20000000.0) {
		tmp = ((fma(t_0, fma((0.0424060604 * fabs(x)), fabs(x), ((0.0072644182 * (t_0 * fabs(x))) * fabs(x))), fma(t_0, 0.1049934947, 1.0)) - fma(-0.0005064034, pow(fabs(x), 8.0), (-0.0001789971 * pow(t_0, 5.0)))) / ((((((1.0 + (0.7715471019 * t_0)) + (0.2909738639 * t_1)) + (0.0694555761 * t_2)) + (0.0140005442 * t_3)) + (0.0008327945 * t_4)) + ((2.0 * 0.0001789971) * (t_4 * t_0)))) * fabs(x);
	} else {
		tmp = (1.0 / fabs(x)) * ((0.2514179000665374 / t_0) - -0.5);
	}
	return 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) <= 20000000.0)
		tmp = Float64(Float64(Float64(fma(t_0, fma(Float64(0.0424060604 * abs(x)), abs(x), Float64(Float64(0.0072644182 * Float64(t_0 * abs(x))) * abs(x))), fma(t_0, 0.1049934947, 1.0)) - fma(-0.0005064034, (abs(x) ^ 8.0), Float64(-0.0001789971 * (t_0 ^ 5.0)))) / 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(1.0 / abs(x)) * Float64(Float64(0.2514179000665374 / t_0) - -0.5));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, 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], 20000000.0], N[(N[(N[(N[(t$95$0 * N[(N[(0.0424060604 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(N[(0.0072644182 * N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.1049934947 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-0.0005064034 * N[Power[N[Abs[x], $MachinePrecision], 8.0], $MachinePrecision] + N[(-0.0001789971 * N[Power[t$95$0, 5.0], $MachinePrecision]), $MachinePrecision]), $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[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(0.2514179000665374 / t$95$0), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e7

   1. Initial program 54.6%

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

   3. Applied rewrites 54.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0424060604 \cdot x, x, \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right), \mathsf{fma}\left(x \cdot x, 0.1049934947, 1\right)\right) - \mathsf{fma}\left(-0.0005064034, {x}^{8}, -0.0001789971 \cdot {\left(x \cdot x\right)}^{5}\right)}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   if 2e7 < x

   1. Initial program 54.6%

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

   2. 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 50.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 50.6%

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. mult-flip-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 50.6%

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

   6. Applied rewrites 50.6%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := t\_0 \cdot \left|x\right|\\ t_2 := {t\_0}^{5}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 40:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_2, 0.0001789971, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(0.0072644182 \cdot t\_1, \left|x\right|, \left(0.0005064034 \cdot t\_1\right) \cdot t\_1\right), \mathsf{fma}\left(t\_0, 0.1049934947 + 0.0424060604 \cdot t\_0, 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({t\_0}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, t\_2, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(0.0694555761 \cdot t\_1, \left|x\right|, \left(0.0140005442 \cdot t\_1\right) \cdot t\_1\right), \mathsf{fma}\left(t\_0, 0.2909738639, 0.7715471019\right) \cdot t\_0 - -1\right)\right)\right)}\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{t\_0} - -0.5}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x)))
       (t_1 (* t_0 (fabs x)))
       (t_2 (pow t_0 5.0)))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 40.0)
     (*
      (*
       (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)))
       (/
        1.0
        (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 0.2909738639 0.7715471019) t_0) -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 t_1 = t_0 * fabs(x);
	double t_2 = pow(t_0, 5.0);
	double tmp;
	if (fabs(x) <= 40.0) {
		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))) * (1.0 / 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, 0.2909738639, 0.7715471019) * t_0) - -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))
	t_1 = Float64(t_0 * abs(x))
	t_2 = t_0 ^ 5.0
	tmp = 0.0
	if (abs(x) <= 40.0)
		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))) * Float64(1.0 / 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)), Float64(Float64(fma(t_0, 0.2909738639, 0.7715471019) * t_0) - -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]}, 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], 40.0], 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[(1.0 / 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[(N[(N[(t$95$0 * 0.2909738639 + 0.7715471019), $MachinePrecision] * t$95$0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 40

   1. Initial program 54.6%

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

   3. Applied rewrites 54.6%

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

      1. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, \frac{1789971}{10000000000}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{36322091}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{2532017}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{1049934947}{10000000000} + \frac{106015151}{2500000000} \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \color{blue}{\left(x \cdot x\right) \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right)\right) + 1}\right)\right)\right)}\right) \cdot x \]

      2. add-flip N/A

         \[\leadsto \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, \frac{1789971}{10000000000}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{36322091}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{2532017}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{1049934947}{10000000000} + \frac{106015151}{2500000000} \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \color{blue}{\left(x \cdot x\right) \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right)\right) - \left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)}\right) \cdot x \]

      3. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, \frac{1789971}{10000000000}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{36322091}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{2532017}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{1049934947}{10000000000} + \frac{106015151}{2500000000} \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(x \cdot x\right) \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right)\right) - \color{blue}{-1}\right)\right)\right)}\right) \cdot x \]

      4. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, \frac{1789971}{10000000000}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{36322091}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{2532017}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{1049934947}{10000000000} + \frac{106015151}{2500000000} \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \color{blue}{\left(x \cdot x\right) \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right)\right) - -1}\right)\right)\right)}\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, \frac{1789971}{10000000000}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{36322091}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{2532017}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{1049934947}{10000000000} + \frac{106015151}{2500000000} \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \color{blue}{\left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} - -1\right)\right)\right)}\right) \cdot x \]

      6. lower-*.f64 54.6%

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

      7. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, \frac{1789971}{10000000000}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{36322091}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{2532017}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{1049934947}{10000000000} + \frac{106015151}{2500000000} \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \color{blue}{\left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right)\right)} \cdot \left(x \cdot x\right) - -1\right)\right)\right)}\right) \cdot x \]

      8. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, \frac{1789971}{10000000000}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{36322091}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{2532017}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{1049934947}{10000000000} + \frac{106015151}{2500000000} \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \color{blue}{\left(\frac{2909738639}{10000000000} \cdot \left(x \cdot x\right) + \frac{7715471019}{10000000000}\right)} \cdot \left(x \cdot x\right) - -1\right)\right)\right)}\right) \cdot x \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, \frac{1789971}{10000000000}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{36322091}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{2532017}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{1049934947}{10000000000} + \frac{106015151}{2500000000} \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\color{blue}{\frac{2909738639}{10000000000} \cdot \left(x \cdot x\right)} + \frac{7715471019}{10000000000}\right) \cdot \left(x \cdot x\right) - -1\right)\right)\right)}\right) \cdot x \]

      10. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, \frac{1789971}{10000000000}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{36322091}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{2532017}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{1049934947}{10000000000} + \frac{106015151}{2500000000} \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \left(\color{blue}{\left(x \cdot x\right) \cdot \frac{2909738639}{10000000000}} + \frac{7715471019}{10000000000}\right) \cdot \left(x \cdot x\right) - -1\right)\right)\right)}\right) \cdot x \]

      11. lower-fma.f64 54.6%

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

   5. Applied rewrites 54.6%

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

   if 40 < x

   1. Initial program 54.6%

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

   2. 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 50.6%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   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 50.6%

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

   6. Applied rewrites 50.6%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := t\_0 \cdot \left|x\right|\\ t_2 := {t\_0}^{5}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 40:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_2, 0.0001789971, \mathsf{fma}\left(\mathsf{fma}\left(t\_1 \cdot 0.0005064034, t\_1, \left(t\_1 \cdot 0.0072644182\right) \cdot \left|x\right|\right), t\_0, \mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.0424060604, 0.1049934947\right), t\_0, 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({t\_0}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, t\_2, \mathsf{fma}\left(\mathsf{fma}\left(t\_1 \cdot 0.0140005442, t\_1, \left(t\_1 \cdot 0.0694555761\right) \cdot \left|x\right|\right), t\_0, \mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.2909738639, 0.7715471019\right), t\_0, 1\right)\right)\right)\right)}\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{t\_0} - -0.5}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x)))
       (t_1 (* t_0 (fabs x)))
       (t_2 (pow t_0 5.0)))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 40.0)
     (*
      (*
       (fma
        t_2
        0.0001789971
        (fma
         (fma
          (* t_1 0.0005064034)
          t_1
          (* (* t_1 0.0072644182) (fabs x)))
         t_0
         (fma (fma t_0 0.0424060604 0.1049934947) t_0 1.0)))
       (/
        1.0
        (fma
         (pow t_0 6.0)
         0.0003579942
         (fma
          0.0008327945
          t_2
          (fma
           (fma
            (* t_1 0.0140005442)
            t_1
            (* (* t_1 0.0694555761) (fabs x)))
           t_0
           (fma (fma t_0 0.2909738639 0.7715471019) t_0 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 t_1 = t_0 * fabs(x);
	double t_2 = pow(t_0, 5.0);
	double tmp;
	if (fabs(x) <= 40.0) {
		tmp = (fma(t_2, 0.0001789971, fma(fma((t_1 * 0.0005064034), t_1, ((t_1 * 0.0072644182) * fabs(x))), t_0, fma(fma(t_0, 0.0424060604, 0.1049934947), t_0, 1.0))) * (1.0 / fma(pow(t_0, 6.0), 0.0003579942, fma(0.0008327945, t_2, fma(fma((t_1 * 0.0140005442), t_1, ((t_1 * 0.0694555761) * fabs(x))), t_0, fma(fma(t_0, 0.2909738639, 0.7715471019), t_0, 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))
	t_1 = Float64(t_0 * abs(x))
	t_2 = t_0 ^ 5.0
	tmp = 0.0
	if (abs(x) <= 40.0)
		tmp = Float64(Float64(fma(t_2, 0.0001789971, fma(fma(Float64(t_1 * 0.0005064034), t_1, Float64(Float64(t_1 * 0.0072644182) * abs(x))), t_0, fma(fma(t_0, 0.0424060604, 0.1049934947), t_0, 1.0))) * Float64(1.0 / fma((t_0 ^ 6.0), 0.0003579942, fma(0.0008327945, t_2, fma(fma(Float64(t_1 * 0.0140005442), t_1, Float64(Float64(t_1 * 0.0694555761) * abs(x))), t_0, fma(fma(t_0, 0.2909738639, 0.7715471019), t_0, 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]}, 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], 40.0], N[(N[(N[(t$95$2 * 0.0001789971 + N[(N[(N[(t$95$1 * 0.0005064034), $MachinePrecision] * t$95$1 + N[(N[(t$95$1 * 0.0072644182), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * 0.0424060604 + 0.1049934947), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Power[t$95$0, 6.0], $MachinePrecision] * 0.0003579942 + N[(0.0008327945 * t$95$2 + N[(N[(N[(t$95$1 * 0.0140005442), $MachinePrecision] * t$95$1 + N[(N[(t$95$1 * 0.0694555761), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * 0.2909738639 + 0.7715471019), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 40

   1. Initial program 54.6%

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

   3. Applied rewrites 54.6%

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

   5. Applied rewrites 54.6%

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

   if 40 < x

   1. Initial program 54.6%

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

   2. 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 50.6%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   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 50.6%

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

   6. Applied rewrites 50.6%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := t\_0 \cdot \left|x\right|\\ t_2 := {t\_0}^{5}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 40:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, 0.0001789971, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(0.0072644182 \cdot t\_1, \left|x\right|, \left(0.0005064034 \cdot t\_1\right) \cdot t\_1\right), \mathsf{fma}\left(t\_0, 0.1049934947 + 0.0424060604 \cdot t\_0, 1\right)\right)\right) \cdot \left|x\right|}{\mathsf{fma}\left({t\_0}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, t\_2, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(0.0694555761 \cdot t\_1, \left|x\right|, \left(0.0140005442 \cdot t\_1\right) \cdot t\_1\right), \mathsf{fma}\left(t\_0, 0.7715471019 + 0.2909738639 \cdot t\_0, 1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{t\_0} - -0.5}{\left|x\right|}\\ \end{array} \end{array} \]

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) 40.0)
     (/
      (*
       (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) <= 40.0) {
		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) <= 40.0)
		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], 40.0], 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 < 40

   1. Initial program 54.6%

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

   3. Applied rewrites 54.6%

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

   if 40 < x

   1. Initial program 54.6%

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

   2. 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 50.6%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   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 50.6%

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

   6. Applied rewrites 50.6%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := t\_0 \cdot \left|x\right|\\ t_2 := {t\_0}^{5}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 40:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, 0.0001789971, \mathsf{fma}\left(\mathsf{fma}\left(t\_1 \cdot 0.0005064034, t\_1, \left(t\_1 \cdot 0.0072644182\right) \cdot \left|x\right|\right), t\_0, \mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.0424060604, 0.1049934947\right), t\_0, 1\right)\right)\right) \cdot \left|x\right|}{\mathsf{fma}\left({t\_0}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, t\_2, \mathsf{fma}\left(\mathsf{fma}\left(t\_1 \cdot 0.0140005442, t\_1, \left(t\_1 \cdot 0.0694555761\right) \cdot \left|x\right|\right), t\_0, \mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.2909738639, 0.7715471019\right), t\_0, 1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{t\_0} - -0.5}{\left|x\right|}\\ \end{array} \end{array} \]

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) 40.0)
     (/
      (*
       (fma
        t_2
        0.0001789971
        (fma
         (fma
          (* t_1 0.0005064034)
          t_1
          (* (* t_1 0.0072644182) (fabs x)))
         t_0
         (fma (fma t_0 0.0424060604 0.1049934947) t_0 1.0)))
       (fabs x))
      (fma
       (pow t_0 6.0)
       0.0003579942
       (fma
        0.0008327945
        t_2
        (fma
         (fma
          (* t_1 0.0140005442)
          t_1
          (* (* t_1 0.0694555761) (fabs x)))
         t_0
         (fma (fma t_0 0.2909738639 0.7715471019) 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) <= 40.0) {
		tmp = (fma(t_2, 0.0001789971, fma(fma((t_1 * 0.0005064034), t_1, ((t_1 * 0.0072644182) * fabs(x))), t_0, fma(fma(t_0, 0.0424060604, 0.1049934947), t_0, 1.0))) * fabs(x)) / fma(pow(t_0, 6.0), 0.0003579942, fma(0.0008327945, t_2, fma(fma((t_1 * 0.0140005442), t_1, ((t_1 * 0.0694555761) * fabs(x))), t_0, fma(fma(t_0, 0.2909738639, 0.7715471019), 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) <= 40.0)
		tmp = Float64(Float64(fma(t_2, 0.0001789971, fma(fma(Float64(t_1 * 0.0005064034), t_1, Float64(Float64(t_1 * 0.0072644182) * abs(x))), t_0, fma(fma(t_0, 0.0424060604, 0.1049934947), t_0, 1.0))) * abs(x)) / fma((t_0 ^ 6.0), 0.0003579942, fma(0.0008327945, t_2, fma(fma(Float64(t_1 * 0.0140005442), t_1, Float64(Float64(t_1 * 0.0694555761) * abs(x))), t_0, fma(fma(t_0, 0.2909738639, 0.7715471019), 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], 40.0], N[(N[(N[(t$95$2 * 0.0001789971 + N[(N[(N[(t$95$1 * 0.0005064034), $MachinePrecision] * t$95$1 + N[(N[(t$95$1 * 0.0072644182), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * 0.0424060604 + 0.1049934947), $MachinePrecision] * t$95$0 + 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[(N[(N[(t$95$1 * 0.0140005442), $MachinePrecision] * t$95$1 + N[(N[(t$95$1 * 0.0694555761), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * 0.2909738639 + 0.7715471019), $MachinePrecision] * t$95$0 + 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 < 40

   1. Initial program 54.6%

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

   3. Applied rewrites 54.6%

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

   5. Applied rewrites 54.6%

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

   if 40 < x

   1. Initial program 54.6%

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

   2. 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 50.6%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   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 50.6%

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

   6. Applied rewrites 50.6%

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

Alternative 6: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := t\_0 \cdot \left|x\right|\\ t_2 := {\left(\left|x\right|\right)}^{10}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 40:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_2, 0.0001789971, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(0.0072644182 \cdot t\_1, \left|x\right|, \left(0.0005064034 \cdot t\_1\right) \cdot t\_1\right), \mathsf{fma}\left(t\_0, 0.1049934947 + 0.0424060604 \cdot t\_0, 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({t\_0}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, t\_2, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(0.0694555761 \cdot t\_1, \left|x\right|, \left(0.0140005442 \cdot t\_1\right) \cdot t\_1\right), \mathsf{fma}\left(t\_0, 0.7715471019 + 0.2909738639 \cdot t\_0, 1\right)\right)\right)\right)}\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{t\_0} - -0.5}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x)))
       (t_1 (* t_0 (fabs x)))
       (t_2 (pow (fabs x) 10.0)))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 40.0)
     (*
      (*
       (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)))
       (/
        1.0
        (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))))))
      (fabs x))
     (/ (- (/ 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(fabs(x), 10.0);
	double tmp;
	if (fabs(x) <= 40.0) {
		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))) * (1.0 / 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)))))) * 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))
	t_1 = Float64(t_0 * abs(x))
	t_2 = abs(x) ^ 10.0
	tmp = 0.0
	if (abs(x) <= 40.0)
		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))) * Float64(1.0 / 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)))))) * 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]}, Block[{t$95$1 = N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[x], $MachinePrecision], 10.0], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 40.0], 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[(1.0 / 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]), $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 < 40

   1. Initial program 54.6%

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

   3. Applied rewrites 54.6%

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

   4. Taylor expanded in x around 0

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{x}^{10}}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)}\right) \cdot x \]
   5. Step-by-step derivation

      1. lower-pow.f64 54.6%

         \[\leadsto \left(\mathsf{fma}\left({x}^{\color{blue}{10}}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)}\right) \cdot x \]

   6. Applied rewrites 54.6%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{x}^{10}}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)}\right) \cdot x \]

   7. Taylor expanded in x around 0

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

      1. lower-pow.f64 54.6%

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

   9. Applied rewrites 54.6%

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

   if 40 < x

   1. Initial program 54.6%

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

   2. 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 50.6%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   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 50.6%

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

   6. Applied rewrites 50.6%

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

Alternative 7: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|x\right|\right)}^{4}\\ t_1 := \left|x\right| \cdot \left|x\right|\\ t_2 := t\_1 \cdot \left|x\right|\\ t_3 := {t\_1}^{5}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 22:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_3, 0.0001789971, \mathsf{fma}\left(t\_1, 0.0072644182 \cdot t\_0, \mathsf{fma}\left(t\_1, 0.1049934947 + 0.0424060604 \cdot t\_1, 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({t\_1}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, t\_3, \mathsf{fma}\left(t\_1, \mathsf{fma}\left(0.0694555761 \cdot t\_2, \left|x\right|, \left(0.0140005442 \cdot t\_2\right) \cdot t\_2\right), \mathsf{fma}\left(t\_1, 0.7715471019 + 0.2909738639 \cdot t\_1, 1\right)\right)\right)\right)}\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \left(\frac{0.15298196345929074}{t\_0} + \mathsf{fma}\left(0.2514179000665374, \frac{1}{{\left(\left|x\right|\right)}^{2}}, 11.259630434457211 \cdot \frac{1}{{\left(\left|x\right|\right)}^{6}}\right)\right)}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (pow (fabs x) 4.0))
       (t_1 (* (fabs x) (fabs x)))
       (t_2 (* t_1 (fabs x)))
       (t_3 (pow t_1 5.0)))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 22.0)
     (*
      (*
       (fma
        t_3
        0.0001789971
        (fma
         t_1
         (* 0.0072644182 t_0)
         (fma t_1 (+ 0.1049934947 (* 0.0424060604 t_1)) 1.0)))
       (/
        1.0
        (fma
         (pow t_1 6.0)
         0.0003579942
         (fma
          0.0008327945
          t_3
          (fma
           t_1
           (fma
            (* 0.0694555761 t_2)
            (fabs x)
            (* (* 0.0140005442 t_2) t_2))
           (fma t_1 (+ 0.7715471019 (* 0.2909738639 t_1)) 1.0))))))
      (fabs x))
     (/
      (+
       0.5
       (+
        (/ 0.15298196345929074 t_0)
        (fma
         0.2514179000665374
         (/ 1.0 (pow (fabs x) 2.0))
         (* 11.259630434457211 (/ 1.0 (pow (fabs x) 6.0))))))
      (fabs x))))))

C

double code(double x) {
	double t_0 = pow(fabs(x), 4.0);
	double t_1 = fabs(x) * fabs(x);
	double t_2 = t_1 * fabs(x);
	double t_3 = pow(t_1, 5.0);
	double tmp;
	if (fabs(x) <= 22.0) {
		tmp = (fma(t_3, 0.0001789971, fma(t_1, (0.0072644182 * t_0), fma(t_1, (0.1049934947 + (0.0424060604 * t_1)), 1.0))) * (1.0 / fma(pow(t_1, 6.0), 0.0003579942, fma(0.0008327945, t_3, fma(t_1, fma((0.0694555761 * t_2), fabs(x), ((0.0140005442 * t_2) * t_2)), fma(t_1, (0.7715471019 + (0.2909738639 * t_1)), 1.0)))))) * fabs(x);
	} else {
		tmp = (0.5 + ((0.15298196345929074 / t_0) + fma(0.2514179000665374, (1.0 / pow(fabs(x), 2.0)), (11.259630434457211 * (1.0 / pow(fabs(x), 6.0)))))) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = abs(x) ^ 4.0
	t_1 = Float64(abs(x) * abs(x))
	t_2 = Float64(t_1 * abs(x))
	t_3 = t_1 ^ 5.0
	tmp = 0.0
	if (abs(x) <= 22.0)
		tmp = Float64(Float64(fma(t_3, 0.0001789971, fma(t_1, Float64(0.0072644182 * t_0), fma(t_1, Float64(0.1049934947 + Float64(0.0424060604 * t_1)), 1.0))) * Float64(1.0 / fma((t_1 ^ 6.0), 0.0003579942, fma(0.0008327945, t_3, fma(t_1, fma(Float64(0.0694555761 * t_2), abs(x), Float64(Float64(0.0140005442 * t_2) * t_2)), fma(t_1, Float64(0.7715471019 + Float64(0.2909738639 * t_1)), 1.0)))))) * abs(x));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.15298196345929074 / t_0) + fma(0.2514179000665374, Float64(1.0 / (abs(x) ^ 2.0)), Float64(11.259630434457211 * Float64(1.0 / (abs(x) ^ 6.0)))))) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Abs[x], $MachinePrecision], 4.0], $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[Power[t$95$1, 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], 22.0], N[(N[(N[(t$95$3 * 0.0001789971 + N[(t$95$1 * N[(0.0072644182 * t$95$0), $MachinePrecision] + N[(t$95$1 * N[(0.1049934947 + N[(0.0424060604 * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Power[t$95$1, 6.0], $MachinePrecision] * 0.0003579942 + N[(0.0008327945 * t$95$3 + N[(t$95$1 * N[(N[(0.0694555761 * t$95$2), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(N[(0.0140005442 * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(0.7715471019 + N[(0.2909738639 * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.15298196345929074 / t$95$0), $MachinePrecision] + N[(0.2514179000665374 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(11.259630434457211 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 22

   1. Initial program 54.6%

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

   3. Applied rewrites 54.6%

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

   4. Taylor expanded in x around 0

      \[\leadsto \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, 0.0001789971, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{36322091}{5000000000} \cdot {x}^{4}}, \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, 0.0003579942, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)}\right) \cdot x \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left({\left(x \cdot x\right)}^{5}, \frac{1789971}{10000000000}, \mathsf{fma}\left(x \cdot x, \frac{36322091}{5000000000} \cdot \color{blue}{{x}^{4}}, \mathsf{fma}\left(x \cdot x, \frac{1049934947}{10000000000} + \frac{106015151}{2500000000} \cdot \left(x \cdot x\right), 1\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left({\left(x \cdot x\right)}^{6}, \frac{1789971}{5000000000}, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)}\right) \cdot x \]

      2. lower-pow.f64 54.0%

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

   6. Applied rewrites 54.0%

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

   if 22 < x

   1. Initial program 54.6%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.5%

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

Alternative 8: 99.3% accurate, 2.7× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.45:\\ \;\;\;\;1 \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \left(\frac{0.15298196345929074}{{\left(\left|x\right|\right)}^{4}} + \mathsf{fma}\left(0.2514179000665374, \frac{1}{{\left(\left|x\right|\right)}^{2}}, 11.259630434457211 \cdot \frac{1}{{\left(\left|x\right|\right)}^{6}}\right)\right)}{\left|x\right|}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (fabs x) 1.45)
   (* 1.0 (fabs x))
   (/
    (+
     0.5
     (+
      (/ 0.15298196345929074 (pow (fabs x) 4.0))
      (fma
       0.2514179000665374
       (/ 1.0 (pow (fabs x) 2.0))
       (* 11.259630434457211 (/ 1.0 (pow (fabs x) 6.0))))))
    (fabs x)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1.45) {
		tmp = 1.0 * fabs(x);
	} else {
		tmp = (0.5 + ((0.15298196345929074 / pow(fabs(x), 4.0)) + fma(0.2514179000665374, (1.0 / pow(fabs(x), 2.0)), (11.259630434457211 * (1.0 / pow(fabs(x), 6.0)))))) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1.45)
		tmp = Float64(1.0 * abs(x));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.15298196345929074 / (abs(x) ^ 4.0)) + fma(0.2514179000665374, Float64(1.0 / (abs(x) ^ 2.0)), Float64(11.259630434457211 * Float64(1.0 / (abs(x) ^ 6.0)))))) / 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], 1.45], N[(1.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.15298196345929074 / N[Power[N[Abs[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.2514179000665374 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(11.259630434457211 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.45

   1. Initial program 54.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 27.8%

         \[\leadsto \frac{0.5}{{x}^{\color{blue}{2}}} \cdot x \]

   4. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{2}}} \cdot x \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 51.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   if 1.45 < x

   1. Initial program 54.6%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.5%

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

Alternative 9: 99.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.95:\\ \;\;\;\;1 \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{0.15298196345929074}{\left(t\_0 \cdot \left|x\right|\right) \cdot t\_0} - \frac{\frac{-0.2514179000665374}{t\_0} - 0.5}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x))))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 0.95)
     (* 1.0 (fabs x))
     (-
      (/ 0.15298196345929074 (* (* t_0 (fabs x)) t_0))
      (/ (- (/ -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) <= 0.95) {
		tmp = 1.0 * fabs(x);
	} else {
		tmp = (0.15298196345929074 / ((t_0 * fabs(x)) * t_0)) - (((-0.2514179000665374 / t_0) - 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 tmp;
	if (Math.abs(x) <= 0.95) {
		tmp = 1.0 * Math.abs(x);
	} else {
		tmp = (0.15298196345929074 / ((t_0 * Math.abs(x)) * t_0)) - (((-0.2514179000665374 / t_0) - 0.5) / Math.abs(x));
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) * math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 0.95:
		tmp = 1.0 * math.fabs(x)
	else:
		tmp = (0.15298196345929074 / ((t_0 * math.fabs(x)) * t_0)) - (((-0.2514179000665374 / t_0) - 0.5) / math.fabs(x))
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 0.95)
		tmp = Float64(1.0 * abs(x));
	else
		tmp = Float64(Float64(0.15298196345929074 / Float64(Float64(t_0 * abs(x)) * t_0)) - Float64(Float64(Float64(-0.2514179000665374 / t_0) - 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);
	tmp = 0.0;
	if (abs(x) <= 0.95)
		tmp = 1.0 * abs(x);
	else
		tmp = (0.15298196345929074 / ((t_0 * abs(x)) * t_0)) - (((-0.2514179000665374 / t_0) - 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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 0.95], N[(1.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(0.15298196345929074 / N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-0.2514179000665374 / t$95$0), $MachinePrecision] - 0.5), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 54.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 27.8%

         \[\leadsto \frac{0.5}{{x}^{\color{blue}{2}}} \cdot x \]

   4. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{2}}} \cdot x \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 51.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   if 0.94999999999999996 < x

   1. Initial program 54.6%

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

   2. 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 50.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. add-flip N/A

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

      5. associate-+l- N/A

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

      6. sub-negate N/A

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

      7. add-flip N/A

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

      8. lift-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. pow2 N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. add-flip N/A

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

   6. Applied rewrites 50.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

   8. Applied rewrites 50.5%

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

Alternative 10: 99.2% accurate, 6.4× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.95:\\ \;\;\;\;1 \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.15298196345929074}{\left|x\right| \cdot \left|x\right|} - -0.2514179000665374}{\left|x\right|}}{\left|x\right|} - -0.5}{\left|x\right|}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (fabs x) 0.95)
   (* 1.0 (fabs x))
   (/
    (-
     (/
      (/
       (-
        (/ 0.15298196345929074 (* (fabs x) (fabs x)))
        -0.2514179000665374)
       (fabs x))
      (fabs x))
     -0.5)
    (fabs x)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.95) {
		tmp = 1.0 * fabs(x);
	} else {
		tmp = (((((0.15298196345929074 / (fabs(x) * fabs(x))) - -0.2514179000665374) / fabs(x)) / fabs(x)) - -0.5) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.95) {
		tmp = 1.0 * Math.abs(x);
	} else {
		tmp = (((((0.15298196345929074 / (Math.abs(x) * Math.abs(x))) - -0.2514179000665374) / Math.abs(x)) / Math.abs(x)) - -0.5) / Math.abs(x);
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.95:
		tmp = 1.0 * math.fabs(x)
	else:
		tmp = (((((0.15298196345929074 / (math.fabs(x) * math.fabs(x))) - -0.2514179000665374) / math.fabs(x)) / math.fabs(x)) - -0.5) / math.fabs(x)
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.95)
		tmp = Float64(1.0 * abs(x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.15298196345929074 / Float64(abs(x) * abs(x))) - -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)
	tmp = 0.0;
	if (abs(x) <= 0.95)
		tmp = 1.0 * abs(x);
	else
		tmp = (((((0.15298196345929074 / (abs(x) * abs(x))) - -0.2514179000665374) / abs(x)) / abs(x)) - -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], 0.95], N[(1.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.15298196345929074 / N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $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 < 0.94999999999999996

   1. Initial program 54.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 27.8%

         \[\leadsto \frac{0.5}{{x}^{\color{blue}{2}}} \cdot x \]

   4. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{2}}} \cdot x \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 51.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   if 0.94999999999999996 < x

   1. Initial program 54.6%

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

   2. 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 50.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. add-flip N/A

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

      5. associate-+l- N/A

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

      6. sub-negate N/A

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

      7. add-flip N/A

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

      8. lift-+.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. pow2 N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. add-flip N/A

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

   6. Applied rewrites 50.5%

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

   8. Applied rewrites 50.5%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 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 50.5%

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

   10. Applied rewrites 50.5%

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

Alternative 11: 99.2% accurate, 9.3× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.9:\\ \;\;\;\;1 \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{\left|x\right| \cdot \left|x\right|} - -0.5}{\left|x\right|}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (fabs x) 0.9)
   (* 1.0 (fabs x))
   (/
    (- (/ 0.2514179000665374 (* (fabs x) (fabs x))) -0.5)
    (fabs x)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.9) {
		tmp = 1.0 * fabs(x);
	} else {
		tmp = ((0.2514179000665374 / (fabs(x) * fabs(x))) - -0.5) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.9) {
		tmp = 1.0 * Math.abs(x);
	} else {
		tmp = ((0.2514179000665374 / (Math.abs(x) * Math.abs(x))) - -0.5) / Math.abs(x);
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.9:
		tmp = 1.0 * math.fabs(x)
	else:
		tmp = ((0.2514179000665374 / (math.fabs(x) * math.fabs(x))) - -0.5) / math.fabs(x)
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.9)
		tmp = Float64(1.0 * abs(x));
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / Float64(abs(x) * abs(x))) - -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) <= 0.9)
		tmp = 1.0 * abs(x);
	else
		tmp = ((0.2514179000665374 / (abs(x) * abs(x))) - -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], 0.9], N[(1.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.2514179000665374 / N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.90000000000000002

   1. Initial program 54.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 27.8%

         \[\leadsto \frac{0.5}{{x}^{\color{blue}{2}}} \cdot x \]

   4. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{2}}} \cdot x \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 51.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   if 0.90000000000000002 < x

   1. Initial program 54.6%

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

   2. 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 50.6%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   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 50.6%

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

   6. Applied rewrites 50.6%

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

Alternative 12: 99.0% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 54.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 27.8%

         \[\leadsto \frac{0.5}{{x}^{\color{blue}{2}}} \cdot x \]

   4. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{2}}} \cdot x \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 51.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   if 0.69999999999999996 < x

   1. Initial program 54.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 50.9%

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

   4. Applied rewrites 50.9%

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

Alternative 13: 51.9% accurate, 63.9× speedup

Math

\[1 \cdot x \]

FPCore

(FPCore (x)
  :precision binary64
  (* 1.0 x))

C

double code(double x) {
	return 1.0 * 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 = 1.0d0 * x
end function

Java

public static double code(double x) {
	return 1.0 * x;
}

Python

def code(x):
	return 1.0 * x

Julia

function code(x)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x)
	tmp = 1.0 * x;
end

Wolfram

code[x_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 27.8%

      \[\leadsto \frac{0.5}{{x}^{\color{blue}{2}}} \cdot x \]

4. Applied rewrites 27.8%

   \[\leadsto \color{blue}{\frac{0.5}{{x}^{2}}} \cdot x \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \cdot x \]
6. Step-by-step derivation

7. Applied rewrites 51.9%

   \[\leadsto \color{blue}{1} \cdot x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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))