Jmat.Real.dawson

Percentage Accurate: 53.8% → 100.0%

Time: 7.7s

Alternatives: 13

Speedup: 31.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot x\_m\\ t_1 := \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.0072644182 \cdot t\_0, x\_m, \left(0.0005064034 \cdot t\_0\right) \cdot t\_0\right), \mathsf{fma}\left(x\_m \cdot x\_m, 0.1049934947 + 0.0424060604 \cdot \left(x\_m \cdot x\_m\right), 1\right)\right)\\ t_2 := {\left(x\_m \cdot x\_m\right)}^{5}\\ t_3 := t\_2 \cdot 0.0001789971\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 58000000:\\ \;\;\;\;\frac{{t\_1}^{2} - {t\_3}^{2}}{\left(t\_1 - t\_3\right) \cdot \mathsf{fma}\left(\left(t\_2 \cdot 0.0003579942\right) \cdot x\_m, x\_m, \mathsf{fma}\left(0.0008327945, t\_2, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.0694555761 \cdot t\_0, x\_m, \left(0.0140005442 \cdot t\_0\right) \cdot t\_0\right), \mathsf{fma}\left(x\_m \cdot x\_m, 0.7715471019 + 0.2909738639 \cdot \left(x\_m \cdot x\_m\right), 1\right)\right)\right)\right)} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (* x_m x_m) x_m))
        (t_1
         (fma
          (* x_m x_m)
          (fma (* 0.0072644182 t_0) x_m (* (* 0.0005064034 t_0) t_0))
          (fma (* x_m x_m) (+ 0.1049934947 (* 0.0424060604 (* x_m x_m))) 1.0)))
        (t_2 (pow (* x_m x_m) 5.0))
        (t_3 (* t_2 0.0001789971)))
   (*
    x_s
    (if (<= x_m 58000000.0)
      (*
       (/
        (- (pow t_1 2.0) (pow t_3 2.0))
        (*
         (- t_1 t_3)
         (fma
          (* (* t_2 0.0003579942) x_m)
          x_m
          (fma
           0.0008327945
           t_2
           (fma
            (* x_m x_m)
            (fma (* 0.0694555761 t_0) x_m (* (* 0.0140005442 t_0) t_0))
            (fma
             (* x_m x_m)
             (+ 0.7715471019 (* 0.2909738639 (* x_m x_m)))
             1.0))))))
       x_m)
      (/ 0.5 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = (x_m * x_m) * x_m;
	double t_1 = fma((x_m * x_m), fma((0.0072644182 * t_0), x_m, ((0.0005064034 * t_0) * t_0)), fma((x_m * x_m), (0.1049934947 + (0.0424060604 * (x_m * x_m))), 1.0));
	double t_2 = pow((x_m * x_m), 5.0);
	double t_3 = t_2 * 0.0001789971;
	double tmp;
	if (x_m <= 58000000.0) {
		tmp = ((pow(t_1, 2.0) - pow(t_3, 2.0)) / ((t_1 - t_3) * fma(((t_2 * 0.0003579942) * x_m), x_m, fma(0.0008327945, t_2, fma((x_m * x_m), fma((0.0694555761 * t_0), x_m, ((0.0140005442 * t_0) * t_0)), fma((x_m * x_m), (0.7715471019 + (0.2909738639 * (x_m * x_m))), 1.0)))))) * x_m;
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(x_m * x_m) * x_m)
	t_1 = fma(Float64(x_m * x_m), fma(Float64(0.0072644182 * t_0), x_m, Float64(Float64(0.0005064034 * t_0) * t_0)), fma(Float64(x_m * x_m), Float64(0.1049934947 + Float64(0.0424060604 * Float64(x_m * x_m))), 1.0))
	t_2 = Float64(x_m * x_m) ^ 5.0
	t_3 = Float64(t_2 * 0.0001789971)
	tmp = 0.0
	if (x_m <= 58000000.0)
		tmp = Float64(Float64(Float64((t_1 ^ 2.0) - (t_3 ^ 2.0)) / Float64(Float64(t_1 - t_3) * fma(Float64(Float64(t_2 * 0.0003579942) * x_m), x_m, fma(0.0008327945, t_2, fma(Float64(x_m * x_m), fma(Float64(0.0694555761 * t_0), x_m, Float64(Float64(0.0140005442 * t_0) * t_0)), fma(Float64(x_m * x_m), Float64(0.7715471019 + Float64(0.2909738639 * Float64(x_m * x_m))), 1.0)))))) * x_m);
	else
		tmp = Float64(0.5 / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.0072644182 * t$95$0), $MachinePrecision] * x$95$m + N[(N[(0.0005064034 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.1049934947 + N[(0.0424060604 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 0.0001789971), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 58000000.0], N[(N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 - t$95$3), $MachinePrecision] * N[(N[(N[(t$95$2 * 0.0003579942), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + N[(0.0008327945 * t$95$2 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.0694555761 * t$95$0), $MachinePrecision] * x$95$m + N[(N[(0.0140005442 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.7715471019 + N[(0.2909738639 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e7

   1. Initial program 53.8%

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

   3. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right)\right)}^{2} - {\left({\left(x \cdot x\right)}^{5} \cdot 0.0001789971\right)}^{2}}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.1049934947 + 0.0424060604 \cdot \left(x \cdot x\right), 1\right)\right) - {\left(x \cdot x\right)}^{5} \cdot 0.0001789971\right) \cdot \mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot 0.0003579942\right) \cdot x, x, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)}} \cdot x \]

   if 5.8e7 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 51.5

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

   4. Applied rewrites 51.5%

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

Alternative 2: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot x\_m\\ t_1 := {\left(x\_m \cdot x\_m\right)}^{5}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 56000000:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 0.0001789971, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.0072644182 \cdot t\_0, x\_m, \left(0.0005064034 \cdot t\_0\right) \cdot t\_0\right), \mathsf{fma}\left(x\_m \cdot x\_m, 0.1049934947 + 0.0424060604 \cdot \left(x\_m \cdot x\_m\right), 1\right)\right)\right) \cdot \frac{x\_m}{\mathsf{fma}\left(\left(t\_1 \cdot 0.0003579942\right) \cdot x\_m, x\_m, \mathsf{fma}\left(0.0008327945, t\_1, \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.0694555761 \cdot t\_0, x\_m, \left(0.0140005442 \cdot t\_0\right) \cdot t\_0\right), \mathsf{fma}\left(x\_m \cdot x\_m, 0.7715471019 + 0.2909738639 \cdot \left(x\_m \cdot x\_m\right), 1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (* x_m x_m) x_m)) (t_1 (pow (* x_m x_m) 5.0)))
   (*
    x_s
    (if (<= x_m 56000000.0)
      (*
       (fma
        t_1
        0.0001789971
        (fma
         (* x_m x_m)
         (fma (* 0.0072644182 t_0) x_m (* (* 0.0005064034 t_0) t_0))
         (fma (* x_m x_m) (+ 0.1049934947 (* 0.0424060604 (* x_m x_m))) 1.0)))
       (/
        x_m
        (fma
         (* (* t_1 0.0003579942) x_m)
         x_m
         (fma
          0.0008327945
          t_1
          (fma
           (* x_m x_m)
           (fma (* 0.0694555761 t_0) x_m (* (* 0.0140005442 t_0) t_0))
           (fma
            (* x_m x_m)
            (+ 0.7715471019 (* 0.2909738639 (* x_m x_m)))
            1.0))))))
      (/ 0.5 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = (x_m * x_m) * x_m;
	double t_1 = pow((x_m * x_m), 5.0);
	double tmp;
	if (x_m <= 56000000.0) {
		tmp = fma(t_1, 0.0001789971, fma((x_m * x_m), fma((0.0072644182 * t_0), x_m, ((0.0005064034 * t_0) * t_0)), fma((x_m * x_m), (0.1049934947 + (0.0424060604 * (x_m * x_m))), 1.0))) * (x_m / fma(((t_1 * 0.0003579942) * x_m), x_m, fma(0.0008327945, t_1, fma((x_m * x_m), fma((0.0694555761 * t_0), x_m, ((0.0140005442 * t_0) * t_0)), fma((x_m * x_m), (0.7715471019 + (0.2909738639 * (x_m * x_m))), 1.0)))));
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(x_m * x_m) * x_m)
	t_1 = Float64(x_m * x_m) ^ 5.0
	tmp = 0.0
	if (x_m <= 56000000.0)
		tmp = Float64(fma(t_1, 0.0001789971, fma(Float64(x_m * x_m), fma(Float64(0.0072644182 * t_0), x_m, Float64(Float64(0.0005064034 * t_0) * t_0)), fma(Float64(x_m * x_m), Float64(0.1049934947 + Float64(0.0424060604 * Float64(x_m * x_m))), 1.0))) * Float64(x_m / fma(Float64(Float64(t_1 * 0.0003579942) * x_m), x_m, fma(0.0008327945, t_1, fma(Float64(x_m * x_m), fma(Float64(0.0694555761 * t_0), x_m, Float64(Float64(0.0140005442 * t_0) * t_0)), fma(Float64(x_m * x_m), Float64(0.7715471019 + Float64(0.2909738639 * Float64(x_m * x_m))), 1.0))))));
	else
		tmp = Float64(0.5 / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 5.0], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 56000000.0], N[(N[(t$95$1 * 0.0001789971 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.0072644182 * t$95$0), $MachinePrecision] * x$95$m + N[(N[(0.0005064034 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.1049934947 + N[(0.0424060604 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / N[(N[(N[(t$95$1 * 0.0003579942), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + N[(0.0008327945 * t$95$1 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.0694555761 * t$95$0), $MachinePrecision] * x$95$m + N[(N[(0.0140005442 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.7715471019 + N[(0.2909738639 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.6e7

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   if 5.6e7 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 51.5

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

   4. Applied rewrites 51.5%

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

Alternative 3: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 60000000:\\ \;\;\;\;\frac{\mathsf{fma}\left({x\_m}^{10}, 0.0001789971, \mathsf{fma}\left(\mathsf{fma}\left(t\_0 \cdot 0.0005064034, t\_0, \left(t\_0 \cdot 0.0072644182\right) \cdot x\_m\right), x\_m \cdot x\_m, \mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.0424060604, 0.1049934947\right), x\_m \cdot x\_m, 1\right)\right)\right)}{\mathsf{fma}\left({x\_m}^{10} \cdot 0.0003579942, x\_m, \frac{\mathsf{fma}\left(0.0008327945, {x\_m}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(t\_0 \cdot 0.0140005442, t\_0, \left(t\_0 \cdot 0.0694555761\right) \cdot x\_m\right), x\_m \cdot x\_m, \mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.2909738639, 0.7715471019\right), x\_m \cdot x\_m, 1\right)\right)\right)}{x\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (* x_m x_m) x_m)))
   (*
    x_s
    (if (<= x_m 60000000.0)
      (/
       (fma
        (pow x_m 10.0)
        0.0001789971
        (fma
         (fma (* t_0 0.0005064034) t_0 (* (* t_0 0.0072644182) x_m))
         (* x_m x_m)
         (fma (fma (* x_m x_m) 0.0424060604 0.1049934947) (* x_m x_m) 1.0)))
       (fma
        (* (pow x_m 10.0) 0.0003579942)
        x_m
        (/
         (fma
          0.0008327945
          (pow x_m 10.0)
          (fma
           (fma (* t_0 0.0140005442) t_0 (* (* t_0 0.0694555761) x_m))
           (* x_m x_m)
           (fma (fma (* x_m x_m) 0.2909738639 0.7715471019) (* x_m x_m) 1.0)))
         x_m)))
      (/ 0.5 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = (x_m * x_m) * x_m;
	double tmp;
	if (x_m <= 60000000.0) {
		tmp = fma(pow(x_m, 10.0), 0.0001789971, fma(fma((t_0 * 0.0005064034), t_0, ((t_0 * 0.0072644182) * x_m)), (x_m * x_m), fma(fma((x_m * x_m), 0.0424060604, 0.1049934947), (x_m * x_m), 1.0))) / fma((pow(x_m, 10.0) * 0.0003579942), x_m, (fma(0.0008327945, pow(x_m, 10.0), fma(fma((t_0 * 0.0140005442), t_0, ((t_0 * 0.0694555761) * x_m)), (x_m * x_m), fma(fma((x_m * x_m), 0.2909738639, 0.7715471019), (x_m * x_m), 1.0))) / x_m));
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(x_m * x_m) * x_m)
	tmp = 0.0
	if (x_m <= 60000000.0)
		tmp = Float64(fma((x_m ^ 10.0), 0.0001789971, fma(fma(Float64(t_0 * 0.0005064034), t_0, Float64(Float64(t_0 * 0.0072644182) * x_m)), Float64(x_m * x_m), fma(fma(Float64(x_m * x_m), 0.0424060604, 0.1049934947), Float64(x_m * x_m), 1.0))) / fma(Float64((x_m ^ 10.0) * 0.0003579942), x_m, Float64(fma(0.0008327945, (x_m ^ 10.0), fma(fma(Float64(t_0 * 0.0140005442), t_0, Float64(Float64(t_0 * 0.0694555761) * x_m)), Float64(x_m * x_m), fma(fma(Float64(x_m * x_m), 0.2909738639, 0.7715471019), Float64(x_m * x_m), 1.0))) / x_m)));
	else
		tmp = Float64(0.5 / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 60000000.0], N[(N[(N[Power[x$95$m, 10.0], $MachinePrecision] * 0.0001789971 + N[(N[(N[(t$95$0 * 0.0005064034), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * 0.0072644182), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0424060604 + 0.1049934947), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[x$95$m, 10.0], $MachinePrecision] * 0.0003579942), $MachinePrecision] * x$95$m + N[(N[(0.0008327945 * N[Power[x$95$m, 10.0], $MachinePrecision] + N[(N[(N[(t$95$0 * 0.0140005442), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * 0.0694555761), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.2909738639 + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6e7

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   5. Applied rewrites 54.0%

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

   if 6e7 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 51.5

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

   4. Applied rewrites 51.5%

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

Alternative 4: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot x\_m\\ t_1 := {\left(x\_m \cdot x\_m\right)}^{5}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.45:\\ \;\;\;\;\left(1 + {x\_m}^{2} \cdot \left(0.1049934947 + {x\_m}^{2} \cdot \left(0.0424060604 + 0.0072644182 \cdot {x\_m}^{2}\right)\right)\right) \cdot \frac{x\_m}{\mathsf{fma}\left(\left(t\_1 \cdot 0.0003579942\right) \cdot x\_m, x\_m, \mathsf{fma}\left(0.0008327945, t\_1, \mathsf{fma}\left(\mathsf{fma}\left(t\_0 \cdot 0.0140005442, t\_0, \left(t\_0 \cdot 0.0694555761\right) \cdot x\_m\right), x\_m \cdot x\_m, \mathsf{fma}\left(0.2909738639, x\_m \cdot x\_m, 0.7715471019\right) \cdot \left(x\_m \cdot x\_m\right)\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.15298196345929074}{t\_0} - \frac{-0.2514179000665374}{x\_m}}{x\_m} - -0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (* x_m x_m) x_m)) (t_1 (pow (* x_m x_m) 5.0)))
   (*
    x_s
    (if (<= x_m 1.45)
      (*
       (+
        1.0
        (*
         (pow x_m 2.0)
         (+
          0.1049934947
          (* (pow x_m 2.0) (+ 0.0424060604 (* 0.0072644182 (pow x_m 2.0)))))))
       (/
        x_m
        (fma
         (* (* t_1 0.0003579942) x_m)
         x_m
         (fma
          0.0008327945
          t_1
          (+
           (fma
            (fma (* t_0 0.0140005442) t_0 (* (* t_0 0.0694555761) x_m))
            (* x_m x_m)
            (* (fma 0.2909738639 (* x_m x_m) 0.7715471019) (* x_m x_m)))
           1.0)))))
      (/
       (-
        (/ (- (/ 0.15298196345929074 t_0) (/ -0.2514179000665374 x_m)) x_m)
        -0.5)
       x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = (x_m * x_m) * x_m;
	double t_1 = pow((x_m * x_m), 5.0);
	double tmp;
	if (x_m <= 1.45) {
		tmp = (1.0 + (pow(x_m, 2.0) * (0.1049934947 + (pow(x_m, 2.0) * (0.0424060604 + (0.0072644182 * pow(x_m, 2.0))))))) * (x_m / fma(((t_1 * 0.0003579942) * x_m), x_m, fma(0.0008327945, t_1, (fma(fma((t_0 * 0.0140005442), t_0, ((t_0 * 0.0694555761) * x_m)), (x_m * x_m), (fma(0.2909738639, (x_m * x_m), 0.7715471019) * (x_m * x_m))) + 1.0))));
	} else {
		tmp = ((((0.15298196345929074 / t_0) - (-0.2514179000665374 / x_m)) / x_m) - -0.5) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(x_m * x_m) * x_m)
	t_1 = Float64(x_m * x_m) ^ 5.0
	tmp = 0.0
	if (x_m <= 1.45)
		tmp = Float64(Float64(1.0 + Float64((x_m ^ 2.0) * Float64(0.1049934947 + Float64((x_m ^ 2.0) * Float64(0.0424060604 + Float64(0.0072644182 * (x_m ^ 2.0))))))) * Float64(x_m / fma(Float64(Float64(t_1 * 0.0003579942) * x_m), x_m, fma(0.0008327945, t_1, Float64(fma(fma(Float64(t_0 * 0.0140005442), t_0, Float64(Float64(t_0 * 0.0694555761) * x_m)), Float64(x_m * x_m), Float64(fma(0.2909738639, Float64(x_m * x_m), 0.7715471019) * Float64(x_m * x_m))) + 1.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.15298196345929074 / t_0) - Float64(-0.2514179000665374 / x_m)) / x_m) - -0.5) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 5.0], $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 1.45], N[(N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.1049934947 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.0424060604 + N[(0.0072644182 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / N[(N[(N[(t$95$1 * 0.0003579942), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + N[(0.0008327945 * t$95$1 + N[(N[(N[(N[(t$95$0 * 0.0140005442), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * 0.0694555761), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(0.2909738639 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.15298196345929074 / t$95$0), $MachinePrecision] - N[(-0.2514179000665374 / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)}\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \color{blue}{\left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)}\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\color{blue}{\frac{1049934947}{10000000000}} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + \color{blue}{{x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)}\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \color{blue}{\left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)}\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\color{blue}{\frac{106015151}{2500000000}} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \color{blue}{\frac{36322091}{5000000000} \cdot {x}^{2}}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot \color{blue}{{x}^{2}}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      9. lower-pow.f64 50.2

         \[\leadsto \left(1 + {x}^{2} \cdot \left(0.1049934947 + {x}^{2} \cdot \left(0.0424060604 + 0.0072644182 \cdot {x}^{\color{blue}{2}}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot 0.0003579942\right) \cdot x, x, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(0.1049934947 + {x}^{2} \cdot \left(0.0424060604 + 0.0072644182 \cdot {x}^{2}\right)\right)\right)} \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot 0.0003579942\right) \cdot x, x, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(\color{blue}{x \cdot x}, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)} \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right)\right) + 1\right)}\right)\right)} \]

      5. associate-+r+ N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \color{blue}{\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right)\right)\right) + 1}\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \color{blue}{\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(\frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right)\right)\right) + 1}\right)\right)} \]

   8. Applied rewrites 50.2%

      \[\leadsto \left(1 + {x}^{2} \cdot \left(0.1049934947 + {x}^{2} \cdot \left(0.0424060604 + 0.0072644182 \cdot {x}^{2}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot 0.0003579942\right) \cdot x, x, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0140005442, \left(x \cdot x\right) \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.0694555761\right) \cdot x\right), x \cdot x, \mathsf{fma}\left(0.2909738639, x \cdot x, 0.7715471019\right) \cdot \left(x \cdot x\right)\right) + 1}\right)\right)} \]

   if 1.44999999999999996 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.3%

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

   6. Applied rewrites 51.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 51.3

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

   8. Applied rewrites 51.3%

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

Alternative 5: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.45:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0072644182, x\_m \cdot x\_m, 0.0424060604\right), x\_m \cdot x\_m, 0.1049934947\right), x\_m \cdot x\_m, 1\right) \cdot x\_m}{\mathsf{fma}\left(\left({x\_m}^{10} \cdot x\_m\right) \cdot x\_m, 0.0003579942, \mathsf{fma}\left(0.0008327945, {x\_m}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(t\_0 \cdot 0.0140005442, t\_0, \left(t\_0 \cdot 0.0694555761\right) \cdot x\_m\right), x\_m \cdot x\_m, \mathsf{fma}\left(\mathsf{fma}\left(0.2909738639, x\_m \cdot x\_m, 0.7715471019\right), x\_m \cdot x\_m, 1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.15298196345929074}{t\_0} - \frac{-0.2514179000665374}{x\_m}}{x\_m} - -0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (* x_m x_m) x_m)))
   (*
    x_s
    (if (<= x_m 1.45)
      (/
       (*
        (fma
         (fma
          (fma 0.0072644182 (* x_m x_m) 0.0424060604)
          (* x_m x_m)
          0.1049934947)
         (* x_m x_m)
         1.0)
        x_m)
       (fma
        (* (* (pow x_m 10.0) x_m) x_m)
        0.0003579942
        (fma
         0.0008327945
         (pow x_m 10.0)
         (fma
          (fma (* t_0 0.0140005442) t_0 (* (* t_0 0.0694555761) x_m))
          (* x_m x_m)
          (fma (fma 0.2909738639 (* x_m x_m) 0.7715471019) (* x_m x_m) 1.0)))))
      (/
       (-
        (/ (- (/ 0.15298196345929074 t_0) (/ -0.2514179000665374 x_m)) x_m)
        -0.5)
       x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = (x_m * x_m) * x_m;
	double tmp;
	if (x_m <= 1.45) {
		tmp = (fma(fma(fma(0.0072644182, (x_m * x_m), 0.0424060604), (x_m * x_m), 0.1049934947), (x_m * x_m), 1.0) * x_m) / fma(((pow(x_m, 10.0) * x_m) * x_m), 0.0003579942, fma(0.0008327945, pow(x_m, 10.0), fma(fma((t_0 * 0.0140005442), t_0, ((t_0 * 0.0694555761) * x_m)), (x_m * x_m), fma(fma(0.2909738639, (x_m * x_m), 0.7715471019), (x_m * x_m), 1.0))));
	} else {
		tmp = ((((0.15298196345929074 / t_0) - (-0.2514179000665374 / x_m)) / x_m) - -0.5) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(x_m * x_m) * x_m)
	tmp = 0.0
	if (x_m <= 1.45)
		tmp = Float64(Float64(fma(fma(fma(0.0072644182, Float64(x_m * x_m), 0.0424060604), Float64(x_m * x_m), 0.1049934947), Float64(x_m * x_m), 1.0) * x_m) / fma(Float64(Float64((x_m ^ 10.0) * x_m) * x_m), 0.0003579942, fma(0.0008327945, (x_m ^ 10.0), fma(fma(Float64(t_0 * 0.0140005442), t_0, Float64(Float64(t_0 * 0.0694555761) * x_m)), Float64(x_m * x_m), fma(fma(0.2909738639, Float64(x_m * x_m), 0.7715471019), Float64(x_m * x_m), 1.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.15298196345929074 / t_0) - Float64(-0.2514179000665374 / x_m)) / x_m) - -0.5) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 1.45], N[(N[(N[(N[(N[(0.0072644182 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0424060604), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.1049934947), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(N[(N[(N[Power[x$95$m, 10.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.0003579942 + N[(0.0008327945 * N[Power[x$95$m, 10.0], $MachinePrecision] + N[(N[(N[(t$95$0 * 0.0140005442), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * 0.0694555761), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(0.2909738639 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.15298196345929074 / t$95$0), $MachinePrecision] - N[(-0.2514179000665374 / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)}\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \color{blue}{\left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)}\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\color{blue}{\frac{1049934947}{10000000000}} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + \color{blue}{{x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)}\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \color{blue}{\left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)}\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\color{blue}{\frac{106015151}{2500000000}} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \color{blue}{\frac{36322091}{5000000000} \cdot {x}^{2}}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot \color{blue}{{x}^{2}}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      9. lower-pow.f64 50.2

         \[\leadsto \left(1 + {x}^{2} \cdot \left(0.1049934947 + {x}^{2} \cdot \left(0.0424060604 + 0.0072644182 \cdot {x}^{\color{blue}{2}}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot 0.0003579942\right) \cdot x, x, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(0.1049934947 + {x}^{2} \cdot \left(0.0424060604 + 0.0072644182 \cdot {x}^{2}\right)\right)\right)} \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot 0.0003579942\right) \cdot x, x, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

   8. Applied rewrites 50.2%

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

   if 1.44999999999999996 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.3%

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

   6. Applied rewrites 51.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 51.3

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

   8. Applied rewrites 51.3%

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

Alternative 6: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.45:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\left({x\_m}^{10} \cdot x\_m\right) \cdot x\_m, 0.0003579942, \mathsf{fma}\left(0.0008327945, {x\_m}^{10}, \mathsf{fma}\left(\mathsf{fma}\left(t\_0 \cdot 0.0140005442, t\_0, \left(t\_0 \cdot 0.0694555761\right) \cdot x\_m\right), x\_m \cdot x\_m, \mathsf{fma}\left(\mathsf{fma}\left(0.2909738639, x\_m \cdot x\_m, 0.7715471019\right), x\_m \cdot x\_m, 1\right)\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0072644182, x\_m \cdot x\_m, 0.0424060604\right), x\_m \cdot x\_m, 0.1049934947\right), x\_m \cdot x\_m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.15298196345929074}{t\_0} - \frac{-0.2514179000665374}{x\_m}}{x\_m} - -0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (* x_m x_m) x_m)))
   (*
    x_s
    (if (<= x_m 1.45)
      (*
       (/
        x_m
        (fma
         (* (* (pow x_m 10.0) x_m) x_m)
         0.0003579942
         (fma
          0.0008327945
          (pow x_m 10.0)
          (fma
           (fma (* t_0 0.0140005442) t_0 (* (* t_0 0.0694555761) x_m))
           (* x_m x_m)
           (fma
            (fma 0.2909738639 (* x_m x_m) 0.7715471019)
            (* x_m x_m)
            1.0)))))
       (fma
        (fma
         (fma 0.0072644182 (* x_m x_m) 0.0424060604)
         (* x_m x_m)
         0.1049934947)
        (* x_m x_m)
        1.0))
      (/
       (-
        (/ (- (/ 0.15298196345929074 t_0) (/ -0.2514179000665374 x_m)) x_m)
        -0.5)
       x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = (x_m * x_m) * x_m;
	double tmp;
	if (x_m <= 1.45) {
		tmp = (x_m / fma(((pow(x_m, 10.0) * x_m) * x_m), 0.0003579942, fma(0.0008327945, pow(x_m, 10.0), fma(fma((t_0 * 0.0140005442), t_0, ((t_0 * 0.0694555761) * x_m)), (x_m * x_m), fma(fma(0.2909738639, (x_m * x_m), 0.7715471019), (x_m * x_m), 1.0))))) * fma(fma(fma(0.0072644182, (x_m * x_m), 0.0424060604), (x_m * x_m), 0.1049934947), (x_m * x_m), 1.0);
	} else {
		tmp = ((((0.15298196345929074 / t_0) - (-0.2514179000665374 / x_m)) / x_m) - -0.5) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(x_m * x_m) * x_m)
	tmp = 0.0
	if (x_m <= 1.45)
		tmp = Float64(Float64(x_m / fma(Float64(Float64((x_m ^ 10.0) * x_m) * x_m), 0.0003579942, fma(0.0008327945, (x_m ^ 10.0), fma(fma(Float64(t_0 * 0.0140005442), t_0, Float64(Float64(t_0 * 0.0694555761) * x_m)), Float64(x_m * x_m), fma(fma(0.2909738639, Float64(x_m * x_m), 0.7715471019), Float64(x_m * x_m), 1.0))))) * fma(fma(fma(0.0072644182, Float64(x_m * x_m), 0.0424060604), Float64(x_m * x_m), 0.1049934947), Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.15298196345929074 / t_0) - Float64(-0.2514179000665374 / x_m)) / x_m) - -0.5) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 1.45], N[(N[(x$95$m / N[(N[(N[(N[Power[x$95$m, 10.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.0003579942 + N[(0.0008327945 * N[Power[x$95$m, 10.0], $MachinePrecision] + N[(N[(N[(t$95$0 * 0.0140005442), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * 0.0694555761), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(N[(0.2909738639 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.7715471019), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.0072644182 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0424060604), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.1049934947), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.15298196345929074 / t$95$0), $MachinePrecision] - N[(-0.2514179000665374 / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 53.8%

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

   3. Applied rewrites 53.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right)} \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)}\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \color{blue}{\left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)}\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\color{blue}{\frac{1049934947}{10000000000}} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + \color{blue}{{x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)}\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \color{blue}{\left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)}\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\color{blue}{\frac{106015151}{2500000000}} + \frac{36322091}{5000000000} \cdot {x}^{2}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \color{blue}{\frac{36322091}{5000000000} \cdot {x}^{2}}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1049934947}{10000000000} + {x}^{2} \cdot \left(\frac{106015151}{2500000000} + \frac{36322091}{5000000000} \cdot \color{blue}{{x}^{2}}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot \frac{1789971}{5000000000}\right) \cdot x, x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000} + \frac{2909738639}{10000000000} \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

      9. lower-pow.f64 50.2

         \[\leadsto \left(1 + {x}^{2} \cdot \left(0.1049934947 + {x}^{2} \cdot \left(0.0424060604 + 0.0072644182 \cdot {x}^{\color{blue}{2}}\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot 0.0003579942\right) \cdot x, x, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

   6. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(0.1049934947 + {x}^{2} \cdot \left(0.0424060604 + 0.0072644182 \cdot {x}^{2}\right)\right)\right)} \cdot \frac{x}{\mathsf{fma}\left(\left({\left(x \cdot x\right)}^{5} \cdot 0.0003579942\right) \cdot x, x, \mathsf{fma}\left(0.0008327945, {\left(x \cdot x\right)}^{5}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot x\right), x, \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.7715471019 + 0.2909738639 \cdot \left(x \cdot x\right), 1\right)\right)\right)\right)} \]

   8. Applied rewrites 50.2%

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

   if 1.44999999999999996 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.3%

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

   6. Applied rewrites 51.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 51.3

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

   8. Applied rewrites 51.3%

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

Alternative 7: 99.7% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 53.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 50.3%

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

   6. Applied rewrites 50.3%

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

   if 1.19999999999999996 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.3%

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

   6. Applied rewrites 51.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 51.3

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

   8. Applied rewrites 51.3%

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

Alternative 8: 99.7% accurate, 8.6× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.15)
    (*
     (fma (fma 0.265709700396151 (* x_m x_m) -0.6665536072) (* x_m x_m) 1.0)
     x_m)
    (/
     (-
      (/
       (-
        (/ 0.15298196345929074 (* (* x_m x_m) x_m))
        (/ -0.2514179000665374 x_m))
       x_m)
      -0.5)
     x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.15) {
		tmp = fma(fma(0.265709700396151, (x_m * x_m), -0.6665536072), (x_m * x_m), 1.0) * x_m;
	} else {
		tmp = ((((0.15298196345929074 / ((x_m * x_m) * x_m)) - (-0.2514179000665374 / x_m)) / x_m) - -0.5) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.15)
		tmp = Float64(fma(fma(0.265709700396151, Float64(x_m * x_m), -0.6665536072), Float64(x_m * x_m), 1.0) * x_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.15298196345929074 / Float64(Float64(x_m * x_m) * x_m)) - Float64(-0.2514179000665374 / x_m)) / x_m) - -0.5) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.15], N[(N[(N[(0.265709700396151 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(N[(N[(0.15298196345929074 / N[(N[(x$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] - N[(-0.2514179000665374 / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 53.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(0.265709700396151 \cdot {x}^{2} - 0.6665536072\right)\right)} \cdot x \]
   5. Step-by-step derivation

   6. Applied rewrites 51.0%

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

   if 1.1499999999999999 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.3%

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

   6. Applied rewrites 51.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 51.3

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

   8. Applied rewrites 51.3%

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

Alternative 9: 99.6% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 53.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(0.265709700396151 \cdot {x}^{2} - 0.6665536072\right)\right)} \cdot x \]
   5. Step-by-step derivation

   6. Applied rewrites 51.0%

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

   if 1.1000000000000001 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 51.4

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

   6. Applied rewrites 51.4%

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

Alternative 10: 99.5% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 53.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.2

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

   4. Applied rewrites 50.2%

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

   6. Applied rewrites 50.2%

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

   if 0.94999999999999996 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 51.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval 51.4

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

   6. Applied rewrites 51.4%

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

Alternative 11: 99.3% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 53.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.2

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

   4. Applied rewrites 50.2%

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

   6. Applied rewrites 50.2%

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

   if 0.80000000000000004 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 51.5

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

   4. Applied rewrites 51.5%

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

Alternative 12: 99.0% accurate, 31.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 53.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.3%

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

   if 0.69999999999999996 < x

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 51.5

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

   4. Applied rewrites 51.5%

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

Alternative 13: 51.3% accurate, 253.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 51.3%

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

Reproduce

herbie shell --seed 2025148 
(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))