Jmat.Real.dawson

Percentage Accurate: 53.5% → 100.0%

Time: 5.8s

Alternatives: 10

Speedup: 10.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := t\_0 \cdot \left|x\right|\\ t_2 := {\left(\left|x\right|\right)}^{10}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, 0.0001789971, \mathsf{fma}\left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(t\_1, \mathsf{fma}\left(0.0005064034 \cdot \left|x\right|, \left|x\right|, 0.0072644182\right), 0.0424060604 \cdot \left|x\right|\right), 0.1049934947\right), t\_0, 1\right)\right)}{\mathsf{fma}\left(0.0003579942 \cdot t\_2, t\_0, \mathsf{fma}\left(0.0008327945, t\_2, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.0694555761 \cdot t\_0, \left|x\right|, \left(0.0140005442 \cdot t\_0\right) \cdot t\_1\right), \mathsf{fma}\left(0.2909738639, t\_0, 0.7715471019\right)\right), 1\right)\right)\right)} \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x)))
       (t_1 (* t_0 (fabs x)))
       (t_2 (pow (fabs x) 10.0)))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 1000000.0)
     (*
      (/
       (fma
        t_2
        0.0001789971
        (fma
         (fma
          (fabs x)
          (fma
           t_1
           (fma (* 0.0005064034 (fabs x)) (fabs x) 0.0072644182)
           (* 0.0424060604 (fabs x)))
          0.1049934947)
         t_0
         1.0))
       (fma
        (* 0.0003579942 t_2)
        t_0
        (fma
         0.0008327945
         t_2
         (fma
          t_0
          (fma
           (fabs x)
           (fma
            (* 0.0694555761 t_0)
            (fabs x)
            (* (* 0.0140005442 t_0) t_1))
           (fma 0.2909738639 t_0 0.7715471019))
          1.0))))
      (fabs x))
     (/ 0.5 (fabs x))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = t_0 * fabs(x);
	double t_2 = pow(fabs(x), 10.0);
	double tmp;
	if (fabs(x) <= 1000000.0) {
		tmp = (fma(t_2, 0.0001789971, fma(fma(fabs(x), fma(t_1, fma((0.0005064034 * fabs(x)), fabs(x), 0.0072644182), (0.0424060604 * fabs(x))), 0.1049934947), t_0, 1.0)) / fma((0.0003579942 * t_2), t_0, fma(0.0008327945, t_2, fma(t_0, fma(fabs(x), fma((0.0694555761 * t_0), fabs(x), ((0.0140005442 * t_0) * t_1)), fma(0.2909738639, t_0, 0.7715471019)), 1.0)))) * fabs(x);
	} else {
		tmp = 0.5 / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = Float64(t_0 * abs(x))
	t_2 = abs(x) ^ 10.0
	tmp = 0.0
	if (abs(x) <= 1000000.0)
		tmp = Float64(Float64(fma(t_2, 0.0001789971, fma(fma(abs(x), fma(t_1, fma(Float64(0.0005064034 * abs(x)), abs(x), 0.0072644182), Float64(0.0424060604 * abs(x))), 0.1049934947), t_0, 1.0)) / fma(Float64(0.0003579942 * t_2), t_0, fma(0.0008327945, t_2, fma(t_0, fma(abs(x), fma(Float64(0.0694555761 * t_0), abs(x), Float64(Float64(0.0140005442 * t_0) * t_1)), fma(0.2909738639, t_0, 0.7715471019)), 1.0)))) * abs(x));
	else
		tmp = Float64(0.5 / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[x], $MachinePrecision], 10.0], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1000000.0], N[(N[(N[(t$95$2 * 0.0001789971 + N[(N[(N[Abs[x], $MachinePrecision] * N[(t$95$1 * N[(N[(0.0005064034 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + 0.0072644182), $MachinePrecision] + N[(0.0424060604 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.1049934947), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.0003579942 * t$95$2), $MachinePrecision] * t$95$0 + N[(0.0008327945 * t$95$2 + N[(t$95$0 * N[(N[Abs[x], $MachinePrecision] * N[(N[(0.0694555761 * t$95$0), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(N[(0.0140005442 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$0 + 0.7715471019), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e6

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \color{blue}{{x}^{10}}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   3. Step-by-step derivation

      1. lower-pow.f64 53.5%

         \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{\color{blue}{10}}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   4. Applied rewrites 53.5%

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \color{blue}{{x}^{10}}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \color{blue}{{x}^{10}}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   6. Step-by-step derivation

      1. lower-pow.f64 53.5%

         \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{\color{blue}{10}}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   7. Applied rewrites 53.5%

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \color{blue}{{x}^{10}}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{10}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\color{blue}{{x}^{10}} \cdot \left(x \cdot x\right)\right)} \cdot x \]
   9. Step-by-step derivation

      1. lower-pow.f64 53.5%

         \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{10}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left({x}^{\color{blue}{10}} \cdot \left(x \cdot x\right)\right)} \cdot x \]

   10. Applied rewrites 53.5%

       \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{10}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\color{blue}{{x}^{10}} \cdot \left(x \cdot x\right)\right)} \cdot x \]

   12. Applied rewrites 53.6%

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

       1. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{36322091}{5000000000} \cdot \left(x \cdot x\right), x, \left(\frac{2532017}{5000000000} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(\frac{106015151}{2500000000}, x \cdot x, \frac{1049934947}{10000000000}\right)\right) + 1}\right)}{\mathsf{fma}\left(\frac{1789971}{5000000000} \cdot {x}^{10}, x \cdot x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(x \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(\frac{2909738639}{10000000000}, x \cdot x, \frac{7715471019}{10000000000}\right)\right), 1\right)\right)\right)} \cdot x \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({x}^{10}, \frac{1789971}{10000000000}, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{36322091}{5000000000} \cdot \left(x \cdot x\right), x, \left(\frac{2532017}{5000000000} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(\frac{106015151}{2500000000}, x \cdot x, \frac{1049934947}{10000000000}\right)\right) \cdot \left(x \cdot x\right)} + 1\right)}{\mathsf{fma}\left(\frac{1789971}{5000000000} \cdot {x}^{10}, x \cdot x, \mathsf{fma}\left(\frac{1665589}{2000000000}, {x}^{10}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{694555761}{10000000000} \cdot \left(x \cdot x\right), x, \left(\frac{70002721}{5000000000} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(\frac{2909738639}{10000000000}, x \cdot x, \frac{7715471019}{10000000000}\right)\right), 1\right)\right)\right)} \cdot x \]

       3. lower-fma.f64 53.6%

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

   14. Applied rewrites 53.5%

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

   if 1e6 < x

   1. Initial program 53.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 52.2%

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

   4. Applied rewrites 52.2%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := t\_0 \cdot \left|x\right|\\ t_2 := {\left(\left|x\right|\right)}^{10}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1000000:\\ \;\;\;\;\mathsf{fma}\left(0.0001789971, t\_2, \mathsf{fma}\left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(t\_1, \mathsf{fma}\left(0.0005064034 \cdot \left|x\right|, \left|x\right|, 0.0072644182\right), 0.0424060604 \cdot \left|x\right|\right), 0.1049934947\right), t\_0, 1\right)\right) \cdot \frac{\left|x\right|}{\mathsf{fma}\left(t\_2 \cdot \left(0.0003579942 \cdot \left|x\right|\right), \left|x\right|, \mathsf{fma}\left(0.0008327945, t\_2, \mathsf{fma}\left(\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(t\_1, \mathsf{fma}\left(0.0140005442 \cdot \left|x\right|, \left|x\right|, 0.0694555761\right), 0.2909738639 \cdot \left|x\right|\right), 0.7715471019\right), t\_0, 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x)))
       (t_1 (* t_0 (fabs x)))
       (t_2 (pow (fabs x) 10.0)))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 1000000.0)
     (*
      (fma
       0.0001789971
       t_2
       (fma
        (fma
         (fabs x)
         (fma
          t_1
          (fma (* 0.0005064034 (fabs x)) (fabs x) 0.0072644182)
          (* 0.0424060604 (fabs x)))
         0.1049934947)
        t_0
        1.0))
      (/
       (fabs x)
       (fma
        (* t_2 (* 0.0003579942 (fabs x)))
        (fabs x)
        (fma
         0.0008327945
         t_2
         (fma
          (fma
           (fabs x)
           (fma
            t_1
            (fma (* 0.0140005442 (fabs x)) (fabs x) 0.0694555761)
            (* 0.2909738639 (fabs x)))
           0.7715471019)
          t_0
          1.0)))))
     (/ 0.5 (fabs x))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = t_0 * fabs(x);
	double t_2 = pow(fabs(x), 10.0);
	double tmp;
	if (fabs(x) <= 1000000.0) {
		tmp = fma(0.0001789971, t_2, fma(fma(fabs(x), fma(t_1, fma((0.0005064034 * fabs(x)), fabs(x), 0.0072644182), (0.0424060604 * fabs(x))), 0.1049934947), t_0, 1.0)) * (fabs(x) / fma((t_2 * (0.0003579942 * fabs(x))), fabs(x), fma(0.0008327945, t_2, fma(fma(fabs(x), fma(t_1, fma((0.0140005442 * fabs(x)), fabs(x), 0.0694555761), (0.2909738639 * fabs(x))), 0.7715471019), t_0, 1.0))));
	} else {
		tmp = 0.5 / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = Float64(t_0 * abs(x))
	t_2 = abs(x) ^ 10.0
	tmp = 0.0
	if (abs(x) <= 1000000.0)
		tmp = Float64(fma(0.0001789971, t_2, fma(fma(abs(x), fma(t_1, fma(Float64(0.0005064034 * abs(x)), abs(x), 0.0072644182), Float64(0.0424060604 * abs(x))), 0.1049934947), t_0, 1.0)) * Float64(abs(x) / fma(Float64(t_2 * Float64(0.0003579942 * abs(x))), abs(x), fma(0.0008327945, t_2, fma(fma(abs(x), fma(t_1, fma(Float64(0.0140005442 * abs(x)), abs(x), 0.0694555761), Float64(0.2909738639 * abs(x))), 0.7715471019), t_0, 1.0)))));
	else
		tmp = Float64(0.5 / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[x], $MachinePrecision], 10.0], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1000000.0], N[(N[(0.0001789971 * t$95$2 + N[(N[(N[Abs[x], $MachinePrecision] * N[(t$95$1 * N[(N[(0.0005064034 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + 0.0072644182), $MachinePrecision] + N[(0.0424060604 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.1049934947), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] / N[(N[(t$95$2 * N[(0.0003579942 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(0.0008327945 * t$95$2 + N[(N[(N[Abs[x], $MachinePrecision] * N[(t$95$1 * N[(N[(0.0140005442 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + 0.0694555761), $MachinePrecision] + N[(0.2909738639 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.7715471019), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e6

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \color{blue}{{x}^{10}}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   3. Step-by-step derivation

      1. lower-pow.f64 53.5%

         \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{\color{blue}{10}}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   4. Applied rewrites 53.5%

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \color{blue}{{x}^{10}}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \color{blue}{{x}^{10}}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   6. Step-by-step derivation

      1. lower-pow.f64 53.5%

         \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{\color{blue}{10}}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   7. Applied rewrites 53.5%

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \color{blue}{{x}^{10}}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{10}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\color{blue}{{x}^{10}} \cdot \left(x \cdot x\right)\right)} \cdot x \]
   9. Step-by-step derivation

      1. lower-pow.f64 53.5%

         \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{10}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left({x}^{\color{blue}{10}} \cdot \left(x \cdot x\right)\right)} \cdot x \]

   10. Applied rewrites 53.5%

       \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{10}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\color{blue}{{x}^{10}} \cdot \left(x \cdot x\right)\right)} \cdot x \]

   12. Applied rewrites 53.6%

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

   14. Applied rewrites 53.5%

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

   if 1e6 < x

   1. Initial program 53.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 52.2%

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

   4. Applied rewrites 52.2%

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

Alternative 3: 99.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, t\_0, 0.265709700396151\right), t\_0, -0.6665536072\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_0, 1\right) \cdot \mathsf{fma}\left(t\_1, t\_0, -1\right)}{t\_1 \cdot t\_0 - 1} \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \left(\frac{0.15298196345929074}{{\left(\left|x\right|\right)}^{4}} + \mathsf{fma}\left(0.2514179000665374, \frac{1}{{\left(\left|x\right|\right)}^{2}}, 11.259630434457211 \cdot \frac{1}{{\left(\left|x\right|\right)}^{6}}\right)\right)}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x)))
       (t_1
        (fma
         (fma -0.0732490286039007 t_0 0.265709700396151)
         t_0
         -0.6665536072)))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 1.5)
     (*
      (/ (* (fma t_1 t_0 1.0) (fma t_1 t_0 -1.0)) (- (* t_1 t_0) 1.0))
      (fabs x))
     (/
      (+
       0.5
       (+
        (/ 0.15298196345929074 (pow (fabs x) 4.0))
        (fma
         0.2514179000665374
         (/ 1.0 (pow (fabs x) 2.0))
         (* 11.259630434457211 (/ 1.0 (pow (fabs x) 6.0))))))
      (fabs x))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = fma(fma(-0.0732490286039007, t_0, 0.265709700396151), t_0, -0.6665536072);
	double tmp;
	if (fabs(x) <= 1.5) {
		tmp = ((fma(t_1, t_0, 1.0) * fma(t_1, t_0, -1.0)) / ((t_1 * t_0) - 1.0)) * fabs(x);
	} else {
		tmp = (0.5 + ((0.15298196345929074 / pow(fabs(x), 4.0)) + fma(0.2514179000665374, (1.0 / pow(fabs(x), 2.0)), (11.259630434457211 * (1.0 / pow(fabs(x), 6.0)))))) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = fma(fma(-0.0732490286039007, t_0, 0.265709700396151), t_0, -0.6665536072)
	tmp = 0.0
	if (abs(x) <= 1.5)
		tmp = Float64(Float64(Float64(fma(t_1, t_0, 1.0) * fma(t_1, t_0, -1.0)) / Float64(Float64(t_1 * t_0) - 1.0)) * abs(x));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.15298196345929074 / (abs(x) ^ 4.0)) + fma(0.2514179000665374, Float64(1.0 / (abs(x) ^ 2.0)), Float64(11.259630434457211 * Float64(1.0 / (abs(x) ^ 6.0)))))) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.0732490286039007 * t$95$0 + 0.265709700396151), $MachinePrecision] * t$95$0 + -0.6665536072), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.5], N[(N[(N[(N[(t$95$1 * t$95$0 + 1.0), $MachinePrecision] * N[(t$95$1 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.15298196345929074 / N[Power[N[Abs[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.2514179000665374 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(11.259630434457211 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 53.5%

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

   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 49.6%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right) - 0.6665536072\right)\right)} \cdot x \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 49.1%

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right) - 1}} \cdot x \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)} - 1} \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right)} \cdot \left(x \cdot x\right) - 1} \cdot x \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \color{blue}{\left(x \cdot x\right)} - 1} \cdot x \]

      4. difference-of-squares-rev N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 49.1%

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

      9. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right), x \cdot x, 1\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right) - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \color{blue}{\left(x \cdot x\right)} - 1} \cdot x \]

      10. sub-flip N/A

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

      11. lift-*.f64 N/A

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

   8. Applied rewrites 49.1%

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

   if 1.5 < x

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \color{blue}{{x}^{10}}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   3. Step-by-step derivation

      1. lower-pow.f64 53.5%

         \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{\color{blue}{10}}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   4. Applied rewrites 53.5%

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \color{blue}{{x}^{10}}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \color{blue}{{x}^{10}}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   6. Step-by-step derivation

      1. lower-pow.f64 53.5%

         \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{\color{blue}{10}}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   7. Applied rewrites 53.5%

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \color{blue}{{x}^{10}}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{10}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\color{blue}{{x}^{10}} \cdot \left(x \cdot x\right)\right)} \cdot x \]
   9. Step-by-step derivation

      1. lower-pow.f64 53.5%

         \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{10}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left({x}^{\color{blue}{10}} \cdot \left(x \cdot x\right)\right)} \cdot x \]

   10. Applied rewrites 53.5%

       \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{10}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\color{blue}{{x}^{10}} \cdot \left(x \cdot x\right)\right)} \cdot x \]

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 51.9%

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

Alternative 4: 99.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, t\_0, 0.265709700396151\right), t\_0, -0.6665536072\right), t\_0, 1\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \left(\frac{0.15298196345929074}{{\left(\left|x\right|\right)}^{4}} + \mathsf{fma}\left(0.2514179000665374, \frac{1}{{\left(\left|x\right|\right)}^{2}}, 11.259630434457211 \cdot \frac{1}{{\left(\left|x\right|\right)}^{6}}\right)\right)}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x))))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 1.5)
     (*
      (fma
       (fma
        (fma -0.0732490286039007 t_0 0.265709700396151)
        t_0
        -0.6665536072)
       t_0
       1.0)
      (fabs x))
     (/
      (+
       0.5
       (+
        (/ 0.15298196345929074 (pow (fabs x) 4.0))
        (fma
         0.2514179000665374
         (/ 1.0 (pow (fabs x) 2.0))
         (* 11.259630434457211 (/ 1.0 (pow (fabs x) 6.0))))))
      (fabs x))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 1.5) {
		tmp = fma(fma(fma(-0.0732490286039007, t_0, 0.265709700396151), t_0, -0.6665536072), t_0, 1.0) * fabs(x);
	} else {
		tmp = (0.5 + ((0.15298196345929074 / pow(fabs(x), 4.0)) + fma(0.2514179000665374, (1.0 / pow(fabs(x), 2.0)), (11.259630434457211 * (1.0 / pow(fabs(x), 6.0)))))) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 1.5)
		tmp = Float64(fma(fma(fma(-0.0732490286039007, t_0, 0.265709700396151), t_0, -0.6665536072), t_0, 1.0) * abs(x));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.15298196345929074 / (abs(x) ^ 4.0)) + fma(0.2514179000665374, Float64(1.0 / (abs(x) ^ 2.0)), Float64(11.259630434457211 * Float64(1.0 / (abs(x) ^ 6.0)))))) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.5], N[(N[(N[(N[(-0.0732490286039007 * t$95$0 + 0.265709700396151), $MachinePrecision] * t$95$0 + -0.6665536072), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.15298196345929074 / N[Power[N[Abs[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.2514179000665374 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(11.259630434457211 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 53.5%

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

   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 49.6%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right) - 0.6665536072\right)\right)} \cdot x \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 49.1%

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right) - 1}} \cdot x \]
   7. Step-by-step derivation

      1. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \color{blue}{\left(x \cdot x\right)} - 1} \cdot x \]

      2. lower-unsound--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)} - 1} \cdot x \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right)} \cdot \left(x \cdot x\right) - 1} \cdot x \]

      4. lower-unsound-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right) - 1}} \cdot x \]

      5. lower-unsound--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right) - \color{blue}{1}} \cdot x \]

      6. flip-+ N/A

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

   8. Applied rewrites 49.6%

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

   if 1.5 < x

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \color{blue}{{x}^{10}}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   3. Step-by-step derivation

      1. lower-pow.f64 53.5%

         \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{\color{blue}{10}}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   4. Applied rewrites 53.5%

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \color{blue}{{x}^{10}}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \color{blue}{{x}^{10}}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
   6. Step-by-step derivation

      1. lower-pow.f64 53.5%

         \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{\color{blue}{10}}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   7. Applied rewrites 53.5%

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \color{blue}{{x}^{10}}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{10}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\color{blue}{{x}^{10}} \cdot \left(x \cdot x\right)\right)} \cdot x \]
   9. Step-by-step derivation

      1. lower-pow.f64 53.5%

         \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{10}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left({x}^{\color{blue}{10}} \cdot \left(x \cdot x\right)\right)} \cdot x \]

   10. Applied rewrites 53.5%

       \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot {x}^{10}\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\color{blue}{{x}^{10}} \cdot \left(x \cdot x\right)\right)} \cdot x \]

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 51.9%

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

Alternative 5: 99.7% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 53.5%

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

   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 49.6%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right) - 0.6665536072\right)\right)} \cdot x \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 49.1%

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right) - 1}} \cdot x \]
   7. Step-by-step derivation

      1. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \color{blue}{\left(x \cdot x\right)} - 1} \cdot x \]

      2. lower-unsound--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)} - 1} \cdot x \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right)} \cdot \left(x \cdot x\right) - 1} \cdot x \]

      4. lower-unsound-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right) - 1}} \cdot x \]

      5. lower-unsound--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right) - \color{blue}{1}} \cdot x \]

      6. flip-+ N/A

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

   8. Applied rewrites 49.6%

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

   if 1.5 < x

   1. Initial program 53.5%

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

   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 51.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 51.9%

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

   6. Applied rewrites 51.9%

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

Alternative 6: 99.6% accurate, 5.6× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x))))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 1.5)
     (*
      (fma
       (fma
        (fma -0.0732490286039007 t_0 0.265709700396151)
        t_0
        -0.6665536072)
       t_0
       1.0)
      (fabs x))
     (/ (- (/ 0.2514179000665374 t_0) -0.5) (fabs x))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 53.5%

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

   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 49.6%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right) - 0.6665536072\right)\right)} \cdot x \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 49.1%

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right) - 1}} \cdot x \]
   7. Step-by-step derivation

      1. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \color{blue}{\left(x \cdot x\right)} - 1} \cdot x \]

      2. lower-unsound--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)} - 1} \cdot x \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right)} \cdot \left(x \cdot x\right) - 1} \cdot x \]

      4. lower-unsound-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right) - 1}} \cdot x \]

      5. lower-unsound--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right) - \color{blue}{1}} \cdot x \]

      6. flip-+ N/A

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

   8. Applied rewrites 49.6%

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

   if 1.5 < x

   1. Initial program 53.5%

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

   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 52.0%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   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 52.0%

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

   6. Applied rewrites 52.0%

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

Alternative 7: 99.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.265709700396151, t\_0, -0.6665536072\right), t\_0, 1\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{t\_0} - -0.5}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x))))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 1.1)
     (*
      (fma (fma 0.265709700396151 t_0 -0.6665536072) t_0 1.0)
      (fabs x))
     (/ (- (/ 0.2514179000665374 t_0) -0.5) (fabs x))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 1.1) {
		tmp = fma(fma(0.265709700396151, t_0, -0.6665536072), t_0, 1.0) * fabs(x);
	} else {
		tmp = ((0.2514179000665374 / t_0) - -0.5) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 1.1)
		tmp = Float64(fma(fma(0.265709700396151, t_0, -0.6665536072), t_0, 1.0) * abs(x));
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / t_0) - -0.5) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.1], N[(N[(N[(0.265709700396151 * t$95$0 + -0.6665536072), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.2514179000665374 / t$95$0), $MachinePrecision] - -0.5), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 53.5%

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

   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 49.6%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right) - 0.6665536072\right)\right)} \cdot x \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 49.1%

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right) - 1}} \cdot x \]
   7. Step-by-step derivation

      1. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \color{blue}{\left(x \cdot x\right)} - 1} \cdot x \]

      2. lower-unsound--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)} - 1} \cdot x \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right)} \cdot \left(x \cdot x\right) - 1} \cdot x \]

      4. lower-unsound-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right) - 1}} \cdot x \]

      5. lower-unsound--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right) - \color{blue}{1}} \cdot x \]

      6. flip-+ N/A

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

   8. Applied rewrites 49.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000}, x \cdot x, -0.6665536072\right), x \cdot x, 1\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 50.3%

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

   if 1.1000000000000001 < x

   1. Initial program 53.5%

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

   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 52.0%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   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 52.0%

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

   6. Applied rewrites 52.0%

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

Alternative 8: 99.5% accurate, 9.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(-0.6665536072, t\_0, 1\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{t\_0} - -0.5}{\left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x))))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 1.2)
     (* (fma -0.6665536072 t_0 1.0) (fabs x))
     (/ (- (/ 0.2514179000665374 t_0) -0.5) (fabs x))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 1.2) {
		tmp = fma(-0.6665536072, t_0, 1.0) * fabs(x);
	} else {
		tmp = ((0.2514179000665374 / t_0) - -0.5) / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 1.2)
		tmp = Float64(fma(-0.6665536072, t_0, 1.0) * abs(x));
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / t_0) - -0.5) / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1.2], N[(N[(-0.6665536072 * t$95$0 + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.2514179000665374 / t$95$0), $MachinePrecision] - -0.5), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2

   1. Initial program 53.5%

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

   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 49.6%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right) - 0.6665536072\right)\right)} \cdot x \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 49.1%

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right) - 1}} \cdot x \]
   7. Step-by-step derivation

      1. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \color{blue}{\left(x \cdot x\right)} - 1} \cdot x \]

      2. lower-unsound--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)} - 1} \cdot x \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right)} \cdot \left(x \cdot x\right) - 1} \cdot x \]

      4. lower-unsound-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right) - 1}} \cdot x \]

      5. lower-unsound--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right) - \color{blue}{1}} \cdot x \]

      6. flip-+ N/A

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

   8. Applied rewrites 49.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{-833192009}{1250000000}, \color{blue}{x} \cdot x, 1\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 49.5%

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

   if 1.2 < x

   1. Initial program 53.5%

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

   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 52.0%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   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 52.0%

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

   6. Applied rewrites 52.0%

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

Alternative 9: 99.3% accurate, 10.1× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.78:\\ \;\;\;\;\mathsf{fma}\left(-0.6665536072, \left|x\right| \cdot \left|x\right|, 1\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left|x\right|}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (fabs x) 0.78)
   (* (fma -0.6665536072 (* (fabs x) (fabs x)) 1.0) (fabs x))
   (/ 0.5 (fabs x)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.78) {
		tmp = fma(-0.6665536072, (fabs(x) * fabs(x)), 1.0) * fabs(x);
	} else {
		tmp = 0.5 / fabs(x);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.78)
		tmp = Float64(fma(-0.6665536072, Float64(abs(x) * abs(x)), 1.0) * abs(x));
	else
		tmp = Float64(0.5 / abs(x));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 0.78], N[(N[(-0.6665536072 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(0.5 / N[Abs[x], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.78000000000000003

   1. Initial program 53.5%

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

   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 49.6%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right) - 0.6665536072\right)\right)} \cdot x \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 49.1%

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right) \cdot \left(x \cdot x\right) - 1}} \cdot x \]
   7. Step-by-step derivation

      1. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \color{blue}{\left(x \cdot x\right)} - 1} \cdot x \]

      2. lower-unsound--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)} - 1} \cdot x \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right)} \cdot \left(x \cdot x\right) - 1} \cdot x \]

      4. lower-unsound-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right) - 1}} \cdot x \]

      5. lower-unsound--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right)\right) - 1 \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right) \cdot \left(x \cdot x\right) - \color{blue}{1}} \cdot x \]

      6. flip-+ N/A

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

   8. Applied rewrites 49.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{-833192009}{1250000000}, \color{blue}{x} \cdot x, 1\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 49.5%

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

   if 0.78000000000000003 < x

   1. Initial program 53.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 52.2%

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

   4. Applied rewrites 52.2%

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

Alternative 10: 52.2% accurate, 53.2× speedup

Math

\[\frac{0.5}{x} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ 0.5 x))

C

double code(double x) {
	return 0.5 / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 0.5d0 / x
end function

Java

public static double code(double x) {
	return 0.5 / x;
}

Python

def code(x):
	return 0.5 / x

Julia

function code(x)
	return Float64(0.5 / x)
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / x;
end

Wolfram

code[x_] := N[(0.5 / x), $MachinePrecision]

Derivation

1. Initial program 53.5%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 52.2%

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

4. Applied rewrites 52.2%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025210 
(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))