Jmat.Real.dawson

Percentage Accurate: 54.2% → 100.0%

Time: 13.9s

Alternatives: 6

Speedup: 21.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\\ t_1 := \left(x\_m \cdot x\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot t\_0\\ t_3 := t\_1 \cdot \left(x\_m \cdot x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2000:\\ \;\;\;\;\frac{\left(0.0001789971 \cdot t\_2 + \left(\left(1 + \left(0.1049934947 \cdot \left(x\_m \cdot x\_m\right) + \left(0.0424060604 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(x\_m \cdot x\_m\right)\right)\right) + \left(0.0072644182 \cdot t\_1 + 0.0005064034 \cdot t\_3\right)\right)\right) \cdot x\_m}{0.0003579942 \cdot \left(t\_3 \cdot {x\_m}^{4}\right) + \left(t\_2 \cdot 0.0008327945 + \left(t\_3 \cdot 0.0140005442 + \left(t\_1 \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x\_m \cdot x\_m\right)\right) + 0.2909738639 \cdot t\_0\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{\frac{0.2514179000665374}{x\_m}}{x\_m}}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m (* x_m x_m))))
        (t_1 (* (* x_m x_m) t_0))
        (t_2 (* t_1 t_0))
        (t_3 (* t_1 (* x_m x_m))))
   (*
    x_s
    (if (<= x_m 2000.0)
      (/
       (*
        (+
         (* 0.0001789971 t_2)
         (+
          (+
           1.0
           (+
            (* 0.1049934947 (* x_m x_m))
            (* (* 0.0424060604 (* x_m x_m)) (* x_m x_m))))
          (+ (* 0.0072644182 t_1) (* 0.0005064034 t_3))))
        x_m)
       (+
        (* 0.0003579942 (* t_3 (pow x_m 4.0)))
        (+
         (* t_2 0.0008327945)
         (+
          (* t_3 0.0140005442)
          (+
           (* t_1 0.0694555761)
           (+ (+ 1.0 (* 0.7715471019 (* x_m x_m))) (* 0.2909738639 t_0)))))))
      (/ (+ 0.5 (/ (/ 0.2514179000665374 x_m) x_m)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = x_m * (x_m * (x_m * x_m));
	double t_1 = (x_m * x_m) * t_0;
	double t_2 = t_1 * t_0;
	double t_3 = t_1 * (x_m * x_m);
	double tmp;
	if (x_m <= 2000.0) {
		tmp = (((0.0001789971 * t_2) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((0.0424060604 * (x_m * x_m)) * (x_m * x_m)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_3)))) * x_m) / ((0.0003579942 * (t_3 * pow(x_m, 4.0))) + ((t_2 * 0.0008327945) + ((t_3 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + (0.7715471019 * (x_m * x_m))) + (0.2909738639 * t_0))))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 / x_m) / x_m)) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = x_m * (x_m * (x_m * x_m))
    t_1 = (x_m * x_m) * t_0
    t_2 = t_1 * t_0
    t_3 = t_1 * (x_m * x_m)
    if (x_m <= 2000.0d0) then
        tmp = (((0.0001789971d0 * t_2) + ((1.0d0 + ((0.1049934947d0 * (x_m * x_m)) + ((0.0424060604d0 * (x_m * x_m)) * (x_m * x_m)))) + ((0.0072644182d0 * t_1) + (0.0005064034d0 * t_3)))) * x_m) / ((0.0003579942d0 * (t_3 * (x_m ** 4.0d0))) + ((t_2 * 0.0008327945d0) + ((t_3 * 0.0140005442d0) + ((t_1 * 0.0694555761d0) + ((1.0d0 + (0.7715471019d0 * (x_m * x_m))) + (0.2909738639d0 * t_0))))))
    else
        tmp = (0.5d0 + ((0.2514179000665374d0 / x_m) / x_m)) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = x_m * (x_m * (x_m * x_m));
	double t_1 = (x_m * x_m) * t_0;
	double t_2 = t_1 * t_0;
	double t_3 = t_1 * (x_m * x_m);
	double tmp;
	if (x_m <= 2000.0) {
		tmp = (((0.0001789971 * t_2) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((0.0424060604 * (x_m * x_m)) * (x_m * x_m)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_3)))) * x_m) / ((0.0003579942 * (t_3 * Math.pow(x_m, 4.0))) + ((t_2 * 0.0008327945) + ((t_3 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + (0.7715471019 * (x_m * x_m))) + (0.2909738639 * t_0))))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 / x_m) / x_m)) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = x_m * (x_m * (x_m * x_m))
	t_1 = (x_m * x_m) * t_0
	t_2 = t_1 * t_0
	t_3 = t_1 * (x_m * x_m)
	tmp = 0
	if x_m <= 2000.0:
		tmp = (((0.0001789971 * t_2) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((0.0424060604 * (x_m * x_m)) * (x_m * x_m)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_3)))) * x_m) / ((0.0003579942 * (t_3 * math.pow(x_m, 4.0))) + ((t_2 * 0.0008327945) + ((t_3 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + (0.7715471019 * (x_m * x_m))) + (0.2909738639 * t_0))))))
	else:
		tmp = (0.5 + ((0.2514179000665374 / x_m) / x_m)) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(x_m * Float64(x_m * Float64(x_m * x_m)))
	t_1 = Float64(Float64(x_m * x_m) * t_0)
	t_2 = Float64(t_1 * t_0)
	t_3 = Float64(t_1 * Float64(x_m * x_m))
	tmp = 0.0
	if (x_m <= 2000.0)
		tmp = Float64(Float64(Float64(Float64(0.0001789971 * t_2) + Float64(Float64(1.0 + Float64(Float64(0.1049934947 * Float64(x_m * x_m)) + Float64(Float64(0.0424060604 * Float64(x_m * x_m)) * Float64(x_m * x_m)))) + Float64(Float64(0.0072644182 * t_1) + Float64(0.0005064034 * t_3)))) * x_m) / Float64(Float64(0.0003579942 * Float64(t_3 * (x_m ^ 4.0))) + Float64(Float64(t_2 * 0.0008327945) + Float64(Float64(t_3 * 0.0140005442) + Float64(Float64(t_1 * 0.0694555761) + Float64(Float64(1.0 + Float64(0.7715471019 * Float64(x_m * x_m))) + Float64(0.2909738639 * t_0)))))));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.2514179000665374 / x_m) / x_m)) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = x_m * (x_m * (x_m * x_m));
	t_1 = (x_m * x_m) * t_0;
	t_2 = t_1 * t_0;
	t_3 = t_1 * (x_m * x_m);
	tmp = 0.0;
	if (x_m <= 2000.0)
		tmp = (((0.0001789971 * t_2) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((0.0424060604 * (x_m * x_m)) * (x_m * x_m)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_3)))) * x_m) / ((0.0003579942 * (t_3 * (x_m ^ 4.0))) + ((t_2 * 0.0008327945) + ((t_3 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + (0.7715471019 * (x_m * x_m))) + (0.2909738639 * t_0))))));
	else
		tmp = (0.5 + ((0.2514179000665374 / x_m) / x_m)) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 2000.0], N[(N[(N[(N[(0.0001789971 * t$95$2), $MachinePrecision] + N[(N[(1.0 + N[(N[(0.1049934947 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0424060604 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0072644182 * t$95$1), $MachinePrecision] + N[(0.0005064034 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(N[(0.0003579942 * N[(t$95$3 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * 0.0008327945), $MachinePrecision] + N[(N[(t$95$3 * 0.0140005442), $MachinePrecision] + N[(N[(t$95$1 * 0.0694555761), $MachinePrecision] + N[(N[(1.0 + N[(0.7715471019 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.2514179000665374 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e3

   1. Initial program 73.6%

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

   3. Simplified 73.7%

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

      1. add-cbrt-cube 73.7%

         \[\leadsto \frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\sqrt[3]{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}}\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)} \]

      2. pow3 73.7%

         \[\leadsto \frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \sqrt[3]{\color{blue}{{\left(x \cdot x\right)}^{3}}}\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)} \]

      3. unpow-prod-down 73.6%

         \[\leadsto \frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \sqrt[3]{\color{blue}{{x}^{3} \cdot {x}^{3}}}\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)} \]

      4. metadata-eval 73.6%

         \[\leadsto \frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \sqrt[3]{{x}^{\color{blue}{\left(\frac{6}{2}\right)}} \cdot {x}^{3}}\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)} \]

      5. metadata-eval 73.6%

         \[\leadsto \frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \sqrt[3]{{x}^{\left(\frac{6}{2}\right)} \cdot {x}^{\color{blue}{\left(\frac{6}{2}\right)}}}\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)} \]

      6. sqr-pow 73.6%

         \[\leadsto \frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \sqrt[3]{\color{blue}{{x}^{6}}}\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)} \]

   6. Applied egg-rr 73.6%

      \[\leadsto \frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\sqrt[3]{{x}^{6}}}\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)} \]

   7. Taylor expanded in x around 0 73.7%

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

   if 2e3 < x

   1. Initial program 8.5%

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

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

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

      1. pow2 100.0%

         \[\leadsto \frac{0.5 + 0.2514179000665374 \cdot \frac{1}{\color{blue}{x \cdot x}}}{x} \]

   7. Applied egg-rr 100.0%

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

      1. un-div-inv 100.0%

         \[\leadsto \frac{0.5 + \color{blue}{\frac{0.2514179000665374}{x \cdot x}}}{x} \]

      2. associate-/r* 100.0%

         \[\leadsto \frac{0.5 + \color{blue}{\frac{\frac{0.2514179000665374}{x}}{x}}}{x} \]

   9. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\\ t_1 := \left(x\_m \cdot x\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot t\_0\\ t_3 := t\_1 \cdot \left(x\_m \cdot x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5000:\\ \;\;\;\;\frac{\left(0.0001789971 \cdot t\_2 + \left(\left(1 + \left(0.1049934947 \cdot \left(x\_m \cdot x\_m\right) + \left(0.0424060604 \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(x\_m \cdot x\_m\right)\right)\right) + \left(0.0072644182 \cdot t\_1 + 0.0005064034 \cdot t\_3\right)\right)\right) \cdot x\_m}{0.0003579942 \cdot \left(t\_3 \cdot t\_0\right) + \left(t\_2 \cdot 0.0008327945 + \left(t\_3 \cdot 0.0140005442 + \left(t\_1 \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x\_m \cdot x\_m\right)\right) + 0.2909738639 \cdot t\_0\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{\frac{0.2514179000665374}{x\_m}}{x\_m}}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m (* x_m x_m))))
        (t_1 (* (* x_m x_m) t_0))
        (t_2 (* t_1 t_0))
        (t_3 (* t_1 (* x_m x_m))))
   (*
    x_s
    (if (<= x_m 5000.0)
      (/
       (*
        (+
         (* 0.0001789971 t_2)
         (+
          (+
           1.0
           (+
            (* 0.1049934947 (* x_m x_m))
            (* (* 0.0424060604 (* x_m x_m)) (* x_m x_m))))
          (+ (* 0.0072644182 t_1) (* 0.0005064034 t_3))))
        x_m)
       (+
        (* 0.0003579942 (* t_3 t_0))
        (+
         (* t_2 0.0008327945)
         (+
          (* t_3 0.0140005442)
          (+
           (* t_1 0.0694555761)
           (+ (+ 1.0 (* 0.7715471019 (* x_m x_m))) (* 0.2909738639 t_0)))))))
      (/ (+ 0.5 (/ (/ 0.2514179000665374 x_m) x_m)) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = x_m * (x_m * (x_m * x_m));
	double t_1 = (x_m * x_m) * t_0;
	double t_2 = t_1 * t_0;
	double t_3 = t_1 * (x_m * x_m);
	double tmp;
	if (x_m <= 5000.0) {
		tmp = (((0.0001789971 * t_2) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((0.0424060604 * (x_m * x_m)) * (x_m * x_m)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_3)))) * x_m) / ((0.0003579942 * (t_3 * t_0)) + ((t_2 * 0.0008327945) + ((t_3 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + (0.7715471019 * (x_m * x_m))) + (0.2909738639 * t_0))))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 / x_m) / x_m)) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = x_m * (x_m * (x_m * x_m))
    t_1 = (x_m * x_m) * t_0
    t_2 = t_1 * t_0
    t_3 = t_1 * (x_m * x_m)
    if (x_m <= 5000.0d0) then
        tmp = (((0.0001789971d0 * t_2) + ((1.0d0 + ((0.1049934947d0 * (x_m * x_m)) + ((0.0424060604d0 * (x_m * x_m)) * (x_m * x_m)))) + ((0.0072644182d0 * t_1) + (0.0005064034d0 * t_3)))) * x_m) / ((0.0003579942d0 * (t_3 * t_0)) + ((t_2 * 0.0008327945d0) + ((t_3 * 0.0140005442d0) + ((t_1 * 0.0694555761d0) + ((1.0d0 + (0.7715471019d0 * (x_m * x_m))) + (0.2909738639d0 * t_0))))))
    else
        tmp = (0.5d0 + ((0.2514179000665374d0 / x_m) / x_m)) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = x_m * (x_m * (x_m * x_m));
	double t_1 = (x_m * x_m) * t_0;
	double t_2 = t_1 * t_0;
	double t_3 = t_1 * (x_m * x_m);
	double tmp;
	if (x_m <= 5000.0) {
		tmp = (((0.0001789971 * t_2) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((0.0424060604 * (x_m * x_m)) * (x_m * x_m)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_3)))) * x_m) / ((0.0003579942 * (t_3 * t_0)) + ((t_2 * 0.0008327945) + ((t_3 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + (0.7715471019 * (x_m * x_m))) + (0.2909738639 * t_0))))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 / x_m) / x_m)) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = x_m * (x_m * (x_m * x_m))
	t_1 = (x_m * x_m) * t_0
	t_2 = t_1 * t_0
	t_3 = t_1 * (x_m * x_m)
	tmp = 0
	if x_m <= 5000.0:
		tmp = (((0.0001789971 * t_2) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((0.0424060604 * (x_m * x_m)) * (x_m * x_m)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_3)))) * x_m) / ((0.0003579942 * (t_3 * t_0)) + ((t_2 * 0.0008327945) + ((t_3 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + (0.7715471019 * (x_m * x_m))) + (0.2909738639 * t_0))))))
	else:
		tmp = (0.5 + ((0.2514179000665374 / x_m) / x_m)) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(x_m * Float64(x_m * Float64(x_m * x_m)))
	t_1 = Float64(Float64(x_m * x_m) * t_0)
	t_2 = Float64(t_1 * t_0)
	t_3 = Float64(t_1 * Float64(x_m * x_m))
	tmp = 0.0
	if (x_m <= 5000.0)
		tmp = Float64(Float64(Float64(Float64(0.0001789971 * t_2) + Float64(Float64(1.0 + Float64(Float64(0.1049934947 * Float64(x_m * x_m)) + Float64(Float64(0.0424060604 * Float64(x_m * x_m)) * Float64(x_m * x_m)))) + Float64(Float64(0.0072644182 * t_1) + Float64(0.0005064034 * t_3)))) * x_m) / Float64(Float64(0.0003579942 * Float64(t_3 * t_0)) + Float64(Float64(t_2 * 0.0008327945) + Float64(Float64(t_3 * 0.0140005442) + Float64(Float64(t_1 * 0.0694555761) + Float64(Float64(1.0 + Float64(0.7715471019 * Float64(x_m * x_m))) + Float64(0.2909738639 * t_0)))))));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.2514179000665374 / x_m) / x_m)) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = x_m * (x_m * (x_m * x_m));
	t_1 = (x_m * x_m) * t_0;
	t_2 = t_1 * t_0;
	t_3 = t_1 * (x_m * x_m);
	tmp = 0.0;
	if (x_m <= 5000.0)
		tmp = (((0.0001789971 * t_2) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((0.0424060604 * (x_m * x_m)) * (x_m * x_m)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_3)))) * x_m) / ((0.0003579942 * (t_3 * t_0)) + ((t_2 * 0.0008327945) + ((t_3 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + (0.7715471019 * (x_m * x_m))) + (0.2909738639 * t_0))))));
	else
		tmp = (0.5 + ((0.2514179000665374 / x_m) / x_m)) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 5000.0], N[(N[(N[(N[(0.0001789971 * t$95$2), $MachinePrecision] + N[(N[(1.0 + N[(N[(0.1049934947 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0424060604 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0072644182 * t$95$1), $MachinePrecision] + N[(0.0005064034 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(N[(0.0003579942 * N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * 0.0008327945), $MachinePrecision] + N[(N[(t$95$3 * 0.0140005442), $MachinePrecision] + N[(N[(t$95$1 * 0.0694555761), $MachinePrecision] + N[(N[(1.0 + N[(0.7715471019 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.2514179000665374 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 73.6%

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

   3. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   if 5e3 < x

   1. Initial program 8.5%

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

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

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

      1. pow2 100.0%

         \[\leadsto \frac{0.5 + 0.2514179000665374 \cdot \frac{1}{\color{blue}{x \cdot x}}}{x} \]

   7. Applied egg-rr 100.0%

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

      1. un-div-inv 100.0%

         \[\leadsto \frac{0.5 + \color{blue}{\frac{0.2514179000665374}{x \cdot x}}}{x} \]

      2. associate-/r* 100.0%

         \[\leadsto \frac{0.5 + \color{blue}{\frac{\frac{0.2514179000665374}{x}}{x}}}{x} \]

   9. Applied egg-rr 100.0%

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

Alternative 3: 99.5% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.95d0) then
        tmp = (1.0d0 + ((-0.6665536072d0) * (x_m * x_m))) * x_m
    else
        tmp = (0.5d0 + ((0.2514179000665374d0 / x_m) / x_m)) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.95) {
		tmp = (1.0 + (-0.6665536072 * (x_m * x_m))) * x_m;
	} else {
		tmp = (0.5 + ((0.2514179000665374 / x_m) / x_m)) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.95:
		tmp = (1.0 + (-0.6665536072 * (x_m * x_m))) * x_m
	else:
		tmp = (0.5 + ((0.2514179000665374 / x_m) / x_m)) / x_m
	return x_s * tmp

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.95)
		tmp = (1.0 + (-0.6665536072 * (x_m * x_m))) * x_m;
	else
		tmp = (0.5 + ((0.2514179000665374 / x_m) / x_m)) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 73.5%

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

   3. Taylor expanded in x around 0 68.5%

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

      1. pow2 68.5%

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

   5. Applied egg-rr 68.5%

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

   if 0.94999999999999996 < x

   1. Initial program 10.0%

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

   3. Simplified 10.0%

      \[\leadsto \color{blue}{\frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.8%

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

      1. pow2 99.8%

         \[\leadsto \frac{0.5 + 0.2514179000665374 \cdot \frac{1}{\color{blue}{x \cdot x}}}{x} \]

   7. Applied egg-rr 99.8%

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

      1. un-div-inv 99.8%

         \[\leadsto \frac{0.5 + \color{blue}{\frac{0.2514179000665374}{x \cdot x}}}{x} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{0.5 + \color{blue}{\frac{\frac{0.2514179000665374}{x}}{x}}}{x} \]

   9. Applied egg-rr 99.8%

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

Alternative 4: 99.2% accurate, 12.3× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.78)
    (* (+ 1.0 (* -0.6665536072 (* x_m x_m))) x_m)
    (/ 0.5 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.78) {
		tmp = (1.0 + (-0.6665536072 * (x_m * x_m))) * x_m;
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.78d0) then
        tmp = (1.0d0 + ((-0.6665536072d0) * (x_m * x_m))) * x_m
    else
        tmp = 0.5d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.78) {
		tmp = (1.0 + (-0.6665536072 * (x_m * x_m))) * x_m;
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.78:
		tmp = (1.0 + (-0.6665536072 * (x_m * x_m))) * x_m
	else:
		tmp = 0.5 / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.78)
		tmp = Float64(Float64(1.0 + Float64(-0.6665536072 * Float64(x_m * x_m))) * x_m);
	else
		tmp = Float64(0.5 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.78)
		tmp = (1.0 + (-0.6665536072 * (x_m * x_m))) * x_m;
	else
		tmp = 0.5 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.78], N[(N[(1.0 + N[(-0.6665536072 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.78000000000000003

   1. Initial program 73.5%

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

   3. Taylor expanded in x around 0 68.5%

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

      1. pow2 68.5%

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

   5. Applied egg-rr 68.5%

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

   if 0.78000000000000003 < x

   1. Initial program 10.0%

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

   3. Simplified 10.0%

      \[\leadsto \color{blue}{\frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.9%

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

Alternative 5: 98.9% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.7d0) then
        tmp = x_m
    else
        tmp = 0.5d0 / x_m
    end if
    code = x_s * tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 73.5%

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

   3. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 68.6%

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

   if 0.69999999999999996 < x

   1. Initial program 10.0%

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

   3. Simplified 10.0%

      \[\leadsto \color{blue}{\frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.9%

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

Alternative 6: 51.4% accurate, 173.0× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.9%

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

3. Simplified 58.9%

   \[\leadsto \color{blue}{\frac{\left(0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{0.0003579942 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 53.7%

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

Reproduce

herbie shell --seed 2024116 -o generate:simplify
(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))