Jmat.Real.dawson

Percentage Accurate: 54.4% → 99.9%

Time: 14.1s

Alternatives: 9

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: 54.4% 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: 99.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5000:\\ \;\;\;\;\frac{x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(0.1049934947 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0424060604 + \left(x\_m \cdot x\_m\right) \cdot 0.0072644182\right)\right) + \left(1 + x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(t\_0 \cdot \left(0.0005064034 + \left(x\_m \cdot x\_m\right) \cdot 0.0001789971\right)\right)\right)\right)\right)\right)}{1 + \left(\left(x\_m \cdot \left(x\_m \cdot \left(t\_0 \cdot t\_0\right)\right)\right) \cdot \left(0.0008327945 + \left(x\_m \cdot x\_m\right) \cdot 0.0003579942\right) + \left(x\_m \cdot x\_m\right) \cdot \left(0.7715471019 + x\_m \cdot \left(x\_m \cdot \left(0.2909738639 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0694555761 + \left(x\_m \cdot x\_m\right) \cdot 0.0140005442\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1.0056716002661497}{x\_m} + x\_m \cdot 2}\\ \end{array} \end{array} \end{array} \]

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)))))
   (*
    x_s
    (if (<= x_m 5000.0)
      (/
       (*
        x_m
        (+
         (*
          (* x_m x_m)
          (+
           0.1049934947
           (* (* x_m x_m) (+ 0.0424060604 (* (* x_m x_m) 0.0072644182)))))
         (+
          1.0
          (*
           x_m
           (*
            x_m
            (*
             (* x_m x_m)
             (* t_0 (+ 0.0005064034 (* (* x_m x_m) 0.0001789971)))))))))
       (+
        1.0
        (+
         (*
          (* x_m (* x_m (* t_0 t_0)))
          (+ 0.0008327945 (* (* x_m x_m) 0.0003579942)))
         (*
          (* x_m x_m)
          (+
           0.7715471019
           (*
            x_m
            (*
             x_m
             (+
              0.2909738639
              (*
               (* x_m x_m)
               (+ 0.0694555761 (* (* x_m x_m) 0.0140005442)))))))))))
      (/ 1.0 (+ (/ -1.0056716002661497 x_m) (* x_m 2.0)))))))

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 tmp;
	if (x_m <= 5000.0) {
		tmp = (x_m * (((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + ((x_m * x_m) * 0.0072644182))))) + (1.0 + (x_m * (x_m * ((x_m * x_m) * (t_0 * (0.0005064034 + ((x_m * x_m) * 0.0001789971))))))))) / (1.0 + (((x_m * (x_m * (t_0 * t_0))) * (0.0008327945 + ((x_m * x_m) * 0.0003579942))) + ((x_m * x_m) * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * 0.0140005442))))))))));
	} else {
		tmp = 1.0 / ((-1.0056716002661497 / x_m) + (x_m * 2.0));
	}
	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) :: tmp
    t_0 = x_m * (x_m * (x_m * x_m))
    if (x_m <= 5000.0d0) then
        tmp = (x_m * (((x_m * x_m) * (0.1049934947d0 + ((x_m * x_m) * (0.0424060604d0 + ((x_m * x_m) * 0.0072644182d0))))) + (1.0d0 + (x_m * (x_m * ((x_m * x_m) * (t_0 * (0.0005064034d0 + ((x_m * x_m) * 0.0001789971d0))))))))) / (1.0d0 + (((x_m * (x_m * (t_0 * t_0))) * (0.0008327945d0 + ((x_m * x_m) * 0.0003579942d0))) + ((x_m * x_m) * (0.7715471019d0 + (x_m * (x_m * (0.2909738639d0 + ((x_m * x_m) * (0.0694555761d0 + ((x_m * x_m) * 0.0140005442d0))))))))))
    else
        tmp = 1.0d0 / (((-1.0056716002661497d0) / x_m) + (x_m * 2.0d0))
    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 tmp;
	if (x_m <= 5000.0) {
		tmp = (x_m * (((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + ((x_m * x_m) * 0.0072644182))))) + (1.0 + (x_m * (x_m * ((x_m * x_m) * (t_0 * (0.0005064034 + ((x_m * x_m) * 0.0001789971))))))))) / (1.0 + (((x_m * (x_m * (t_0 * t_0))) * (0.0008327945 + ((x_m * x_m) * 0.0003579942))) + ((x_m * x_m) * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * 0.0140005442))))))))));
	} else {
		tmp = 1.0 / ((-1.0056716002661497 / x_m) + (x_m * 2.0));
	}
	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))
	tmp = 0
	if x_m <= 5000.0:
		tmp = (x_m * (((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + ((x_m * x_m) * 0.0072644182))))) + (1.0 + (x_m * (x_m * ((x_m * x_m) * (t_0 * (0.0005064034 + ((x_m * x_m) * 0.0001789971))))))))) / (1.0 + (((x_m * (x_m * (t_0 * t_0))) * (0.0008327945 + ((x_m * x_m) * 0.0003579942))) + ((x_m * x_m) * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * 0.0140005442))))))))))
	else:
		tmp = 1.0 / ((-1.0056716002661497 / x_m) + (x_m * 2.0))
	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)))
	tmp = 0.0
	if (x_m <= 5000.0)
		tmp = Float64(Float64(x_m * Float64(Float64(Float64(x_m * x_m) * Float64(0.1049934947 + Float64(Float64(x_m * x_m) * Float64(0.0424060604 + Float64(Float64(x_m * x_m) * 0.0072644182))))) + Float64(1.0 + Float64(x_m * Float64(x_m * Float64(Float64(x_m * x_m) * Float64(t_0 * Float64(0.0005064034 + Float64(Float64(x_m * x_m) * 0.0001789971))))))))) / Float64(1.0 + Float64(Float64(Float64(x_m * Float64(x_m * Float64(t_0 * t_0))) * Float64(0.0008327945 + Float64(Float64(x_m * x_m) * 0.0003579942))) + Float64(Float64(x_m * x_m) * Float64(0.7715471019 + Float64(x_m * Float64(x_m * Float64(0.2909738639 + Float64(Float64(x_m * x_m) * Float64(0.0694555761 + Float64(Float64(x_m * x_m) * 0.0140005442)))))))))));
	else
		tmp = Float64(1.0 / Float64(Float64(-1.0056716002661497 / x_m) + Float64(x_m * 2.0)));
	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));
	tmp = 0.0;
	if (x_m <= 5000.0)
		tmp = (x_m * (((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + ((x_m * x_m) * 0.0072644182))))) + (1.0 + (x_m * (x_m * ((x_m * x_m) * (t_0 * (0.0005064034 + ((x_m * x_m) * 0.0001789971))))))))) / (1.0 + (((x_m * (x_m * (t_0 * t_0))) * (0.0008327945 + ((x_m * x_m) * 0.0003579942))) + ((x_m * x_m) * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * 0.0140005442))))))))));
	else
		tmp = 1.0 / ((-1.0056716002661497 / x_m) + (x_m * 2.0));
	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]}, N[(x$95$s * If[LessEqual[x$95$m, 5000.0], N[(N[(x$95$m * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.1049934947 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0424060604 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0072644182), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(t$95$0 * N[(0.0005064034 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0001789971), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[(x$95$m * N[(x$95$m * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.0008327945 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0003579942), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.7715471019 + N[(x$95$m * N[(x$95$m * N[(0.2909738639 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0694555761 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-1.0056716002661497 / x$95$m), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 66.5%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(\frac{7715471019}{10000000000} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \color{blue}{\left({x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{2909738639}{10000000000}} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \color{blue}{\left({x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{694555761}{10000000000}} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{694555761}{10000000000}} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \color{blue}{\left(\frac{70002721}{5000000000} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \left({x}^{2} \cdot \color{blue}{\frac{70002721}{5000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{70002721}{5000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 66.5%

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

   if 5e3 < x

   1. Initial program 3.9%

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

   3. Simplified 3.8%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \color{blue}{x}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}{x}} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{\frac{1}{2} + \frac{\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}}{x \cdot x}}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}}{x \cdot x}\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}}{x \cdot x}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}\right), \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \left(\frac{\frac{1307076337763}{8543989815576}}{x \cdot x}\right)\right), \left(\color{blue}{x} \cdot x\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \mathsf{/.f64}\left(\frac{1307076337763}{8543989815576}, \left(x \cdot x\right)\right)\right), \left(x \cdot x\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \mathsf{/.f64}\left(\frac{1307076337763}{8543989815576}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot x\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \mathsf{/.f64}\left(\frac{1307076337763}{8543989815576}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right) + \color{blue}{2}\right)\right)\right) \]

       3. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\left(0 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right) + 2\right)\right)\right) \]

       4. associate-+l- N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(0 - \color{blue}{\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}} - 2\right)}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\mathsf{neg}\left(\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}} - 2\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}} - 2\right)\right)\right)}\right)\right) \]

       7. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(0 - \color{blue}{\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}} - 2\right)}\right)\right)\right) \]

       8. associate-+l- N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(0 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right) + \color{blue}{2}\right)\right)\right) \]

       9. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right) + 2\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(2 + \color{blue}{\left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \color{blue}{\left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right)\right) \]

       12. associate-*r/ N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{\frac{600041}{596657} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{\frac{600041}{596657}}{{x}^{2}}\right)\right)\right)\right)\right) \]

       14. distribute-neg-frac N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\mathsf{neg}\left(\frac{600041}{596657}\right)}{\color{blue}{{x}^{2}}}\right)\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{600041}{596657}\right)\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-600041}{596657}, \left({\color{blue}{x}}^{2}\right)\right)\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-600041}{596657}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       18. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-600041}{596657}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   12. Simplified 100.0%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(2 + \frac{-1.0056716002661497}{x \cdot x}\right)}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\frac{\frac{-600041}{596657}}{x \cdot x} + \color{blue}{2}\right)\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{-600041}{596657}}{x \cdot x} \cdot x + \color{blue}{2 \cdot x}\right)\right) \]

       3. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{-600041}{596657} \cdot \frac{1}{x \cdot x}\right) \cdot x + 2 \cdot x\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot \left(\frac{1}{x \cdot x} \cdot x\right) + \color{blue}{2} \cdot x\right)\right) \]

       5. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot \left(\frac{1}{{x}^{2}} \cdot x\right) + 2 \cdot x\right)\right) \]

       6. pow-flip N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot \left({x}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot x\right) + 2 \cdot x\right)\right) \]

       7. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot {x}^{\left(\left(\mathsf{neg}\left(2\right)\right) + 1\right)} + 2 \cdot x\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot {x}^{\left(-2 + 1\right)} + 2 \cdot x\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot {x}^{-1} + 2 \cdot x\right)\right) \]

       10. inv-pow N/A

           \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot \frac{1}{x} + 2 \cdot x\right)\right) \]

       11. div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{-600041}{596657}}{x} + \color{blue}{2} \cdot x\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{-600041}{596657}}{x}\right), \color{blue}{\left(2 \cdot x\right)}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-600041}{596657}, x\right), \left(\color{blue}{2} \cdot x\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-600041}{596657}, x\right), \left(x \cdot \color{blue}{2}\right)\right)\right) \]

       15. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-600041}{596657}, x\right), \mathsf{*.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   14. Applied egg-rr 100.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{-1.0056716002661497}{x} + x \cdot 2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot x\_m\right)\\ t_1 := x\_m \cdot t\_0\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 200:\\ \;\;\;\;\frac{x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(0.1049934947 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0424060604 + \left(x\_m \cdot x\_m\right) \cdot 0.0072644182\right)\right) + \left(1 + t\_1 \cdot \left(t\_0 \cdot \left(x\_m \cdot \left(0.0005064034 + \left(x\_m \cdot x\_m\right) \cdot 0.0001789971\right)\right)\right)\right)\right)}{1 + x\_m \cdot \left(\left(\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot \left(x\_m \cdot t\_1\right)\right)\right) \cdot \left(x\_m \cdot \left(0.0008327945 + x\_m \cdot \left(x\_m \cdot 0.0003579942\right)\right)\right) + x\_m \cdot \left(0.7715471019 + \left(x\_m \cdot x\_m\right) \cdot \left(0.2909738639 + x\_m \cdot \left(x\_m \cdot \left(0.0694555761 + x\_m \cdot \left(x\_m \cdot 0.0140005442\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 + \frac{0.2514179000665374}{x\_m \cdot x\_m}\right) + \frac{0.15298196345929074 + \frac{11.259630434457211}{x\_m \cdot x\_m}}{t\_1}}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m x_m))) (t_1 (* x_m t_0)))
   (*
    x_s
    (if (<= x_m 200.0)
      (/
       (*
        x_m
        (+
         (*
          (* x_m x_m)
          (+
           0.1049934947
           (* (* x_m x_m) (+ 0.0424060604 (* (* x_m x_m) 0.0072644182)))))
         (+
          1.0
          (*
           t_1
           (* t_0 (* x_m (+ 0.0005064034 (* (* x_m x_m) 0.0001789971))))))))
       (+
        1.0
        (*
         x_m
         (+
          (*
           (* (* x_m x_m) (* x_m (* x_m t_1)))
           (* x_m (+ 0.0008327945 (* x_m (* x_m 0.0003579942)))))
          (*
           x_m
           (+
            0.7715471019
            (*
             (* x_m x_m)
             (+
              0.2909738639
              (*
               x_m
               (* x_m (+ 0.0694555761 (* x_m (* x_m 0.0140005442)))))))))))))
      (/
       (+
        (+ 0.5 (/ 0.2514179000665374 (* x_m x_m)))
        (/ (+ 0.15298196345929074 (/ 11.259630434457211 (* x_m x_m))) t_1))
       x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = x_m * (x_m * x_m);
	double t_1 = x_m * t_0;
	double tmp;
	if (x_m <= 200.0) {
		tmp = (x_m * (((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + ((x_m * x_m) * 0.0072644182))))) + (1.0 + (t_1 * (t_0 * (x_m * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))))))) / (1.0 + (x_m * ((((x_m * x_m) * (x_m * (x_m * t_1))) * (x_m * (0.0008327945 + (x_m * (x_m * 0.0003579942))))) + (x_m * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + (x_m * (x_m * (0.0694555761 + (x_m * (x_m * 0.0140005442))))))))))));
	} else {
		tmp = ((0.5 + (0.2514179000665374 / (x_m * x_m))) + ((0.15298196345929074 + (11.259630434457211 / (x_m * x_m))) / t_1)) / 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) :: tmp
    t_0 = x_m * (x_m * x_m)
    t_1 = x_m * t_0
    if (x_m <= 200.0d0) then
        tmp = (x_m * (((x_m * x_m) * (0.1049934947d0 + ((x_m * x_m) * (0.0424060604d0 + ((x_m * x_m) * 0.0072644182d0))))) + (1.0d0 + (t_1 * (t_0 * (x_m * (0.0005064034d0 + ((x_m * x_m) * 0.0001789971d0)))))))) / (1.0d0 + (x_m * ((((x_m * x_m) * (x_m * (x_m * t_1))) * (x_m * (0.0008327945d0 + (x_m * (x_m * 0.0003579942d0))))) + (x_m * (0.7715471019d0 + ((x_m * x_m) * (0.2909738639d0 + (x_m * (x_m * (0.0694555761d0 + (x_m * (x_m * 0.0140005442d0))))))))))))
    else
        tmp = ((0.5d0 + (0.2514179000665374d0 / (x_m * x_m))) + ((0.15298196345929074d0 + (11.259630434457211d0 / (x_m * x_m))) / t_1)) / 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);
	double t_1 = x_m * t_0;
	double tmp;
	if (x_m <= 200.0) {
		tmp = (x_m * (((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + ((x_m * x_m) * 0.0072644182))))) + (1.0 + (t_1 * (t_0 * (x_m * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))))))) / (1.0 + (x_m * ((((x_m * x_m) * (x_m * (x_m * t_1))) * (x_m * (0.0008327945 + (x_m * (x_m * 0.0003579942))))) + (x_m * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + (x_m * (x_m * (0.0694555761 + (x_m * (x_m * 0.0140005442))))))))))));
	} else {
		tmp = ((0.5 + (0.2514179000665374 / (x_m * x_m))) + ((0.15298196345929074 + (11.259630434457211 / (x_m * x_m))) / t_1)) / 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)
	t_1 = x_m * t_0
	tmp = 0
	if x_m <= 200.0:
		tmp = (x_m * (((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + ((x_m * x_m) * 0.0072644182))))) + (1.0 + (t_1 * (t_0 * (x_m * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))))))) / (1.0 + (x_m * ((((x_m * x_m) * (x_m * (x_m * t_1))) * (x_m * (0.0008327945 + (x_m * (x_m * 0.0003579942))))) + (x_m * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + (x_m * (x_m * (0.0694555761 + (x_m * (x_m * 0.0140005442))))))))))))
	else:
		tmp = ((0.5 + (0.2514179000665374 / (x_m * x_m))) + ((0.15298196345929074 + (11.259630434457211 / (x_m * x_m))) / t_1)) / 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 * x_m))
	t_1 = Float64(x_m * t_0)
	tmp = 0.0
	if (x_m <= 200.0)
		tmp = Float64(Float64(x_m * Float64(Float64(Float64(x_m * x_m) * Float64(0.1049934947 + Float64(Float64(x_m * x_m) * Float64(0.0424060604 + Float64(Float64(x_m * x_m) * 0.0072644182))))) + Float64(1.0 + Float64(t_1 * Float64(t_0 * Float64(x_m * Float64(0.0005064034 + Float64(Float64(x_m * x_m) * 0.0001789971)))))))) / Float64(1.0 + Float64(x_m * Float64(Float64(Float64(Float64(x_m * x_m) * Float64(x_m * Float64(x_m * t_1))) * Float64(x_m * Float64(0.0008327945 + Float64(x_m * Float64(x_m * 0.0003579942))))) + Float64(x_m * Float64(0.7715471019 + Float64(Float64(x_m * x_m) * Float64(0.2909738639 + Float64(x_m * Float64(x_m * Float64(0.0694555761 + Float64(x_m * Float64(x_m * 0.0140005442)))))))))))));
	else
		tmp = Float64(Float64(Float64(0.5 + Float64(0.2514179000665374 / Float64(x_m * x_m))) + Float64(Float64(0.15298196345929074 + Float64(11.259630434457211 / Float64(x_m * x_m))) / t_1)) / 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);
	t_1 = x_m * t_0;
	tmp = 0.0;
	if (x_m <= 200.0)
		tmp = (x_m * (((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + ((x_m * x_m) * 0.0072644182))))) + (1.0 + (t_1 * (t_0 * (x_m * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))))))) / (1.0 + (x_m * ((((x_m * x_m) * (x_m * (x_m * t_1))) * (x_m * (0.0008327945 + (x_m * (x_m * 0.0003579942))))) + (x_m * (0.7715471019 + ((x_m * x_m) * (0.2909738639 + (x_m * (x_m * (0.0694555761 + (x_m * (x_m * 0.0140005442))))))))))));
	else
		tmp = ((0.5 + (0.2514179000665374 / (x_m * x_m))) + ((0.15298196345929074 + (11.259630434457211 / (x_m * x_m))) / t_1)) / 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 * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m * t$95$0), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 200.0], N[(N[(x$95$m * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.1049934947 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0424060604 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0072644182), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(t$95$1 * N[(t$95$0 * N[(x$95$m * N[(0.0005064034 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0001789971), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x$95$m * N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * N[(0.0008327945 + N[(x$95$m * N[(x$95$m * 0.0003579942), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(0.7715471019 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.2909738639 + N[(x$95$m * N[(x$95$m * N[(0.0694555761 + N[(x$95$m * N[(x$95$m * 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 + N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.15298196345929074 + N[(11.259630434457211 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 200

   1. Initial program 66.5%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(\frac{7715471019}{10000000000} + {x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \color{blue}{\left({x}^{2} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{2909738639}{10000000000}} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \color{blue}{\left({x}^{2} \cdot \left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{694555761}{10000000000} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{694555761}{10000000000}} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{694555761}{10000000000}} + \frac{70002721}{5000000000} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \color{blue}{\left(\frac{70002721}{5000000000} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \left({x}^{2} \cdot \color{blue}{\frac{70002721}{5000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{70002721}{5000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1049934947}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{106015151}{2500000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{36322091}{5000000000}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{2532017}{5000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{10000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1665589}{2000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1789971}{5000000000}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{7715471019}{10000000000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2909738639}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{694555761}{10000000000}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{70002721}{5000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 66.5%

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

   9. Applied egg-rr 66.5%

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

   if 200 < x

   1. Initial program 3.9%

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

   3. Simplified 3.8%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}{\color{blue}{x}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} + \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)}{x} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)\right) + \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)}{x} \]

      5. distribute-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} + \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)}{x} \]

      6. remove-double-neg N/A

         \[\leadsto \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} + \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} + \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)\right), \color{blue}{x}\right) \]

   7. Simplified 100.0%

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

Alternative 3: 99.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.16:\\ \;\;\;\;x\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 + \frac{0.2514179000665374}{x\_m \cdot x\_m}\right) + \frac{0.15298196345929074 + \frac{11.259630434457211}{x\_m \cdot x\_m}}{x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.16)
    (* x_m (+ 1.0 (* (* x_m x_m) -0.6665536072)))
    (/
     (+
      (+ 0.5 (/ 0.2514179000665374 (* x_m x_m)))
      (/
       (+ 0.15298196345929074 (/ 11.259630434457211 (* x_m x_m)))
       (* x_m (* x_m (* 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 <= 1.16) {
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	} else {
		tmp = ((0.5 + (0.2514179000665374 / (x_m * x_m))) + ((0.15298196345929074 + (11.259630434457211 / (x_m * x_m))) / (x_m * (x_m * (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 <= 1.16d0) then
        tmp = x_m * (1.0d0 + ((x_m * x_m) * (-0.6665536072d0)))
    else
        tmp = ((0.5d0 + (0.2514179000665374d0 / (x_m * x_m))) + ((0.15298196345929074d0 + (11.259630434457211d0 / (x_m * x_m))) / (x_m * (x_m * (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 <= 1.16) {
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	} else {
		tmp = ((0.5 + (0.2514179000665374 / (x_m * x_m))) + ((0.15298196345929074 + (11.259630434457211 / (x_m * x_m))) / (x_m * (x_m * (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 <= 1.16:
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072))
	else:
		tmp = ((0.5 + (0.2514179000665374 / (x_m * x_m))) + ((0.15298196345929074 + (11.259630434457211 / (x_m * x_m))) / (x_m * (x_m * (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 <= 1.16)
		tmp = Float64(x_m * Float64(1.0 + Float64(Float64(x_m * x_m) * -0.6665536072)));
	else
		tmp = Float64(Float64(Float64(0.5 + Float64(0.2514179000665374 / Float64(x_m * x_m))) + Float64(Float64(0.15298196345929074 + Float64(11.259630434457211 / Float64(x_m * x_m))) / Float64(x_m * Float64(x_m * Float64(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 <= 1.16)
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	else
		tmp = ((0.5 + (0.2514179000665374 / (x_m * x_m))) + ((0.15298196345929074 + (11.259630434457211 / (x_m * x_m))) / (x_m * (x_m * (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, 1.16], N[(x$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 + N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.15298196345929074 + N[(11.259630434457211 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.15999999999999992

   1. Initial program 66.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-833192009}{1250000000} \cdot {x}^{2}\right)}, x\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-833192009}{1250000000} \cdot {x}^{2}\right)\right), x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-833192009}{1250000000}\right)\right), x\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

      5. *-lowering-*.f64 63.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

   5. Simplified 63.3%

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

   if 1.15999999999999992 < x

   1. Initial program 5.7%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}{\color{blue}{x}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} + \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)}{x} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)\right) + \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)}{x} \]

      5. distribute-neg-out N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} + \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)}{x} \]

      6. remove-double-neg N/A

         \[\leadsto \frac{\frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} + \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} + \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)\right), \color{blue}{x}\right) \]

   7. Simplified 98.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.16:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 + \frac{0.2514179000665374}{x \cdot x}\right) + \frac{0.15298196345929074 + \frac{11.259630434457211}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 7.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.02:\\ \;\;\;\;x\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(2 + \frac{-1.0056716002661497 + \frac{-0.10624017004622396}{x\_m \cdot x\_m}}{x\_m \cdot x\_m}\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.02)
    (* x_m (+ 1.0 (* (* x_m x_m) -0.6665536072)))
    (/
     1.0
     (*
      x_m
      (+
       2.0
       (/
        (+ -1.0056716002661497 (/ -0.10624017004622396 (* x_m 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 <= 1.02) {
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	} else {
		tmp = 1.0 / (x_m * (2.0 + ((-1.0056716002661497 + (-0.10624017004622396 / (x_m * 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 <= 1.02d0) then
        tmp = x_m * (1.0d0 + ((x_m * x_m) * (-0.6665536072d0)))
    else
        tmp = 1.0d0 / (x_m * (2.0d0 + (((-1.0056716002661497d0) + ((-0.10624017004622396d0) / (x_m * 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 <= 1.02) {
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	} else {
		tmp = 1.0 / (x_m * (2.0 + ((-1.0056716002661497 + (-0.10624017004622396 / (x_m * 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 <= 1.02:
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072))
	else:
		tmp = 1.0 / (x_m * (2.0 + ((-1.0056716002661497 + (-0.10624017004622396 / (x_m * 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 <= 1.02)
		tmp = Float64(x_m * Float64(1.0 + Float64(Float64(x_m * x_m) * -0.6665536072)));
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(2.0 + Float64(Float64(-1.0056716002661497 + Float64(-0.10624017004622396 / Float64(x_m * x_m))) / Float64(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 <= 1.02)
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	else
		tmp = 1.0 / (x_m * (2.0 + ((-1.0056716002661497 + (-0.10624017004622396 / (x_m * 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, 1.02], N[(x$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(2.0 + N[(N[(-1.0056716002661497 + N[(-0.10624017004622396 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.02

   1. Initial program 66.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-833192009}{1250000000} \cdot {x}^{2}\right)}, x\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-833192009}{1250000000} \cdot {x}^{2}\right)\right), x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-833192009}{1250000000}\right)\right), x\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

      5. *-lowering-*.f64 63.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

   5. Simplified 63.3%

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

   if 1.02 < x

   1. Initial program 5.7%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \color{blue}{x}\right) \]

   7. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}{x}} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{\frac{1}{2} + \frac{\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}}{x \cdot x}}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}}{x \cdot x}\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}}{x \cdot x}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}\right), \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \left(\frac{\frac{1307076337763}{8543989815576}}{x \cdot x}\right)\right), \left(\color{blue}{x} \cdot x\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \mathsf{/.f64}\left(\frac{1307076337763}{8543989815576}, \left(x \cdot x\right)\right)\right), \left(x \cdot x\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \mathsf{/.f64}\left(\frac{1307076337763}{8543989815576}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot x\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \mathsf{/.f64}\left(\frac{1307076337763}{8543989815576}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(2 + -1 \cdot \frac{\frac{600041}{596657} + \frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(2 + -1 \cdot \frac{\frac{600041}{596657} + \frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{\frac{600041}{596657} + \frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{-1 \cdot \left(\frac{600041}{596657} + \frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right)}{\color{blue}{{x}^{2}}}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{600041}{596657} + \frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right)\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{600041}{596657} \cdot -1 + \left(\frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right) \cdot -1\right), \left({\color{blue}{x}}^{2}\right)\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{-600041}{596657} + \left(\frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right) \cdot -1\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{600041}{596657}\right)\right) + \left(\frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right) \cdot -1\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{600041}{596657}\right)\right), \left(\left(\frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right) \cdot -1\right)\right), \left({\color{blue}{x}}^{2}\right)\right)\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \left(\left(\frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right) \cdot -1\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \left(\frac{\frac{113464366360}{1067998726947} \cdot 1}{{x}^{2}} \cdot -1\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \left(\frac{\frac{113464366360}{1067998726947}}{{x}^{2}} \cdot -1\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       12. associate-*l/ N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \left(\frac{\frac{113464366360}{1067998726947} \cdot -1}{{x}^{2}}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \left(\frac{\frac{-113464366360}{1067998726947}}{{x}^{2}}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \left(\frac{\mathsf{neg}\left(\frac{113464366360}{1067998726947}\right)}{{x}^{2}}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{113464366360}{1067998726947}\right)\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \left(x \cdot x\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       20. *-lowering-*.f64 98.7%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   12. Simplified 98.7%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(2 + \frac{-1.0056716002661497 + \frac{-0.10624017004622396}{x \cdot x}}{x \cdot x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(2 + \frac{-1.0056716002661497 + \frac{-0.10624017004622396}{x \cdot x}}{x \cdot x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.02:\\ \;\;\;\;x\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot 2 + \frac{-1.0056716002661497 + \frac{-0.10624017004622396}{x\_m \cdot x\_m}}{x\_m}}\\ \end{array} \end{array} \]

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 1.02)
    (* x_m (+ 1.0 (* (* x_m x_m) -0.6665536072)))
    (/
     1.0
     (+
      (* x_m 2.0)
      (/ (+ -1.0056716002661497 (/ -0.10624017004622396 (* 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 <= 1.02) {
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	} else {
		tmp = 1.0 / ((x_m * 2.0) + ((-1.0056716002661497 + (-0.10624017004622396 / (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 <= 1.02d0) then
        tmp = x_m * (1.0d0 + ((x_m * x_m) * (-0.6665536072d0)))
    else
        tmp = 1.0d0 / ((x_m * 2.0d0) + (((-1.0056716002661497d0) + ((-0.10624017004622396d0) / (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 <= 1.02) {
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	} else {
		tmp = 1.0 / ((x_m * 2.0) + ((-1.0056716002661497 + (-0.10624017004622396 / (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 <= 1.02:
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072))
	else:
		tmp = 1.0 / ((x_m * 2.0) + ((-1.0056716002661497 + (-0.10624017004622396 / (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 <= 1.02)
		tmp = Float64(x_m * Float64(1.0 + Float64(Float64(x_m * x_m) * -0.6665536072)));
	else
		tmp = Float64(1.0 / Float64(Float64(x_m * 2.0) + Float64(Float64(-1.0056716002661497 + Float64(-0.10624017004622396 / Float64(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 <= 1.02)
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	else
		tmp = 1.0 / ((x_m * 2.0) + ((-1.0056716002661497 + (-0.10624017004622396 / (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, 1.02], N[(x$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(N[(-1.0056716002661497 + N[(-0.10624017004622396 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.02

   1. Initial program 66.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-833192009}{1250000000} \cdot {x}^{2}\right)}, x\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-833192009}{1250000000} \cdot {x}^{2}\right)\right), x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-833192009}{1250000000}\right)\right), x\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

      5. *-lowering-*.f64 63.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

   5. Simplified 63.3%

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

   if 1.02 < x

   1. Initial program 5.7%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \color{blue}{x}\right) \]

   7. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}{x}} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{\frac{1}{2} + \frac{\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}}{x \cdot x}}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}}{x \cdot x}\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}}{x \cdot x}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}\right), \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \left(\frac{\frac{1307076337763}{8543989815576}}{x \cdot x}\right)\right), \left(\color{blue}{x} \cdot x\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \mathsf{/.f64}\left(\frac{1307076337763}{8543989815576}, \left(x \cdot x\right)\right)\right), \left(x \cdot x\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \mathsf{/.f64}\left(\frac{1307076337763}{8543989815576}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot x\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \mathsf{/.f64}\left(\frac{1307076337763}{8543989815576}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(2 + -1 \cdot \frac{\frac{600041}{596657} + \frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(2 + -1 \cdot \frac{\frac{600041}{596657} + \frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \color{blue}{\left(-1 \cdot \frac{\frac{600041}{596657} + \frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)}\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{-1 \cdot \left(\frac{600041}{596657} + \frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right)}{\color{blue}{{x}^{2}}}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{600041}{596657} + \frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right)\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       5. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{600041}{596657} \cdot -1 + \left(\frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right) \cdot -1\right), \left({\color{blue}{x}}^{2}\right)\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{-600041}{596657} + \left(\frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right) \cdot -1\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{600041}{596657}\right)\right) + \left(\frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right) \cdot -1\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{600041}{596657}\right)\right), \left(\left(\frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right) \cdot -1\right)\right), \left({\color{blue}{x}}^{2}\right)\right)\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \left(\left(\frac{113464366360}{1067998726947} \cdot \frac{1}{{x}^{2}}\right) \cdot -1\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \left(\frac{\frac{113464366360}{1067998726947} \cdot 1}{{x}^{2}} \cdot -1\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \left(\frac{\frac{113464366360}{1067998726947}}{{x}^{2}} \cdot -1\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       12. associate-*l/ N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \left(\frac{\frac{113464366360}{1067998726947} \cdot -1}{{x}^{2}}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \left(\frac{\frac{-113464366360}{1067998726947}}{{x}^{2}}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \left(\frac{\mathsf{neg}\left(\frac{113464366360}{1067998726947}\right)}{{x}^{2}}\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{113464366360}{1067998726947}\right)\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \left(x \cdot x\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({x}^{2}\right)\right)\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       20. *-lowering-*.f64 98.7%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   12. Simplified 98.7%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(2 + \frac{-1.0056716002661497 + \frac{-0.10624017004622396}{x \cdot x}}{x \cdot x}\right)}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\frac{\frac{-600041}{596657} + \frac{\frac{-113464366360}{1067998726947}}{x \cdot x}}{x \cdot x} + \color{blue}{2}\right)\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{-600041}{596657} + \frac{\frac{-113464366360}{1067998726947}}{x \cdot x}}{x \cdot x} \cdot x + \color{blue}{2 \cdot x}\right)\right) \]

       3. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(\frac{-600041}{596657} + \frac{\frac{-113464366360}{1067998726947}}{x \cdot x}\right) \cdot \frac{1}{x \cdot x}\right) \cdot x + 2 \cdot x\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{-600041}{596657} + \frac{\frac{-113464366360}{1067998726947}}{x \cdot x}\right) \cdot \left(\frac{1}{x \cdot x} \cdot x\right) + \color{blue}{2} \cdot x\right)\right) \]

       5. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{-600041}{596657} + \frac{\frac{-113464366360}{1067998726947}}{x \cdot x}\right) \cdot \left(\frac{1}{{x}^{2}} \cdot x\right) + 2 \cdot x\right)\right) \]

       6. pow-flip N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{-600041}{596657} + \frac{\frac{-113464366360}{1067998726947}}{x \cdot x}\right) \cdot \left({x}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot x\right) + 2 \cdot x\right)\right) \]

       7. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{-600041}{596657} + \frac{\frac{-113464366360}{1067998726947}}{x \cdot x}\right) \cdot {x}^{\left(\left(\mathsf{neg}\left(2\right)\right) + 1\right)} + 2 \cdot x\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{-600041}{596657} + \frac{\frac{-113464366360}{1067998726947}}{x \cdot x}\right) \cdot {x}^{\left(-2 + 1\right)} + 2 \cdot x\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{-600041}{596657} + \frac{\frac{-113464366360}{1067998726947}}{x \cdot x}\right) \cdot {x}^{-1} + 2 \cdot x\right)\right) \]

       10. inv-pow N/A

           \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{-600041}{596657} + \frac{\frac{-113464366360}{1067998726947}}{x \cdot x}\right) \cdot \frac{1}{x} + 2 \cdot x\right)\right) \]

       11. div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{-600041}{596657} + \frac{\frac{-113464366360}{1067998726947}}{x \cdot x}}{x} + \color{blue}{2} \cdot x\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{-600041}{596657} + \frac{\frac{-113464366360}{1067998726947}}{x \cdot x}}{x}\right), \color{blue}{\left(2 \cdot x\right)}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-600041}{596657} + \frac{\frac{-113464366360}{1067998726947}}{x \cdot x}\right), x\right), \left(\color{blue}{2} \cdot x\right)\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \left(\frac{\frac{-113464366360}{1067998726947}}{x \cdot x}\right)\right), x\right), \left(2 \cdot x\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \left(x \cdot x\right)\right)\right), x\right), \left(2 \cdot x\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right), \left(2 \cdot x\right)\right)\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right), \left(x \cdot \color{blue}{2}\right)\right)\right) \]

       18. *-lowering-*.f64 98.7%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-600041}{596657}, \mathsf{/.f64}\left(\frac{-113464366360}{1067998726947}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right), \mathsf{*.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   14. Applied egg-rr 98.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{-1.0056716002661497 + \frac{-0.10624017004622396}{x \cdot x}}{x} + x \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 2 + \frac{-1.0056716002661497 + \frac{-0.10624017004622396}{x \cdot x}}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 12.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.02:\\ \;\;\;\;x\_m \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1.0056716002661497}{x\_m} + x\_m \cdot 2}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.02)
    (* x_m (+ 1.0 (* (* x_m x_m) -0.6665536072)))
    (/ 1.0 (+ (/ -1.0056716002661497 x_m) (* x_m 2.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.02) {
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	} else {
		tmp = 1.0 / ((-1.0056716002661497 / x_m) + (x_m * 2.0));
	}
	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 <= 1.02d0) then
        tmp = x_m * (1.0d0 + ((x_m * x_m) * (-0.6665536072d0)))
    else
        tmp = 1.0d0 / (((-1.0056716002661497d0) / x_m) + (x_m * 2.0d0))
    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 <= 1.02) {
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	} else {
		tmp = 1.0 / ((-1.0056716002661497 / x_m) + (x_m * 2.0));
	}
	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 <= 1.02:
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072))
	else:
		tmp = 1.0 / ((-1.0056716002661497 / x_m) + (x_m * 2.0))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.02)
		tmp = Float64(x_m * Float64(1.0 + Float64(Float64(x_m * x_m) * -0.6665536072)));
	else
		tmp = Float64(1.0 / Float64(Float64(-1.0056716002661497 / x_m) + Float64(x_m * 2.0)));
	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 <= 1.02)
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	else
		tmp = 1.0 / ((-1.0056716002661497 / x_m) + (x_m * 2.0));
	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, 1.02], N[(x$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-1.0056716002661497 / x$95$m), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.02

   1. Initial program 66.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-833192009}{1250000000} \cdot {x}^{2}\right)}, x\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-833192009}{1250000000} \cdot {x}^{2}\right)\right), x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-833192009}{1250000000}\right)\right), x\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

      5. *-lowering-*.f64 63.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

   5. Simplified 63.3%

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

   if 1.02 < x

   1. Initial program 5.7%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \frac{1}{2}}{x}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{600041}{2386628} + \frac{1307076337763}{8543989815576} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \color{blue}{x}\right) \]

   7. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}{x}} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{\frac{1}{2} + \frac{\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}}{x \cdot x}}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}}{x \cdot x}\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}}{x \cdot x}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{600041}{2386628} + \frac{\frac{1307076337763}{8543989815576}}{x \cdot x}\right), \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \left(\frac{\frac{1307076337763}{8543989815576}}{x \cdot x}\right)\right), \left(\color{blue}{x} \cdot x\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \mathsf{/.f64}\left(\frac{1307076337763}{8543989815576}, \left(x \cdot x\right)\right)\right), \left(x \cdot x\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \mathsf{/.f64}\left(\frac{1307076337763}{8543989815576}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(x \cdot x\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{600041}{2386628}, \mathsf{/.f64}\left(\frac{1307076337763}{8543989815576}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{0.5 + \frac{0.2514179000665374 + \frac{0.15298196345929074}{x \cdot x}}{x \cdot x}}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(2 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right) + \color{blue}{2}\right)\right)\right) \]

       3. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\left(0 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right) + 2\right)\right)\right) \]

       4. associate-+l- N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(0 - \color{blue}{\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}} - 2\right)}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\mathsf{neg}\left(\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}} - 2\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}} - 2\right)\right)\right)}\right)\right) \]

       7. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(0 - \color{blue}{\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}} - 2\right)}\right)\right)\right) \]

       8. associate-+l- N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(0 - \frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right) + \color{blue}{2}\right)\right)\right) \]

       9. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right) + 2\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(2 + \color{blue}{\left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \color{blue}{\left(\mathsf{neg}\left(\frac{600041}{596657} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right)\right) \]

       12. associate-*r/ N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{\frac{600041}{596657} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{\frac{600041}{596657}}{{x}^{2}}\right)\right)\right)\right)\right) \]

       14. distribute-neg-frac N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\mathsf{neg}\left(\frac{600041}{596657}\right)}{\color{blue}{{x}^{2}}}\right)\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{600041}{596657}\right)\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-600041}{596657}, \left({\color{blue}{x}}^{2}\right)\right)\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-600041}{596657}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       18. *-lowering-*.f64 98.7%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-600041}{596657}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   12. Simplified 98.7%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(2 + \frac{-1.0056716002661497}{x \cdot x}\right)}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\frac{\frac{-600041}{596657}}{x \cdot x} + \color{blue}{2}\right)\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{-600041}{596657}}{x \cdot x} \cdot x + \color{blue}{2 \cdot x}\right)\right) \]

       3. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{-600041}{596657} \cdot \frac{1}{x \cdot x}\right) \cdot x + 2 \cdot x\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot \left(\frac{1}{x \cdot x} \cdot x\right) + \color{blue}{2} \cdot x\right)\right) \]

       5. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot \left(\frac{1}{{x}^{2}} \cdot x\right) + 2 \cdot x\right)\right) \]

       6. pow-flip N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot \left({x}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot x\right) + 2 \cdot x\right)\right) \]

       7. pow-plus N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot {x}^{\left(\left(\mathsf{neg}\left(2\right)\right) + 1\right)} + 2 \cdot x\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot {x}^{\left(-2 + 1\right)} + 2 \cdot x\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot {x}^{-1} + 2 \cdot x\right)\right) \]

       10. inv-pow N/A

           \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{-600041}{596657} \cdot \frac{1}{x} + 2 \cdot x\right)\right) \]

       11. div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{-600041}{596657}}{x} + \color{blue}{2} \cdot x\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{-600041}{596657}}{x}\right), \color{blue}{\left(2 \cdot x\right)}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-600041}{596657}, x\right), \left(\color{blue}{2} \cdot x\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-600041}{596657}, x\right), \left(x \cdot \color{blue}{2}\right)\right)\right) \]

       15. *-lowering-*.f64 98.7%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-600041}{596657}, x\right), \mathsf{*.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   14. Applied egg-rr 98.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{-1.0056716002661497}{x} + x \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1.0056716002661497}{x} + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 12.3× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.8) {
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	} 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 <= 0.8d0) then
        tmp = x_m * (1.0d0 + ((x_m * x_m) * (-0.6665536072d0)))
    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 <= 0.8) {
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	} 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 <= 0.8:
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072))
	else:
		tmp = 0.5 / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.8)
		tmp = Float64(x_m * Float64(1.0 + Float64(Float64(x_m * x_m) * -0.6665536072)));
	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 <= 0.8)
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	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, 0.8], N[(x$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 66.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-833192009}{1250000000} \cdot {x}^{2}\right)}, x\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-833192009}{1250000000} \cdot {x}^{2}\right)\right), x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-833192009}{1250000000}\right)\right), x\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

      5. *-lowering-*.f64 63.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-833192009}{1250000000}\right)\right), x\right) \]

   5. Simplified 63.3%

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

   if 0.80000000000000004 < x

   1. Initial program 5.7%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 98.6%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

   7. Simplified 98.6%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.8% 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 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 64.2%

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

   if 0.69999999999999996 < x

   1. Initial program 5.7%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 98.6%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

   7. Simplified 98.6%

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

Alternative 9: 51.5% 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 53.5%

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

3. Simplified 53.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 51.5%

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

Reproduce

herbie shell --seed 2024163 
(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))