Jmat.Real.dawson

Percentage Accurate: 54.8% → 98.3%

Time: 5.6s

Alternatives: 5

Speedup: 24.2×

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: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(x \cdot x\right) \cdot 0.7715471019\\ t_1 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_2 := 0.1049934947 \cdot \left(x \cdot x\right)\\ t_3 := \left(x \cdot x\right) \cdot t_1\\ t_4 := \left(x \cdot x\right) \cdot t_3\\ t_5 := \left(x \cdot x\right) \cdot t_4\\ t_6 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_7 := \left(x \cdot x\right) \cdot t_6\\ t_8 := t_6 \cdot t_7\\ t_9 := \left(x \cdot x\right) \cdot t_7\\ \mathbf{if}\;x \cdot \frac{\left(\left(\left(\left(1 + t_2\right) + 0.0424060604 \cdot t_1\right) + 0.0072644182 \cdot t_3\right) + 0.0005064034 \cdot t_4\right) + 0.0001789971 \cdot t_5}{\left(\left(\left(\left(t_0 + t_1 \cdot 0.2909738639\right) + t_3 \cdot 0.0694555761\right) + t_4 \cdot 0.0140005442\right) + t_5 \cdot 0.0008327945\right) + 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot t_5\right)} \leq 10^{-6}:\\ \;\;\;\;x \cdot \frac{0.0001789971 \cdot t_8 + \left(\left(1 + \left(t_2 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0424060604\right)\right)\right) + \left(0.0072644182 \cdot t_7 + 0.0005064034 \cdot t_9\right)\right)}{0.0003579942 \cdot \left(t_6 \cdot t_9\right) + \left(0.0008327945 \cdot t_8 + \left(0.0140005442 \cdot t_9 + \left(0.0694555761 \cdot t_7 + \left(t_0 + 0.2909738639 \cdot t_6\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (* x x) 0.7715471019)))
        (t_1 (* (* x x) (* x x)))
        (t_2 (* 0.1049934947 (* x x)))
        (t_3 (* (* x x) t_1))
        (t_4 (* (* x x) t_3))
        (t_5 (* (* x x) t_4))
        (t_6 (* x (* x (* x x))))
        (t_7 (* (* x x) t_6))
        (t_8 (* t_6 t_7))
        (t_9 (* (* x x) t_7)))
   (if (<=
        (*
         x
         (/
          (+
           (+
            (+ (+ (+ 1.0 t_2) (* 0.0424060604 t_1)) (* 0.0072644182 t_3))
            (* 0.0005064034 t_4))
           (* 0.0001789971 t_5))
          (+
           (+
            (+
             (+ (+ t_0 (* t_1 0.2909738639)) (* t_3 0.0694555761))
             (* t_4 0.0140005442))
            (* t_5 0.0008327945))
           (* 0.0003579942 (* (* x x) t_5)))))
        1e-6)
     (*
      x
      (/
       (+
        (* 0.0001789971 t_8)
        (+
         (+ 1.0 (+ t_2 (* (* x x) (* (* x x) 0.0424060604))))
         (+ (* 0.0072644182 t_7) (* 0.0005064034 t_9))))
       (+
        (* 0.0003579942 (* t_6 t_9))
        (+
         (* 0.0008327945 t_8)
         (+
          (* 0.0140005442 t_9)
          (+ (* 0.0694555761 t_7) (+ t_0 (* 0.2909738639 t_6))))))))
     (+ (/ 0.5 x) (/ 0.2514179000665374 (pow x 3.0))))))

C

double code(double x) {
	double t_0 = 1.0 + ((x * x) * 0.7715471019);
	double t_1 = (x * x) * (x * x);
	double t_2 = 0.1049934947 * (x * x);
	double t_3 = (x * x) * t_1;
	double t_4 = (x * x) * t_3;
	double t_5 = (x * x) * t_4;
	double t_6 = x * (x * (x * x));
	double t_7 = (x * x) * t_6;
	double t_8 = t_6 * t_7;
	double t_9 = (x * x) * t_7;
	double tmp;
	if ((x * ((((((1.0 + t_2) + (0.0424060604 * t_1)) + (0.0072644182 * t_3)) + (0.0005064034 * t_4)) + (0.0001789971 * t_5)) / (((((t_0 + (t_1 * 0.2909738639)) + (t_3 * 0.0694555761)) + (t_4 * 0.0140005442)) + (t_5 * 0.0008327945)) + (0.0003579942 * ((x * x) * t_5))))) <= 1e-6) {
		tmp = x * (((0.0001789971 * t_8) + ((1.0 + (t_2 + ((x * x) * ((x * x) * 0.0424060604)))) + ((0.0072644182 * t_7) + (0.0005064034 * t_9)))) / ((0.0003579942 * (t_6 * t_9)) + ((0.0008327945 * t_8) + ((0.0140005442 * t_9) + ((0.0694555761 * t_7) + (t_0 + (0.2909738639 * t_6)))))));
	} else {
		tmp = (0.5 / x) + (0.2514179000665374 / pow(x, 3.0));
	}
	return tmp;
}

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
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: t_9
    real(8) :: tmp
    t_0 = 1.0d0 + ((x * x) * 0.7715471019d0)
    t_1 = (x * x) * (x * x)
    t_2 = 0.1049934947d0 * (x * x)
    t_3 = (x * x) * t_1
    t_4 = (x * x) * t_3
    t_5 = (x * x) * t_4
    t_6 = x * (x * (x * x))
    t_7 = (x * x) * t_6
    t_8 = t_6 * t_7
    t_9 = (x * x) * t_7
    if ((x * ((((((1.0d0 + t_2) + (0.0424060604d0 * t_1)) + (0.0072644182d0 * t_3)) + (0.0005064034d0 * t_4)) + (0.0001789971d0 * t_5)) / (((((t_0 + (t_1 * 0.2909738639d0)) + (t_3 * 0.0694555761d0)) + (t_4 * 0.0140005442d0)) + (t_5 * 0.0008327945d0)) + (0.0003579942d0 * ((x * x) * t_5))))) <= 1d-6) then
        tmp = x * (((0.0001789971d0 * t_8) + ((1.0d0 + (t_2 + ((x * x) * ((x * x) * 0.0424060604d0)))) + ((0.0072644182d0 * t_7) + (0.0005064034d0 * t_9)))) / ((0.0003579942d0 * (t_6 * t_9)) + ((0.0008327945d0 * t_8) + ((0.0140005442d0 * t_9) + ((0.0694555761d0 * t_7) + (t_0 + (0.2909738639d0 * t_6)))))))
    else
        tmp = (0.5d0 / x) + (0.2514179000665374d0 / (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + ((x * x) * 0.7715471019);
	double t_1 = (x * x) * (x * x);
	double t_2 = 0.1049934947 * (x * x);
	double t_3 = (x * x) * t_1;
	double t_4 = (x * x) * t_3;
	double t_5 = (x * x) * t_4;
	double t_6 = x * (x * (x * x));
	double t_7 = (x * x) * t_6;
	double t_8 = t_6 * t_7;
	double t_9 = (x * x) * t_7;
	double tmp;
	if ((x * ((((((1.0 + t_2) + (0.0424060604 * t_1)) + (0.0072644182 * t_3)) + (0.0005064034 * t_4)) + (0.0001789971 * t_5)) / (((((t_0 + (t_1 * 0.2909738639)) + (t_3 * 0.0694555761)) + (t_4 * 0.0140005442)) + (t_5 * 0.0008327945)) + (0.0003579942 * ((x * x) * t_5))))) <= 1e-6) {
		tmp = x * (((0.0001789971 * t_8) + ((1.0 + (t_2 + ((x * x) * ((x * x) * 0.0424060604)))) + ((0.0072644182 * t_7) + (0.0005064034 * t_9)))) / ((0.0003579942 * (t_6 * t_9)) + ((0.0008327945 * t_8) + ((0.0140005442 * t_9) + ((0.0694555761 * t_7) + (t_0 + (0.2909738639 * t_6)))))));
	} else {
		tmp = (0.5 / x) + (0.2514179000665374 / Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + ((x * x) * 0.7715471019)
	t_1 = (x * x) * (x * x)
	t_2 = 0.1049934947 * (x * x)
	t_3 = (x * x) * t_1
	t_4 = (x * x) * t_3
	t_5 = (x * x) * t_4
	t_6 = x * (x * (x * x))
	t_7 = (x * x) * t_6
	t_8 = t_6 * t_7
	t_9 = (x * x) * t_7
	tmp = 0
	if (x * ((((((1.0 + t_2) + (0.0424060604 * t_1)) + (0.0072644182 * t_3)) + (0.0005064034 * t_4)) + (0.0001789971 * t_5)) / (((((t_0 + (t_1 * 0.2909738639)) + (t_3 * 0.0694555761)) + (t_4 * 0.0140005442)) + (t_5 * 0.0008327945)) + (0.0003579942 * ((x * x) * t_5))))) <= 1e-6:
		tmp = x * (((0.0001789971 * t_8) + ((1.0 + (t_2 + ((x * x) * ((x * x) * 0.0424060604)))) + ((0.0072644182 * t_7) + (0.0005064034 * t_9)))) / ((0.0003579942 * (t_6 * t_9)) + ((0.0008327945 * t_8) + ((0.0140005442 * t_9) + ((0.0694555761 * t_7) + (t_0 + (0.2909738639 * t_6)))))))
	else:
		tmp = (0.5 / x) + (0.2514179000665374 / math.pow(x, 3.0))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(Float64(x * x) * 0.7715471019))
	t_1 = Float64(Float64(x * x) * Float64(x * x))
	t_2 = Float64(0.1049934947 * Float64(x * x))
	t_3 = Float64(Float64(x * x) * t_1)
	t_4 = Float64(Float64(x * x) * t_3)
	t_5 = Float64(Float64(x * x) * t_4)
	t_6 = Float64(x * Float64(x * Float64(x * x)))
	t_7 = Float64(Float64(x * x) * t_6)
	t_8 = Float64(t_6 * t_7)
	t_9 = Float64(Float64(x * x) * t_7)
	tmp = 0.0
	if (Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(1.0 + t_2) + Float64(0.0424060604 * t_1)) + Float64(0.0072644182 * t_3)) + Float64(0.0005064034 * t_4)) + Float64(0.0001789971 * t_5)) / Float64(Float64(Float64(Float64(Float64(t_0 + Float64(t_1 * 0.2909738639)) + Float64(t_3 * 0.0694555761)) + Float64(t_4 * 0.0140005442)) + Float64(t_5 * 0.0008327945)) + Float64(0.0003579942 * Float64(Float64(x * x) * t_5))))) <= 1e-6)
		tmp = Float64(x * Float64(Float64(Float64(0.0001789971 * t_8) + Float64(Float64(1.0 + Float64(t_2 + Float64(Float64(x * x) * Float64(Float64(x * x) * 0.0424060604)))) + Float64(Float64(0.0072644182 * t_7) + Float64(0.0005064034 * t_9)))) / Float64(Float64(0.0003579942 * Float64(t_6 * t_9)) + Float64(Float64(0.0008327945 * t_8) + Float64(Float64(0.0140005442 * t_9) + Float64(Float64(0.0694555761 * t_7) + Float64(t_0 + Float64(0.2909738639 * t_6))))))));
	else
		tmp = Float64(Float64(0.5 / x) + Float64(0.2514179000665374 / (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + ((x * x) * 0.7715471019);
	t_1 = (x * x) * (x * x);
	t_2 = 0.1049934947 * (x * x);
	t_3 = (x * x) * t_1;
	t_4 = (x * x) * t_3;
	t_5 = (x * x) * t_4;
	t_6 = x * (x * (x * x));
	t_7 = (x * x) * t_6;
	t_8 = t_6 * t_7;
	t_9 = (x * x) * t_7;
	tmp = 0.0;
	if ((x * ((((((1.0 + t_2) + (0.0424060604 * t_1)) + (0.0072644182 * t_3)) + (0.0005064034 * t_4)) + (0.0001789971 * t_5)) / (((((t_0 + (t_1 * 0.2909738639)) + (t_3 * 0.0694555761)) + (t_4 * 0.0140005442)) + (t_5 * 0.0008327945)) + (0.0003579942 * ((x * x) * t_5))))) <= 1e-6)
		tmp = x * (((0.0001789971 * t_8) + ((1.0 + (t_2 + ((x * x) * ((x * x) * 0.0424060604)))) + ((0.0072644182 * t_7) + (0.0005064034 * t_9)))) / ((0.0003579942 * (t_6 * t_9)) + ((0.0008327945 * t_8) + ((0.0140005442 * t_9) + ((0.0694555761 * t_7) + (t_0 + (0.2909738639 * t_6)))))));
	else
		tmp = (0.5 / x) + (0.2514179000665374 / (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.7715471019), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x * x), $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x * x), $MachinePrecision] * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(x * x), $MachinePrecision] * t$95$6), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$6 * t$95$7), $MachinePrecision]}, Block[{t$95$9 = N[(N[(x * x), $MachinePrecision] * t$95$7), $MachinePrecision]}, If[LessEqual[N[(x * N[(N[(N[(N[(N[(N[(1.0 + t$95$2), $MachinePrecision] + N[(0.0424060604 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$5), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(t$95$0 + N[(t$95$1 * 0.2909738639), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * 0.0694555761), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * 0.0140005442), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 * 0.0008327945), $MachinePrecision]), $MachinePrecision] + N[(0.0003579942 * N[(N[(x * x), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-6], N[(x * N[(N[(N[(0.0001789971 * t$95$8), $MachinePrecision] + N[(N[(1.0 + N[(t$95$2 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.0424060604), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0072644182 * t$95$7), $MachinePrecision] + N[(0.0005064034 * t$95$9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.0003579942 * N[(t$95$6 * t$95$9), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0008327945 * t$95$8), $MachinePrecision] + N[(N[(0.0140005442 * t$95$9), $MachinePrecision] + N[(N[(0.0694555761 * t$95$7), $MachinePrecision] + N[(t$95$0 + N[(0.2909738639 * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] + N[(0.2514179000665374 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 1 (*.f64 1049934947/10000000000 (*.f64 x x))) (*.f64 106015151/2500000000 (*.f64 (*.f64 x x) (*.f64 x x)))) (*.f64 36322091/5000000000 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)))) (*.f64 2532017/5000000000 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (*.f64 1789971/10000000000 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 1 (*.f64 7715471019/10000000000 (*.f64 x x))) (*.f64 2909738639/10000000000 (*.f64 (*.f64 x x) (*.f64 x x)))) (*.f64 694555761/10000000000 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)))) (*.f64 70002721/5000000000 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (*.f64 1665589/2000000000 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (*.f64 (*.f64 2 1789971/10000000000) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x))))) x) < 9.99999999999999955e-7

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   if 9.99999999999999955e-7 < (*.f64 (/.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 1 (*.f64 1049934947/10000000000 (*.f64 x x))) (*.f64 106015151/2500000000 (*.f64 (*.f64 x x) (*.f64 x x)))) (*.f64 36322091/5000000000 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)))) (*.f64 2532017/5000000000 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (*.f64 1789971/10000000000 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 1 (*.f64 7715471019/10000000000 (*.f64 x x))) (*.f64 2909738639/10000000000 (*.f64 (*.f64 x x) (*.f64 x x)))) (*.f64 694555761/10000000000 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)))) (*.f64 70002721/5000000000 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (*.f64 1665589/2000000000 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (*.f64 (*.f64 2 1789971/10000000000) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x))))) x)

   1. Initial program 0.9%

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

   3. Simplified 0.9%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1}{x} + 0.2514179000665374 \cdot \frac{1}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{x}} + 0.2514179000665374 \cdot \frac{1}{{x}^{3}} \]

      2. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{0.5}}{x} + 0.2514179000665374 \cdot \frac{1}{{x}^{3}} \]

      3. associate-*r/ 100.0%

         \[\leadsto \frac{0.5}{x} + \color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{3}}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{0.5}{x} + \frac{\color{blue}{0.2514179000665374}}{{x}^{3}} \]

   6. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \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(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) + 0.0001789971 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{\left(\left(\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.2909738639\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0694555761\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0140005442\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 0.0008327945\right) + 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)} \leq 10^{-6}:\\ \;\;\;\;x \cdot \frac{0.0001789971 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0424060604\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)}{0.0003579942 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + \left(0.0008327945 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.95 \lor \neg \left(x \leq 0.95\right):\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x + {x}^{3} \cdot -0.6665536072\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.95) (not (<= x 0.95)))
   (+ (/ 0.5 x) (/ 0.2514179000665374 (pow x 3.0)))
   (+ x (* (pow x 3.0) -0.6665536072))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.95) || !(x <= 0.95)) {
		tmp = (0.5 / x) + (0.2514179000665374 / pow(x, 3.0));
	} else {
		tmp = x + (pow(x, 3.0) * -0.6665536072);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.95d0)) .or. (.not. (x <= 0.95d0))) then
        tmp = (0.5d0 / x) + (0.2514179000665374d0 / (x ** 3.0d0))
    else
        tmp = x + ((x ** 3.0d0) * (-0.6665536072d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.95) || !(x <= 0.95)) {
		tmp = (0.5 / x) + (0.2514179000665374 / Math.pow(x, 3.0));
	} else {
		tmp = x + (Math.pow(x, 3.0) * -0.6665536072);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.95) or not (x <= 0.95):
		tmp = (0.5 / x) + (0.2514179000665374 / math.pow(x, 3.0))
	else:
		tmp = x + (math.pow(x, 3.0) * -0.6665536072)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.95) || !(x <= 0.95))
		tmp = Float64(Float64(0.5 / x) + Float64(0.2514179000665374 / (x ^ 3.0)));
	else
		tmp = Float64(x + Float64((x ^ 3.0) * -0.6665536072));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.95) || ~((x <= 0.95)))
		tmp = (0.5 / x) + (0.2514179000665374 / (x ^ 3.0));
	else
		tmp = x + ((x ^ 3.0) * -0.6665536072);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.95], N[Not[LessEqual[x, 0.95]], $MachinePrecision]], N[(N[(0.5 / x), $MachinePrecision] + N[(0.2514179000665374 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.94999999999999996 or 0.94999999999999996 < 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 5.7%

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

   4. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1}{x} + 0.2514179000665374 \cdot \frac{1}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{x}} + 0.2514179000665374 \cdot \frac{1}{{x}^{3}} \]

      2. metadata-eval 99.9%

         \[\leadsto \frac{\color{blue}{0.5}}{x} + 0.2514179000665374 \cdot \frac{1}{{x}^{3}} \]

      3. associate-*r/ 99.9%

         \[\leadsto \frac{0.5}{x} + \color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{3}}} \]

      4. metadata-eval 99.9%

         \[\leadsto \frac{0.5}{x} + \frac{\color{blue}{0.2514179000665374}}{{x}^{3}} \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}} \]

   if -0.94999999999999996 < x < 0.94999999999999996

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{x + -0.6665536072 \cdot {x}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.95 \lor \neg \left(x \leq 0.95\right):\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x + {x}^{3} \cdot -0.6665536072\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.8 \lor \neg \left(x \leq 0.78\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x + {x}^{3} \cdot -0.6665536072\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.8) (not (<= x 0.78)))
   (/ 0.5 x)
   (+ x (* (pow x 3.0) -0.6665536072))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.8) || !(x <= 0.78)) {
		tmp = 0.5 / x;
	} else {
		tmp = x + (pow(x, 3.0) * -0.6665536072);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.8d0)) .or. (.not. (x <= 0.78d0))) then
        tmp = 0.5d0 / x
    else
        tmp = x + ((x ** 3.0d0) * (-0.6665536072d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.8) || !(x <= 0.78)) {
		tmp = 0.5 / x;
	} else {
		tmp = x + (Math.pow(x, 3.0) * -0.6665536072);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.8) or not (x <= 0.78):
		tmp = 0.5 / x
	else:
		tmp = x + (math.pow(x, 3.0) * -0.6665536072)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.8) || !(x <= 0.78))
		tmp = Float64(0.5 / x);
	else
		tmp = Float64(x + Float64((x ^ 3.0) * -0.6665536072));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.8) || ~((x <= 0.78)))
		tmp = 0.5 / x;
	else
		tmp = x + ((x ^ 3.0) * -0.6665536072);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.8], N[Not[LessEqual[x, 0.78]], $MachinePrecision]], N[(0.5 / x), $MachinePrecision], N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.80000000000000004 or 0.78000000000000003 < 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 5.7%

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

   4. Taylor expanded in x around inf 99.3%

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

   if -0.80000000000000004 < x < 0.78000000000000003

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{x + -0.6665536072 \cdot {x}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.8 \lor \neg \left(x \leq 0.78\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x + {x}^{3} \cdot -0.6665536072\\ \end{array} \]

Alternative 4: 99.0% accurate, 24.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.72\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.72) (not (<= x 0.72))) (/ 0.5 x) x))

C

double code(double x) {
	double tmp;
	if ((x <= -0.72) || !(x <= 0.72)) {
		tmp = 0.5 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.72d0)) .or. (.not. (x <= 0.72d0))) then
        tmp = 0.5d0 / x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.72) || !(x <= 0.72)) {
		tmp = 0.5 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.72) or not (x <= 0.72):
		tmp = 0.5 / x
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.72) || !(x <= 0.72))
		tmp = Float64(0.5 / x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.72) || ~((x <= 0.72)))
		tmp = 0.5 / x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.72], N[Not[LessEqual[x, 0.72]], $MachinePrecision]], N[(0.5 / x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -0.71999999999999997 or 0.71999999999999997 < 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 5.7%

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

   4. Taylor expanded in x around inf 99.3%

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

   if -0.71999999999999997 < x < 0.71999999999999997

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.72\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 52.6% accurate, 173.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 55.5%

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

3. Simplified 55.4%

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

4. Taylor expanded in x around 0 54.0%

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

5. Final simplification 54.0%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023305 
(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))