Jmat.Real.dawson

Percentage Accurate: 54.7% → 100.0%

Time: 10.9s

Alternatives: 8

Speedup: 34.2×

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: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x_m \cdot \left(x_m \cdot \left(x_m \cdot x_m\right)\right)\\ t_1 := \left(x_m \cdot x_m\right) \cdot t_0\\ t_2 := \left(x_m \cdot x_m\right) \cdot t_1\\ t_3 := t_0 \cdot t_1\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 85:\\ \;\;\;\;x_m \cdot \frac{0.0001789971 \cdot t_3 + \left(0.0005064034 \cdot t_2 + \left(t_1 \cdot 0.0072644182 + \left(1 + \left(\log \left(e^{{x_m}^{2} \cdot 0.1049934947}\right) + \left(x_m \cdot x_m\right) \cdot \left(\left(x_m \cdot x_m\right) \cdot 0.0424060604\right)\right)\right)\right)\right)}{0.0003579942 \cdot \left(t_0 \cdot t_2\right) + \left(t_3 \cdot 0.0008327945 + \left(t_2 \cdot 0.0140005442 + \left(t_1 \cdot 0.0694555761 + \left(\left(1 + \left(x_m \cdot x_m\right) \cdot 0.7715471019\right) + t_0 \cdot 0.2909738639\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x_m} + \left(\frac{0.2514179000665374}{{x_m}^{3}} + \frac{0.15298196345929074}{{x_m}^{5}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m (* x_m x_m))))
        (t_1 (* (* x_m x_m) t_0))
        (t_2 (* (* x_m x_m) t_1))
        (t_3 (* t_0 t_1)))
   (*
    x_s
    (if (<= x_m 85.0)
      (*
       x_m
       (/
        (+
         (* 0.0001789971 t_3)
         (+
          (* 0.0005064034 t_2)
          (+
           (* t_1 0.0072644182)
           (+
            1.0
            (+
             (log (exp (* (pow x_m 2.0) 0.1049934947)))
             (* (* x_m x_m) (* (* x_m x_m) 0.0424060604)))))))
        (+
         (* 0.0003579942 (* t_0 t_2))
         (+
          (* t_3 0.0008327945)
          (+
           (* t_2 0.0140005442)
           (+
            (* t_1 0.0694555761)
            (+ (+ 1.0 (* (* x_m x_m) 0.7715471019)) (* t_0 0.2909738639))))))))
      (+
       (/ 0.5 x_m)
       (+
        (/ 0.2514179000665374 (pow x_m 3.0))
        (/ 0.15298196345929074 (pow x_m 5.0))))))))

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 t_1 = (x_m * x_m) * t_0;
	double t_2 = (x_m * x_m) * t_1;
	double t_3 = t_0 * t_1;
	double tmp;
	if (x_m <= 85.0) {
		tmp = x_m * (((0.0001789971 * t_3) + ((0.0005064034 * t_2) + ((t_1 * 0.0072644182) + (1.0 + (log(exp((pow(x_m, 2.0) * 0.1049934947))) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604))))))) / ((0.0003579942 * (t_0 * t_2)) + ((t_3 * 0.0008327945) + ((t_2 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (t_0 * 0.2909738639)))))));
	} else {
		tmp = (0.5 / x_m) + ((0.2514179000665374 / pow(x_m, 3.0)) + (0.15298196345929074 / pow(x_m, 5.0)));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = x_m * (x_m * (x_m * x_m))
    t_1 = (x_m * x_m) * t_0
    t_2 = (x_m * x_m) * t_1
    t_3 = t_0 * t_1
    if (x_m <= 85.0d0) then
        tmp = x_m * (((0.0001789971d0 * t_3) + ((0.0005064034d0 * t_2) + ((t_1 * 0.0072644182d0) + (1.0d0 + (log(exp(((x_m ** 2.0d0) * 0.1049934947d0))) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604d0))))))) / ((0.0003579942d0 * (t_0 * t_2)) + ((t_3 * 0.0008327945d0) + ((t_2 * 0.0140005442d0) + ((t_1 * 0.0694555761d0) + ((1.0d0 + ((x_m * x_m) * 0.7715471019d0)) + (t_0 * 0.2909738639d0)))))))
    else
        tmp = (0.5d0 / x_m) + ((0.2514179000665374d0 / (x_m ** 3.0d0)) + (0.15298196345929074d0 / (x_m ** 5.0d0)))
    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 t_1 = (x_m * x_m) * t_0;
	double t_2 = (x_m * x_m) * t_1;
	double t_3 = t_0 * t_1;
	double tmp;
	if (x_m <= 85.0) {
		tmp = x_m * (((0.0001789971 * t_3) + ((0.0005064034 * t_2) + ((t_1 * 0.0072644182) + (1.0 + (Math.log(Math.exp((Math.pow(x_m, 2.0) * 0.1049934947))) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604))))))) / ((0.0003579942 * (t_0 * t_2)) + ((t_3 * 0.0008327945) + ((t_2 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (t_0 * 0.2909738639)))))));
	} else {
		tmp = (0.5 / x_m) + ((0.2514179000665374 / Math.pow(x_m, 3.0)) + (0.15298196345929074 / Math.pow(x_m, 5.0)));
	}
	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))
	t_1 = (x_m * x_m) * t_0
	t_2 = (x_m * x_m) * t_1
	t_3 = t_0 * t_1
	tmp = 0
	if x_m <= 85.0:
		tmp = x_m * (((0.0001789971 * t_3) + ((0.0005064034 * t_2) + ((t_1 * 0.0072644182) + (1.0 + (math.log(math.exp((math.pow(x_m, 2.0) * 0.1049934947))) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604))))))) / ((0.0003579942 * (t_0 * t_2)) + ((t_3 * 0.0008327945) + ((t_2 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (t_0 * 0.2909738639)))))))
	else:
		tmp = (0.5 / x_m) + ((0.2514179000665374 / math.pow(x_m, 3.0)) + (0.15298196345929074 / math.pow(x_m, 5.0)))
	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)))
	t_1 = Float64(Float64(x_m * x_m) * t_0)
	t_2 = Float64(Float64(x_m * x_m) * t_1)
	t_3 = Float64(t_0 * t_1)
	tmp = 0.0
	if (x_m <= 85.0)
		tmp = Float64(x_m * Float64(Float64(Float64(0.0001789971 * t_3) + Float64(Float64(0.0005064034 * t_2) + Float64(Float64(t_1 * 0.0072644182) + Float64(1.0 + Float64(log(exp(Float64((x_m ^ 2.0) * 0.1049934947))) + Float64(Float64(x_m * x_m) * Float64(Float64(x_m * x_m) * 0.0424060604))))))) / Float64(Float64(0.0003579942 * Float64(t_0 * t_2)) + Float64(Float64(t_3 * 0.0008327945) + Float64(Float64(t_2 * 0.0140005442) + Float64(Float64(t_1 * 0.0694555761) + Float64(Float64(1.0 + Float64(Float64(x_m * x_m) * 0.7715471019)) + Float64(t_0 * 0.2909738639))))))));
	else
		tmp = Float64(Float64(0.5 / x_m) + Float64(Float64(0.2514179000665374 / (x_m ^ 3.0)) + Float64(0.15298196345929074 / (x_m ^ 5.0))));
	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));
	t_1 = (x_m * x_m) * t_0;
	t_2 = (x_m * x_m) * t_1;
	t_3 = t_0 * t_1;
	tmp = 0.0;
	if (x_m <= 85.0)
		tmp = x_m * (((0.0001789971 * t_3) + ((0.0005064034 * t_2) + ((t_1 * 0.0072644182) + (1.0 + (log(exp(((x_m ^ 2.0) * 0.1049934947))) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604))))))) / ((0.0003579942 * (t_0 * t_2)) + ((t_3 * 0.0008327945) + ((t_2 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (t_0 * 0.2909738639)))))));
	else
		tmp = (0.5 / x_m) + ((0.2514179000665374 / (x_m ^ 3.0)) + (0.15298196345929074 / (x_m ^ 5.0)));
	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]}, Block[{t$95$1 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$1), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 85.0], N[(x$95$m * N[(N[(N[(0.0001789971 * t$95$3), $MachinePrecision] + N[(N[(0.0005064034 * t$95$2), $MachinePrecision] + N[(N[(t$95$1 * 0.0072644182), $MachinePrecision] + N[(1.0 + N[(N[Log[N[Exp[N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.1049934947), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0424060604), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.0003579942 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 * 0.0008327945), $MachinePrecision] + N[(N[(t$95$2 * 0.0140005442), $MachinePrecision] + N[(N[(t$95$1 * 0.0694555761), $MachinePrecision] + N[(N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.7715471019), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.2909738639), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x$95$m), $MachinePrecision] + N[(N[(0.2514179000665374 / N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.15298196345929074 / N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 85

   1. Initial program 71.2%

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

   3. Simplified 71.2%

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

      1. add-log-exp 69.5%

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

      2. *-commutative 69.5%

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

      3. pow2 69.5%

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

   5. Applied egg-rr 69.5%

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

   if 85 < x

   1. Initial program 11.3%

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

   3. Simplified 11.2%

      \[\leadsto \color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.1049934947, x \cdot x, \mathsf{fma}\left(0.0424060604, {x}^{4}, 1\right)\right)\right) + {x}^{6} \cdot \left(0.0072644182 + 0.0001789971 \cdot {x}^{4}\right)\right)}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x}^{4}, 1\right)\right)\right) + {x}^{6} \cdot \left(0.0694555761 + 0.0008327945 \cdot {x}^{4}\right)\right)}} \]

   5. Applied egg-rr 5.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\frac{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0008327945, {x}^{4}, 0.0694555761\right), \mathsf{fma}\left(0.0140005442, {x}^{8}, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x}^{4}, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0001789971, {x}^{4}, 0.0072644182\right), \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left({x}^{2}, 0.1049934947, \mathsf{fma}\left(0.0424060604, {x}^{4}, 1\right)\right)\right)\right)}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 11.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\frac{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0008327945, {x}^{4}, 0.0694555761\right), \mathsf{fma}\left(0.0140005442, {x}^{8}, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x}^{4}, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0001789971, {x}^{4}, 0.0072644182\right), \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left({x}^{2}, 0.1049934947, \mathsf{fma}\left(0.0424060604, {x}^{4}, 1\right)\right)\right)\right)}}\right)\right)} \]

      2. expm1-log1p 11.3%

         \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0008327945, {x}^{4}, 0.0694555761\right), \mathsf{fma}\left(0.0140005442, {x}^{8}, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x}^{4}, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0001789971, {x}^{4}, 0.0072644182\right), \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left({x}^{2}, 0.1049934947, \mathsf{fma}\left(0.0424060604, {x}^{4}, 1\right)\right)\right)\right)}}} \]

      3. associate-/r/ 11.3%

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0008327945, {x}^{4}, 0.0694555761\right), \mathsf{fma}\left(0.0140005442, {x}^{8}, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x}^{4}, 1\right)\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0001789971, {x}^{4}, 0.0072644182\right), \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left({x}^{2}, 0.1049934947, \mathsf{fma}\left(0.0424060604, {x}^{4}, 1\right)\right)\right)\right)} \]

   7. Simplified 11.3%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0008327945, {x}^{4}, 0.0694555761\right), \mathsf{fma}\left(0.0140005442, {x}^{8}, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x}^{4}, 1\right)\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0001789971, {x}^{4}, 0.0072644182\right), \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left({x}^{2}, 0.1049934947, \mathsf{fma}\left(0.0424060604, {x}^{4}, 1\right)\right)\right)\right)} \]

   8. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{0.15298196345929074 \cdot \frac{1}{{x}^{5}} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right) + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}} \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.2514179000665374 \cdot \frac{1}{{x}^{3}}\right)} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}} \]

      3. associate-+l+ 99.9%

         \[\leadsto \color{blue}{0.5 \cdot \frac{1}{x} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right)} \]

      4. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{x}} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right) \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{\color{blue}{0.5}}{x} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right) \]

      6. associate-*r/ 99.9%

         \[\leadsto \frac{0.5}{x} + \left(\color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{3}}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right) \]

      7. metadata-eval 99.9%

         \[\leadsto \frac{0.5}{x} + \left(\frac{\color{blue}{0.2514179000665374}}{{x}^{3}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right) \]

      8. associate-*r/ 99.9%

         \[\leadsto \frac{0.5}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \color{blue}{\frac{0.15298196345929074 \cdot 1}{{x}^{5}}}\right) \]

      9. metadata-eval 99.9%

         \[\leadsto \frac{0.5}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \frac{\color{blue}{0.15298196345929074}}{{x}^{5}}\right) \]

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{0.5}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \frac{0.15298196345929074}{{x}^{5}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 85:\\ \;\;\;\;x \cdot \frac{0.0001789971 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0072644182 + \left(1 + \left(\log \left(e^{{x}^{2} \cdot 0.1049934947}\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0424060604\right)\right)\right)\right)\right)}{0.0003579942 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + \left(\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.2909738639\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \frac{0.15298196345929074}{{x}^{5}}\right)\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x_m \cdot \left(x_m \cdot \left(x_m \cdot x_m\right)\right)\\ t_1 := \left(x_m \cdot x_m\right) \cdot t_0\\ t_2 := \left(x_m \cdot x_m\right) \cdot t_1\\ t_3 := t_0 \cdot t_1\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1000:\\ \;\;\;\;x_m \cdot \frac{0.0001789971 \cdot t_3 + \left(0.0005064034 \cdot t_2 + \left(t_1 \cdot 0.0072644182 + \left(1 + \left(\left(x_m \cdot x_m\right) \cdot \left(\left(x_m \cdot x_m\right) \cdot 0.0424060604\right) + \left(x_m \cdot x_m\right) \cdot 0.1049934947\right)\right)\right)\right)}{0.0003579942 \cdot \left(t_0 \cdot t_2\right) + \left(t_3 \cdot 0.0008327945 + \left(t_2 \cdot 0.0140005442 + \left(t_1 \cdot 0.0694555761 + \left(\left(1 + \left(x_m \cdot x_m\right) \cdot 0.7715471019\right) + t_0 \cdot 0.2909738639\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x_m} + \left(\frac{0.2514179000665374}{{x_m}^{3}} + \frac{0.15298196345929074}{{x_m}^{5}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m (* x_m x_m))))
        (t_1 (* (* x_m x_m) t_0))
        (t_2 (* (* x_m x_m) t_1))
        (t_3 (* t_0 t_1)))
   (*
    x_s
    (if (<= x_m 1000.0)
      (*
       x_m
       (/
        (+
         (* 0.0001789971 t_3)
         (+
          (* 0.0005064034 t_2)
          (+
           (* t_1 0.0072644182)
           (+
            1.0
            (+
             (* (* x_m x_m) (* (* x_m x_m) 0.0424060604))
             (* (* x_m x_m) 0.1049934947))))))
        (+
         (* 0.0003579942 (* t_0 t_2))
         (+
          (* t_3 0.0008327945)
          (+
           (* t_2 0.0140005442)
           (+
            (* t_1 0.0694555761)
            (+ (+ 1.0 (* (* x_m x_m) 0.7715471019)) (* t_0 0.2909738639))))))))
      (+
       (/ 0.5 x_m)
       (+
        (/ 0.2514179000665374 (pow x_m 3.0))
        (/ 0.15298196345929074 (pow x_m 5.0))))))))

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 t_1 = (x_m * x_m) * t_0;
	double t_2 = (x_m * x_m) * t_1;
	double t_3 = t_0 * t_1;
	double tmp;
	if (x_m <= 1000.0) {
		tmp = x_m * (((0.0001789971 * t_3) + ((0.0005064034 * t_2) + ((t_1 * 0.0072644182) + (1.0 + (((x_m * x_m) * ((x_m * x_m) * 0.0424060604)) + ((x_m * x_m) * 0.1049934947)))))) / ((0.0003579942 * (t_0 * t_2)) + ((t_3 * 0.0008327945) + ((t_2 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (t_0 * 0.2909738639)))))));
	} else {
		tmp = (0.5 / x_m) + ((0.2514179000665374 / pow(x_m, 3.0)) + (0.15298196345929074 / pow(x_m, 5.0)));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = x_m * (x_m * (x_m * x_m))
    t_1 = (x_m * x_m) * t_0
    t_2 = (x_m * x_m) * t_1
    t_3 = t_0 * t_1
    if (x_m <= 1000.0d0) then
        tmp = x_m * (((0.0001789971d0 * t_3) + ((0.0005064034d0 * t_2) + ((t_1 * 0.0072644182d0) + (1.0d0 + (((x_m * x_m) * ((x_m * x_m) * 0.0424060604d0)) + ((x_m * x_m) * 0.1049934947d0)))))) / ((0.0003579942d0 * (t_0 * t_2)) + ((t_3 * 0.0008327945d0) + ((t_2 * 0.0140005442d0) + ((t_1 * 0.0694555761d0) + ((1.0d0 + ((x_m * x_m) * 0.7715471019d0)) + (t_0 * 0.2909738639d0)))))))
    else
        tmp = (0.5d0 / x_m) + ((0.2514179000665374d0 / (x_m ** 3.0d0)) + (0.15298196345929074d0 / (x_m ** 5.0d0)))
    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 t_1 = (x_m * x_m) * t_0;
	double t_2 = (x_m * x_m) * t_1;
	double t_3 = t_0 * t_1;
	double tmp;
	if (x_m <= 1000.0) {
		tmp = x_m * (((0.0001789971 * t_3) + ((0.0005064034 * t_2) + ((t_1 * 0.0072644182) + (1.0 + (((x_m * x_m) * ((x_m * x_m) * 0.0424060604)) + ((x_m * x_m) * 0.1049934947)))))) / ((0.0003579942 * (t_0 * t_2)) + ((t_3 * 0.0008327945) + ((t_2 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (t_0 * 0.2909738639)))))));
	} else {
		tmp = (0.5 / x_m) + ((0.2514179000665374 / Math.pow(x_m, 3.0)) + (0.15298196345929074 / Math.pow(x_m, 5.0)));
	}
	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))
	t_1 = (x_m * x_m) * t_0
	t_2 = (x_m * x_m) * t_1
	t_3 = t_0 * t_1
	tmp = 0
	if x_m <= 1000.0:
		tmp = x_m * (((0.0001789971 * t_3) + ((0.0005064034 * t_2) + ((t_1 * 0.0072644182) + (1.0 + (((x_m * x_m) * ((x_m * x_m) * 0.0424060604)) + ((x_m * x_m) * 0.1049934947)))))) / ((0.0003579942 * (t_0 * t_2)) + ((t_3 * 0.0008327945) + ((t_2 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (t_0 * 0.2909738639)))))))
	else:
		tmp = (0.5 / x_m) + ((0.2514179000665374 / math.pow(x_m, 3.0)) + (0.15298196345929074 / math.pow(x_m, 5.0)))
	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)))
	t_1 = Float64(Float64(x_m * x_m) * t_0)
	t_2 = Float64(Float64(x_m * x_m) * t_1)
	t_3 = Float64(t_0 * t_1)
	tmp = 0.0
	if (x_m <= 1000.0)
		tmp = Float64(x_m * Float64(Float64(Float64(0.0001789971 * t_3) + Float64(Float64(0.0005064034 * t_2) + Float64(Float64(t_1 * 0.0072644182) + Float64(1.0 + Float64(Float64(Float64(x_m * x_m) * Float64(Float64(x_m * x_m) * 0.0424060604)) + Float64(Float64(x_m * x_m) * 0.1049934947)))))) / Float64(Float64(0.0003579942 * Float64(t_0 * t_2)) + Float64(Float64(t_3 * 0.0008327945) + Float64(Float64(t_2 * 0.0140005442) + Float64(Float64(t_1 * 0.0694555761) + Float64(Float64(1.0 + Float64(Float64(x_m * x_m) * 0.7715471019)) + Float64(t_0 * 0.2909738639))))))));
	else
		tmp = Float64(Float64(0.5 / x_m) + Float64(Float64(0.2514179000665374 / (x_m ^ 3.0)) + Float64(0.15298196345929074 / (x_m ^ 5.0))));
	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));
	t_1 = (x_m * x_m) * t_0;
	t_2 = (x_m * x_m) * t_1;
	t_3 = t_0 * t_1;
	tmp = 0.0;
	if (x_m <= 1000.0)
		tmp = x_m * (((0.0001789971 * t_3) + ((0.0005064034 * t_2) + ((t_1 * 0.0072644182) + (1.0 + (((x_m * x_m) * ((x_m * x_m) * 0.0424060604)) + ((x_m * x_m) * 0.1049934947)))))) / ((0.0003579942 * (t_0 * t_2)) + ((t_3 * 0.0008327945) + ((t_2 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (t_0 * 0.2909738639)))))));
	else
		tmp = (0.5 / x_m) + ((0.2514179000665374 / (x_m ^ 3.0)) + (0.15298196345929074 / (x_m ^ 5.0)));
	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]}, Block[{t$95$1 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$1), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 1000.0], N[(x$95$m * N[(N[(N[(0.0001789971 * t$95$3), $MachinePrecision] + N[(N[(0.0005064034 * t$95$2), $MachinePrecision] + N[(N[(t$95$1 * 0.0072644182), $MachinePrecision] + N[(1.0 + N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0424060604), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.1049934947), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.0003579942 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 * 0.0008327945), $MachinePrecision] + N[(N[(t$95$2 * 0.0140005442), $MachinePrecision] + N[(N[(t$95$1 * 0.0694555761), $MachinePrecision] + N[(N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.7715471019), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.2909738639), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x$95$m), $MachinePrecision] + N[(N[(0.2514179000665374 / N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.15298196345929074 / N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e3

   1. Initial program 71.3%

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

   3. Simplified 71.3%

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

   if 1e3 < x

   1. Initial program 10.2%

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

   3. Simplified 10.1%

      \[\leadsto \color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.1049934947, x \cdot x, \mathsf{fma}\left(0.0424060604, {x}^{4}, 1\right)\right)\right) + {x}^{6} \cdot \left(0.0072644182 + 0.0001789971 \cdot {x}^{4}\right)\right)}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x}^{4}, 1\right)\right)\right) + {x}^{6} \cdot \left(0.0694555761 + 0.0008327945 \cdot {x}^{4}\right)\right)}} \]

   5. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\frac{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0008327945, {x}^{4}, 0.0694555761\right), \mathsf{fma}\left(0.0140005442, {x}^{8}, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x}^{4}, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0001789971, {x}^{4}, 0.0072644182\right), \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left({x}^{2}, 0.1049934947, \mathsf{fma}\left(0.0424060604, {x}^{4}, 1\right)\right)\right)\right)}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 10.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\frac{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0008327945, {x}^{4}, 0.0694555761\right), \mathsf{fma}\left(0.0140005442, {x}^{8}, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x}^{4}, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0001789971, {x}^{4}, 0.0072644182\right), \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left({x}^{2}, 0.1049934947, \mathsf{fma}\left(0.0424060604, {x}^{4}, 1\right)\right)\right)\right)}}\right)\right)} \]

      2. expm1-log1p 10.1%

         \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0008327945, {x}^{4}, 0.0694555761\right), \mathsf{fma}\left(0.0140005442, {x}^{8}, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x}^{4}, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0001789971, {x}^{4}, 0.0072644182\right), \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left({x}^{2}, 0.1049934947, \mathsf{fma}\left(0.0424060604, {x}^{4}, 1\right)\right)\right)\right)}}} \]

      3. associate-/r/ 10.2%

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0008327945, {x}^{4}, 0.0694555761\right), \mathsf{fma}\left(0.0140005442, {x}^{8}, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x}^{4}, 1\right)\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0001789971, {x}^{4}, 0.0072644182\right), \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left({x}^{2}, 0.1049934947, \mathsf{fma}\left(0.0424060604, {x}^{4}, 1\right)\right)\right)\right)} \]

   7. Simplified 10.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0008327945, {x}^{4}, 0.0694555761\right), \mathsf{fma}\left(0.0140005442, {x}^{8}, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x}^{4}, 1\right)\right)\right)\right)\right)} \cdot \mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(0.0001789971, {x}^{4}, 0.0072644182\right), \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left({x}^{2}, 0.1049934947, \mathsf{fma}\left(0.0424060604, {x}^{4}, 1\right)\right)\right)\right)} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{0.15298196345929074 \cdot \frac{1}{{x}^{5}} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right) + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.2514179000665374 \cdot \frac{1}{{x}^{3}}\right)} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}} \]

      3. associate-+l+ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{1}{x} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right)} \]

      4. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{x}} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right) \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{0.5}}{x} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{3}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right) \]

      6. associate-*r/ 100.0%

         \[\leadsto \frac{0.5}{x} + \left(\color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{3}}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right) \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{0.5}{x} + \left(\frac{\color{blue}{0.2514179000665374}}{{x}^{3}} + 0.15298196345929074 \cdot \frac{1}{{x}^{5}}\right) \]

      8. associate-*r/ 100.0%

         \[\leadsto \frac{0.5}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \color{blue}{\frac{0.15298196345929074 \cdot 1}{{x}^{5}}}\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \frac{0.5}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \frac{\color{blue}{0.15298196345929074}}{{x}^{5}}\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{0.5}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \frac{0.15298196345929074}{{x}^{5}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1000:\\ \;\;\;\;x \cdot \frac{0.0001789971 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0072644182 + \left(1 + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0424060604\right) + \left(x \cdot x\right) \cdot 0.1049934947\right)\right)\right)\right)}{0.0003579942 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + \left(\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.2909738639\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665374}{{x}^{3}} + \frac{0.15298196345929074}{{x}^{5}}\right)\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x_m \cdot \left(x_m \cdot \left(x_m \cdot x_m\right)\right)\\ t_1 := \left(x_m \cdot x_m\right) \cdot t_0\\ t_2 := \left(x_m \cdot x_m\right) \cdot t_1\\ t_3 := t_0 \cdot t_1\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 200000:\\ \;\;\;\;x_m \cdot \frac{0.0001789971 \cdot t_3 + \left(0.0005064034 \cdot t_2 + \left(t_1 \cdot 0.0072644182 + \left(1 + \left(\left(x_m \cdot x_m\right) \cdot \left(\left(x_m \cdot x_m\right) \cdot 0.0424060604\right) + \left(x_m \cdot x_m\right) \cdot 0.1049934947\right)\right)\right)\right)}{0.0003579942 \cdot \left(t_0 \cdot t_2\right) + \left(t_3 \cdot 0.0008327945 + \left(t_2 \cdot 0.0140005442 + \left(t_1 \cdot 0.0694555761 + \left(\left(1 + \left(x_m \cdot x_m\right) \cdot 0.7715471019\right) + t_0 \cdot 0.2909738639\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x_m} + \frac{0.2514179000665374}{{x_m}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m (* x_m x_m))))
        (t_1 (* (* x_m x_m) t_0))
        (t_2 (* (* x_m x_m) t_1))
        (t_3 (* t_0 t_1)))
   (*
    x_s
    (if (<= x_m 200000.0)
      (*
       x_m
       (/
        (+
         (* 0.0001789971 t_3)
         (+
          (* 0.0005064034 t_2)
          (+
           (* t_1 0.0072644182)
           (+
            1.0
            (+
             (* (* x_m x_m) (* (* x_m x_m) 0.0424060604))
             (* (* x_m x_m) 0.1049934947))))))
        (+
         (* 0.0003579942 (* t_0 t_2))
         (+
          (* t_3 0.0008327945)
          (+
           (* t_2 0.0140005442)
           (+
            (* t_1 0.0694555761)
            (+ (+ 1.0 (* (* x_m x_m) 0.7715471019)) (* t_0 0.2909738639))))))))
      (+ (/ 0.5 x_m) (/ 0.2514179000665374 (pow x_m 3.0)))))))

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 t_1 = (x_m * x_m) * t_0;
	double t_2 = (x_m * x_m) * t_1;
	double t_3 = t_0 * t_1;
	double tmp;
	if (x_m <= 200000.0) {
		tmp = x_m * (((0.0001789971 * t_3) + ((0.0005064034 * t_2) + ((t_1 * 0.0072644182) + (1.0 + (((x_m * x_m) * ((x_m * x_m) * 0.0424060604)) + ((x_m * x_m) * 0.1049934947)))))) / ((0.0003579942 * (t_0 * t_2)) + ((t_3 * 0.0008327945) + ((t_2 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (t_0 * 0.2909738639)))))));
	} else {
		tmp = (0.5 / x_m) + (0.2514179000665374 / pow(x_m, 3.0));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = x_m * (x_m * (x_m * x_m))
    t_1 = (x_m * x_m) * t_0
    t_2 = (x_m * x_m) * t_1
    t_3 = t_0 * t_1
    if (x_m <= 200000.0d0) then
        tmp = x_m * (((0.0001789971d0 * t_3) + ((0.0005064034d0 * t_2) + ((t_1 * 0.0072644182d0) + (1.0d0 + (((x_m * x_m) * ((x_m * x_m) * 0.0424060604d0)) + ((x_m * x_m) * 0.1049934947d0)))))) / ((0.0003579942d0 * (t_0 * t_2)) + ((t_3 * 0.0008327945d0) + ((t_2 * 0.0140005442d0) + ((t_1 * 0.0694555761d0) + ((1.0d0 + ((x_m * x_m) * 0.7715471019d0)) + (t_0 * 0.2909738639d0)))))))
    else
        tmp = (0.5d0 / x_m) + (0.2514179000665374d0 / (x_m ** 3.0d0))
    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 t_1 = (x_m * x_m) * t_0;
	double t_2 = (x_m * x_m) * t_1;
	double t_3 = t_0 * t_1;
	double tmp;
	if (x_m <= 200000.0) {
		tmp = x_m * (((0.0001789971 * t_3) + ((0.0005064034 * t_2) + ((t_1 * 0.0072644182) + (1.0 + (((x_m * x_m) * ((x_m * x_m) * 0.0424060604)) + ((x_m * x_m) * 0.1049934947)))))) / ((0.0003579942 * (t_0 * t_2)) + ((t_3 * 0.0008327945) + ((t_2 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (t_0 * 0.2909738639)))))));
	} else {
		tmp = (0.5 / x_m) + (0.2514179000665374 / Math.pow(x_m, 3.0));
	}
	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))
	t_1 = (x_m * x_m) * t_0
	t_2 = (x_m * x_m) * t_1
	t_3 = t_0 * t_1
	tmp = 0
	if x_m <= 200000.0:
		tmp = x_m * (((0.0001789971 * t_3) + ((0.0005064034 * t_2) + ((t_1 * 0.0072644182) + (1.0 + (((x_m * x_m) * ((x_m * x_m) * 0.0424060604)) + ((x_m * x_m) * 0.1049934947)))))) / ((0.0003579942 * (t_0 * t_2)) + ((t_3 * 0.0008327945) + ((t_2 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (t_0 * 0.2909738639)))))))
	else:
		tmp = (0.5 / x_m) + (0.2514179000665374 / math.pow(x_m, 3.0))
	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)))
	t_1 = Float64(Float64(x_m * x_m) * t_0)
	t_2 = Float64(Float64(x_m * x_m) * t_1)
	t_3 = Float64(t_0 * t_1)
	tmp = 0.0
	if (x_m <= 200000.0)
		tmp = Float64(x_m * Float64(Float64(Float64(0.0001789971 * t_3) + Float64(Float64(0.0005064034 * t_2) + Float64(Float64(t_1 * 0.0072644182) + Float64(1.0 + Float64(Float64(Float64(x_m * x_m) * Float64(Float64(x_m * x_m) * 0.0424060604)) + Float64(Float64(x_m * x_m) * 0.1049934947)))))) / Float64(Float64(0.0003579942 * Float64(t_0 * t_2)) + Float64(Float64(t_3 * 0.0008327945) + Float64(Float64(t_2 * 0.0140005442) + Float64(Float64(t_1 * 0.0694555761) + Float64(Float64(1.0 + Float64(Float64(x_m * x_m) * 0.7715471019)) + Float64(t_0 * 0.2909738639))))))));
	else
		tmp = Float64(Float64(0.5 / x_m) + Float64(0.2514179000665374 / (x_m ^ 3.0)));
	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));
	t_1 = (x_m * x_m) * t_0;
	t_2 = (x_m * x_m) * t_1;
	t_3 = t_0 * t_1;
	tmp = 0.0;
	if (x_m <= 200000.0)
		tmp = x_m * (((0.0001789971 * t_3) + ((0.0005064034 * t_2) + ((t_1 * 0.0072644182) + (1.0 + (((x_m * x_m) * ((x_m * x_m) * 0.0424060604)) + ((x_m * x_m) * 0.1049934947)))))) / ((0.0003579942 * (t_0 * t_2)) + ((t_3 * 0.0008327945) + ((t_2 * 0.0140005442) + ((t_1 * 0.0694555761) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (t_0 * 0.2909738639)))))));
	else
		tmp = (0.5 / x_m) + (0.2514179000665374 / (x_m ^ 3.0));
	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]}, Block[{t$95$1 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$1), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 200000.0], N[(x$95$m * N[(N[(N[(0.0001789971 * t$95$3), $MachinePrecision] + N[(N[(0.0005064034 * t$95$2), $MachinePrecision] + N[(N[(t$95$1 * 0.0072644182), $MachinePrecision] + N[(1.0 + N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0424060604), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.1049934947), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.0003579942 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 * 0.0008327945), $MachinePrecision] + N[(N[(t$95$2 * 0.0140005442), $MachinePrecision] + N[(N[(t$95$1 * 0.0694555761), $MachinePrecision] + N[(N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.7715471019), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.2909738639), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x$95$m), $MachinePrecision] + N[(0.2514179000665374 / N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e5

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   if 2e5 < x

   1. Initial program 7.8%

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

   3. Simplified 7.8%

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

   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 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 200000:\\ \;\;\;\;x \cdot \frac{0.0001789971 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0072644182 + \left(1 + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0424060604\right) + \left(x \cdot x\right) \cdot 0.1049934947\right)\right)\right)\right)}{0.0003579942 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + \left(\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 0.0008327945 + \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot 0.0140005442 + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0694555761 + \left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.2909738639\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \]

Alternative 4: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x_m \cdot x_m\right) \cdot \left(x_m \cdot x_m\right)\\ t_1 := \left(x_m \cdot x_m\right) \cdot t_0\\ t_2 := \left(x_m \cdot x_m\right) \cdot t_1\\ t_3 := \left(x_m \cdot x_m\right) \cdot t_2\\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1000000:\\ \;\;\;\;x_m \cdot \frac{\left(\left(\left(\left(1 + \left(x_m \cdot x_m\right) \cdot 0.1049934947\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot t_1\right) + 0.0005064034 \cdot t_2\right) + 0.0001789971 \cdot t_3}{\left(\left(\left(\left(\left(1 + \left(x_m \cdot x_m\right) \cdot 0.7715471019\right) + 0.2909738639 \cdot t_0\right) + 0.0694555761 \cdot t_1\right) + 0.0140005442 \cdot t_2\right) + 0.0008327945 \cdot t_3\right) + 0.0003579942 \cdot \left(\left(x_m \cdot x_m\right) \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x_m} + \frac{0.2514179000665374}{{x_m}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* (* x_m x_m) (* x_m x_m)))
        (t_1 (* (* x_m x_m) t_0))
        (t_2 (* (* x_m x_m) t_1))
        (t_3 (* (* x_m x_m) t_2)))
   (*
    x_s
    (if (<= x_m 1000000.0)
      (*
       x_m
       (/
        (+
         (+
          (+
           (+ (+ 1.0 (* (* x_m x_m) 0.1049934947)) (* 0.0424060604 t_0))
           (* 0.0072644182 t_1))
          (* 0.0005064034 t_2))
         (* 0.0001789971 t_3))
        (+
         (+
          (+
           (+
            (+ (+ 1.0 (* (* x_m x_m) 0.7715471019)) (* 0.2909738639 t_0))
            (* 0.0694555761 t_1))
           (* 0.0140005442 t_2))
          (* 0.0008327945 t_3))
         (* 0.0003579942 (* (* x_m x_m) t_3)))))
      (+ (/ 0.5 x_m) (/ 0.2514179000665374 (pow x_m 3.0)))))))

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 t_1 = (x_m * x_m) * t_0;
	double t_2 = (x_m * x_m) * t_1;
	double t_3 = (x_m * x_m) * t_2;
	double tmp;
	if (x_m <= 1000000.0) {
		tmp = x_m * ((((((1.0 + ((x_m * x_m) * 0.1049934947)) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + ((x_m * x_m) * 0.7715471019)) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + (0.0003579942 * ((x_m * x_m) * t_3))));
	} else {
		tmp = (0.5 / x_m) + (0.2514179000665374 / pow(x_m, 3.0));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (x_m * x_m) * (x_m * x_m)
    t_1 = (x_m * x_m) * t_0
    t_2 = (x_m * x_m) * t_1
    t_3 = (x_m * x_m) * t_2
    if (x_m <= 1000000.0d0) then
        tmp = x_m * ((((((1.0d0 + ((x_m * x_m) * 0.1049934947d0)) + (0.0424060604d0 * t_0)) + (0.0072644182d0 * t_1)) + (0.0005064034d0 * t_2)) + (0.0001789971d0 * t_3)) / ((((((1.0d0 + ((x_m * x_m) * 0.7715471019d0)) + (0.2909738639d0 * t_0)) + (0.0694555761d0 * t_1)) + (0.0140005442d0 * t_2)) + (0.0008327945d0 * t_3)) + (0.0003579942d0 * ((x_m * x_m) * t_3))))
    else
        tmp = (0.5d0 / x_m) + (0.2514179000665374d0 / (x_m ** 3.0d0))
    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 t_1 = (x_m * x_m) * t_0;
	double t_2 = (x_m * x_m) * t_1;
	double t_3 = (x_m * x_m) * t_2;
	double tmp;
	if (x_m <= 1000000.0) {
		tmp = x_m * ((((((1.0 + ((x_m * x_m) * 0.1049934947)) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + ((x_m * x_m) * 0.7715471019)) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + (0.0003579942 * ((x_m * x_m) * t_3))));
	} else {
		tmp = (0.5 / x_m) + (0.2514179000665374 / Math.pow(x_m, 3.0));
	}
	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)
	t_1 = (x_m * x_m) * t_0
	t_2 = (x_m * x_m) * t_1
	t_3 = (x_m * x_m) * t_2
	tmp = 0
	if x_m <= 1000000.0:
		tmp = x_m * ((((((1.0 + ((x_m * x_m) * 0.1049934947)) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + ((x_m * x_m) * 0.7715471019)) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + (0.0003579942 * ((x_m * x_m) * t_3))))
	else:
		tmp = (0.5 / x_m) + (0.2514179000665374 / math.pow(x_m, 3.0))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(Float64(x_m * x_m) * Float64(x_m * x_m))
	t_1 = Float64(Float64(x_m * x_m) * t_0)
	t_2 = Float64(Float64(x_m * x_m) * t_1)
	t_3 = Float64(Float64(x_m * x_m) * t_2)
	tmp = 0.0
	if (x_m <= 1000000.0)
		tmp = Float64(x_m * Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(Float64(x_m * x_m) * 0.1049934947)) + 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(Float64(x_m * x_m) * 0.7715471019)) + Float64(0.2909738639 * t_0)) + Float64(0.0694555761 * t_1)) + Float64(0.0140005442 * t_2)) + Float64(0.0008327945 * t_3)) + Float64(0.0003579942 * Float64(Float64(x_m * x_m) * t_3)))));
	else
		tmp = Float64(Float64(0.5 / x_m) + Float64(0.2514179000665374 / (x_m ^ 3.0)));
	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);
	t_1 = (x_m * x_m) * t_0;
	t_2 = (x_m * x_m) * t_1;
	t_3 = (x_m * x_m) * t_2;
	tmp = 0.0;
	if (x_m <= 1000000.0)
		tmp = x_m * ((((((1.0 + ((x_m * x_m) * 0.1049934947)) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + ((x_m * x_m) * 0.7715471019)) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + (0.0003579942 * ((x_m * x_m) * t_3))));
	else
		tmp = (0.5 / x_m) + (0.2514179000665374 / (x_m ^ 3.0));
	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[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 1000000.0], N[(x$95$m * N[(N[(N[(N[(N[(N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.1049934947), $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[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.7715471019), $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[(0.0003579942 * N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x$95$m), $MachinePrecision] + N[(0.2514179000665374 / N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e6

   1. Initial program 71.8%

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

   if 1e6 < x

   1. Initial program 6.6%

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

   3. Simplified 6.6%

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

   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 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1000000:\\ \;\;\;\;x \cdot \frac{\left(\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) + 0.0001789971 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{\left(\left(\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) + 0.0008327945 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.95:\\ \;\;\;\;x_m + {x_m}^{3} \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x_m} + \frac{0.2514179000665374}{{x_m}^{3}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.95)
    (+ x_m (* (pow x_m 3.0) -0.6665536072))
    (+ (/ 0.5 x_m) (/ 0.2514179000665374 (pow x_m 3.0))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.95) {
		tmp = x_m + (pow(x_m, 3.0) * -0.6665536072);
	} else {
		tmp = (0.5 / x_m) + (0.2514179000665374 / pow(x_m, 3.0));
	}
	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 <= 0.95d0) then
        tmp = x_m + ((x_m ** 3.0d0) * (-0.6665536072d0))
    else
        tmp = (0.5d0 / x_m) + (0.2514179000665374d0 / (x_m ** 3.0d0))
    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 <= 0.95) {
		tmp = x_m + (Math.pow(x_m, 3.0) * -0.6665536072);
	} else {
		tmp = (0.5 / x_m) + (0.2514179000665374 / Math.pow(x_m, 3.0));
	}
	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 <= 0.95:
		tmp = x_m + (math.pow(x_m, 3.0) * -0.6665536072)
	else:
		tmp = (0.5 / x_m) + (0.2514179000665374 / math.pow(x_m, 3.0))
	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 <= 0.95)
		tmp = Float64(x_m + Float64((x_m ^ 3.0) * -0.6665536072));
	else
		tmp = Float64(Float64(0.5 / x_m) + Float64(0.2514179000665374 / (x_m ^ 3.0)));
	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 <= 0.95)
		tmp = x_m + ((x_m ^ 3.0) * -0.6665536072);
	else
		tmp = (0.5 / x_m) + (0.2514179000665374 / (x_m ^ 3.0));
	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, 0.95], N[(x$95$m + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x$95$m), $MachinePrecision] + N[(0.2514179000665374 / N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in x around 0 69.6%

      \[\leadsto \color{blue}{x + -0.6665536072 \cdot {x}^{3}} \]

   if 0.94999999999999996 < x

   1. Initial program 11.3%

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

   3. Simplified 11.3%

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

   4. Taylor expanded in x around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

      3. associate-*r/ 99.6%

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

      4. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 78.9%

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

Alternative 6: 99.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.8:\\ \;\;\;\;x_m + {x_m}^{3} \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 0.8) (+ x_m (* (pow x_m 3.0) -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 + (pow(x_m, 3.0) * -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 + ((x_m ** 3.0d0) * (-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 + (Math.pow(x_m, 3.0) * -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 + (math.pow(x_m, 3.0) * -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((x_m ^ 3.0) * -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 + ((x_m ^ 3.0) * -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[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.6665536072), $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 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in x around 0 69.6%

      \[\leadsto \color{blue}{x + -0.6665536072 \cdot {x}^{3}} \]

   if 0.80000000000000004 < x

   1. Initial program 11.3%

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

   3. Simplified 11.3%

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

   4. Taylor expanded in x around inf 98.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;x + {x}^{3} \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 7: 99.0% accurate, 34.2× 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 1 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 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in x around 0 69.4%

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

   if 0.69999999999999996 < x

   1. Initial program 11.3%

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

   3. Simplified 11.3%

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

   4. Taylor expanded in x around inf 98.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 8: 52.2% 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 1 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 52.7%

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

3. Simplified 52.7%

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

4. Taylor expanded in x around 0 49.2%

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

5. Final simplification 49.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023326 
(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))