Jmat.Real.dawson

Percentage Accurate: 53.5% → 99.9%

Time: 17.6s

Alternatives: 10

Speedup: 21.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 200000:\\ \;\;\;\;\frac{x\_m \cdot \left(-1 - \left(x\_m \cdot x\_m\right) \cdot \left(0.1049934947 + x\_m \cdot \left(x\_m \cdot \left(0.0424060604 + \left(x\_m \cdot \left(x\_m \cdot 0.0072644182\right) + x\_m \cdot \left(\left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(0.0005064034 + \left(x\_m \cdot x\_m\right) \cdot 0.0001789971\right)\right)\right)\right)\right)\right)\right)}{-1 - \left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot \left(x\_m \cdot \left(0.2909738639 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0694555761 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0140005442 + x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.0003579942 + 0.0008327945\right)\right)\right)\right)\right)\right) + 0.7715471019\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 200000.0)
    (/
     (*
      x_m
      (-
       -1.0
       (*
        (* x_m x_m)
        (+
         0.1049934947
         (*
          x_m
          (*
           x_m
           (+
            0.0424060604
            (+
             (* x_m (* x_m 0.0072644182))
             (*
              x_m
              (*
               (* x_m (* x_m x_m))
               (+ 0.0005064034 (* (* x_m x_m) 0.0001789971))))))))))))
     (-
      -1.0
      (*
       (* x_m x_m)
       (+
        (*
         x_m
         (*
          x_m
          (+
           0.2909738639
           (*
            (* x_m x_m)
            (+
             0.0694555761
             (*
              (* x_m x_m)
              (+
               0.0140005442
               (*
                x_m
                (* x_m (+ (* (* x_m x_m) 0.0003579942) 0.0008327945))))))))))
        0.7715471019))))
    (/ 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 <= 200000.0) {
		tmp = (x_m * (-1.0 - ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * (0.0424060604 + ((x_m * (x_m * 0.0072644182)) + (x_m * ((x_m * (x_m * x_m)) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))))))))))) / (-1.0 - ((x_m * x_m) * ((x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * (0.0140005442 + (x_m * (x_m * (((x_m * x_m) * 0.0003579942) + 0.0008327945)))))))))) + 0.7715471019)));
	} 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 <= 200000.0d0) then
        tmp = (x_m * ((-1.0d0) - ((x_m * x_m) * (0.1049934947d0 + (x_m * (x_m * (0.0424060604d0 + ((x_m * (x_m * 0.0072644182d0)) + (x_m * ((x_m * (x_m * x_m)) * (0.0005064034d0 + ((x_m * x_m) * 0.0001789971d0)))))))))))) / ((-1.0d0) - ((x_m * x_m) * ((x_m * (x_m * (0.2909738639d0 + ((x_m * x_m) * (0.0694555761d0 + ((x_m * x_m) * (0.0140005442d0 + (x_m * (x_m * (((x_m * x_m) * 0.0003579942d0) + 0.0008327945d0)))))))))) + 0.7715471019d0)))
    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 <= 200000.0) {
		tmp = (x_m * (-1.0 - ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * (0.0424060604 + ((x_m * (x_m * 0.0072644182)) + (x_m * ((x_m * (x_m * x_m)) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))))))))))) / (-1.0 - ((x_m * x_m) * ((x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * (0.0140005442 + (x_m * (x_m * (((x_m * x_m) * 0.0003579942) + 0.0008327945)))))))))) + 0.7715471019)));
	} 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 <= 200000.0:
		tmp = (x_m * (-1.0 - ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * (0.0424060604 + ((x_m * (x_m * 0.0072644182)) + (x_m * ((x_m * (x_m * x_m)) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))))))))))) / (-1.0 - ((x_m * x_m) * ((x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * (0.0140005442 + (x_m * (x_m * (((x_m * x_m) * 0.0003579942) + 0.0008327945)))))))))) + 0.7715471019)))
	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 <= 200000.0)
		tmp = Float64(Float64(x_m * Float64(-1.0 - Float64(Float64(x_m * x_m) * Float64(0.1049934947 + Float64(x_m * Float64(x_m * Float64(0.0424060604 + Float64(Float64(x_m * Float64(x_m * 0.0072644182)) + Float64(x_m * Float64(Float64(x_m * Float64(x_m * x_m)) * Float64(0.0005064034 + Float64(Float64(x_m * x_m) * 0.0001789971)))))))))))) / Float64(-1.0 - Float64(Float64(x_m * x_m) * Float64(Float64(x_m * Float64(x_m * Float64(0.2909738639 + Float64(Float64(x_m * x_m) * Float64(0.0694555761 + Float64(Float64(x_m * x_m) * Float64(0.0140005442 + Float64(x_m * Float64(x_m * Float64(Float64(Float64(x_m * x_m) * 0.0003579942) + 0.0008327945)))))))))) + 0.7715471019))));
	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 <= 200000.0)
		tmp = (x_m * (-1.0 - ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * (0.0424060604 + ((x_m * (x_m * 0.0072644182)) + (x_m * ((x_m * (x_m * x_m)) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))))))))))) / (-1.0 - ((x_m * x_m) * ((x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * (0.0140005442 + (x_m * (x_m * (((x_m * x_m) * 0.0003579942) + 0.0008327945)))))))))) + 0.7715471019)));
	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, 200000.0], N[(N[(x$95$m * N[(-1.0 - N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.1049934947 + N[(x$95$m * N[(x$95$m * N[(0.0424060604 + N[(N[(x$95$m * N[(x$95$m * 0.0072644182), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.0005064034 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0001789971), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * N[(x$95$m * N[(0.2909738639 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0694555761 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0140005442 + N[(x$95$m * N[(x$95$m * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0003579942), $MachinePrecision] + 0.0008327945), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.7715471019), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e5

   1. Initial program 65.8%

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

   4. Applied egg-rr 5.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0001789971 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{\left(\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0694555761\right) + 0.0140005442 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)} - 1} \]

   6. Simplified 66.3%

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

   8. Applied egg-rr 5.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot \left(1 + \left(x \cdot \left(x \cdot \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right)\right)\right) + \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)\right)}{\left(1 + \left(x \cdot x\right) \cdot \left(0.7715471019 + x \cdot \left(x \cdot \left(0.2909738639 + x \cdot \left(x \cdot 0.0694555761\right)\right)\right)\right)\right) + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.0003579942\right) + \left(0.0140005442 + \left(x \cdot x\right) \cdot 0.0008327945\right)\right)}\right)} - 1} \]

   10. Simplified 66.3%

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

   12. Applied egg-rr 62.7%

       \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1 + x \cdot \left(x \cdot \left(\left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + \left(x \cdot x\right) \cdot 0.0072644182\right)\right) + \left(x \cdot x\right) \cdot \left(\left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}{\left(x \cdot x\right) \cdot 0.7715471019 + \left(1 + x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(0.2909738639 + \left(x \cdot x\right) \cdot 0.0694555761\right) + x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0140005442 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.0003579942\right) + 0.0008327945\right)\right)\right)\right)\right)\right)}\right)} - 1\right)} \]

   14. Simplified 66.2%

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

   16. Applied egg-rr 66.3%

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

   if 2e5 < x

   1. Initial program 1.5%

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

   3. Taylor expanded in x around inf 100.0%

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

Alternative 2: 100.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 100000000:\\ \;\;\;\;\left(-1 - \left(x\_m \cdot x\_m\right) \cdot \left(0.1049934947 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0424060604 + x\_m \cdot \left(x\_m \cdot \left(0.0072644182 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0005064034 + x\_m \cdot \left(x\_m \cdot 0.0001789971\right)\right)\right)\right)\right)\right)\right) \cdot \frac{x\_m}{-1 - x\_m \cdot \left(x\_m \cdot \left(0.7715471019 + x\_m \cdot \left(x\_m \cdot \left(0.2909738639 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0694555761 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0140005442 + x\_m \cdot \left(x\_m \cdot \left(0.0008327945 + x\_m \cdot \left(x\_m \cdot 0.0003579942\right)\right)\right)\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 100000000.0)
    (*
     (-
      -1.0
      (*
       (* x_m x_m)
       (+
        0.1049934947
        (*
         (* x_m x_m)
         (+
          0.0424060604
          (*
           x_m
           (*
            x_m
            (+
             0.0072644182
             (*
              (* x_m x_m)
              (+ 0.0005064034 (* x_m (* x_m 0.0001789971))))))))))))
     (/
      x_m
      (-
       -1.0
       (*
        x_m
        (*
         x_m
         (+
          0.7715471019
          (*
           x_m
           (*
            x_m
            (+
             0.2909738639
             (*
              (* x_m x_m)
              (+
               0.0694555761
               (*
                (* x_m x_m)
                (+
                 0.0140005442
                 (*
                  x_m
                  (*
                   x_m
                   (+ 0.0008327945 (* x_m (* x_m 0.0003579942))))))))))))))))))
    (/ 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 <= 100000000.0) {
		tmp = (-1.0 - ((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + (x_m * (x_m * (0.0072644182 + ((x_m * x_m) * (0.0005064034 + (x_m * (x_m * 0.0001789971)))))))))))) * (x_m / (-1.0 - (x_m * (x_m * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * (0.0140005442 + (x_m * (x_m * (0.0008327945 + (x_m * (x_m * 0.0003579942)))))))))))))))));
	} 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 <= 100000000.0d0) then
        tmp = ((-1.0d0) - ((x_m * x_m) * (0.1049934947d0 + ((x_m * x_m) * (0.0424060604d0 + (x_m * (x_m * (0.0072644182d0 + ((x_m * x_m) * (0.0005064034d0 + (x_m * (x_m * 0.0001789971d0)))))))))))) * (x_m / ((-1.0d0) - (x_m * (x_m * (0.7715471019d0 + (x_m * (x_m * (0.2909738639d0 + ((x_m * x_m) * (0.0694555761d0 + ((x_m * x_m) * (0.0140005442d0 + (x_m * (x_m * (0.0008327945d0 + (x_m * (x_m * 0.0003579942d0)))))))))))))))))
    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 <= 100000000.0) {
		tmp = (-1.0 - ((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + (x_m * (x_m * (0.0072644182 + ((x_m * x_m) * (0.0005064034 + (x_m * (x_m * 0.0001789971)))))))))))) * (x_m / (-1.0 - (x_m * (x_m * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * (0.0140005442 + (x_m * (x_m * (0.0008327945 + (x_m * (x_m * 0.0003579942)))))))))))))))));
	} 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 <= 100000000.0:
		tmp = (-1.0 - ((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + (x_m * (x_m * (0.0072644182 + ((x_m * x_m) * (0.0005064034 + (x_m * (x_m * 0.0001789971)))))))))))) * (x_m / (-1.0 - (x_m * (x_m * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * (0.0140005442 + (x_m * (x_m * (0.0008327945 + (x_m * (x_m * 0.0003579942)))))))))))))))))
	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 <= 100000000.0)
		tmp = Float64(Float64(-1.0 - Float64(Float64(x_m * x_m) * Float64(0.1049934947 + Float64(Float64(x_m * x_m) * Float64(0.0424060604 + Float64(x_m * Float64(x_m * Float64(0.0072644182 + Float64(Float64(x_m * x_m) * Float64(0.0005064034 + Float64(x_m * Float64(x_m * 0.0001789971)))))))))))) * Float64(x_m / Float64(-1.0 - Float64(x_m * Float64(x_m * Float64(0.7715471019 + Float64(x_m * Float64(x_m * Float64(0.2909738639 + Float64(Float64(x_m * x_m) * Float64(0.0694555761 + Float64(Float64(x_m * x_m) * Float64(0.0140005442 + Float64(x_m * Float64(x_m * Float64(0.0008327945 + Float64(x_m * Float64(x_m * 0.0003579942))))))))))))))))));
	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 <= 100000000.0)
		tmp = (-1.0 - ((x_m * x_m) * (0.1049934947 + ((x_m * x_m) * (0.0424060604 + (x_m * (x_m * (0.0072644182 + ((x_m * x_m) * (0.0005064034 + (x_m * (x_m * 0.0001789971)))))))))))) * (x_m / (-1.0 - (x_m * (x_m * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + ((x_m * x_m) * (0.0140005442 + (x_m * (x_m * (0.0008327945 + (x_m * (x_m * 0.0003579942)))))))))))))))));
	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, 100000000.0], N[(N[(-1.0 - N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.1049934947 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0424060604 + N[(x$95$m * N[(x$95$m * N[(0.0072644182 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0005064034 + N[(x$95$m * N[(x$95$m * 0.0001789971), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / N[(-1.0 - N[(x$95$m * N[(x$95$m * N[(0.7715471019 + N[(x$95$m * N[(x$95$m * N[(0.2909738639 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0694555761 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0140005442 + N[(x$95$m * N[(x$95$m * N[(0.0008327945 + N[(x$95$m * N[(x$95$m * 0.0003579942), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e8

   1. Initial program 65.8%

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

   4. Applied egg-rr 5.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0001789971 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{\left(\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0694555761\right) + 0.0140005442 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)} - 1} \]

   6. Simplified 66.3%

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

   8. Applied egg-rr 5.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot \left(1 + \left(x \cdot \left(x \cdot \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right)\right)\right) + \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)\right)}{\left(1 + \left(x \cdot x\right) \cdot \left(0.7715471019 + x \cdot \left(x \cdot \left(0.2909738639 + x \cdot \left(x \cdot 0.0694555761\right)\right)\right)\right)\right) + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.0003579942\right) + \left(0.0140005442 + \left(x \cdot x\right) \cdot 0.0008327945\right)\right)}\right)} - 1} \]

   10. Simplified 66.3%

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

   12. Applied egg-rr 62.7%

       \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1 + x \cdot \left(x \cdot \left(\left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + \left(x \cdot x\right) \cdot 0.0072644182\right)\right) + \left(x \cdot x\right) \cdot \left(\left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}{\left(x \cdot x\right) \cdot 0.7715471019 + \left(1 + x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(0.2909738639 + \left(x \cdot x\right) \cdot 0.0694555761\right) + x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0140005442 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.0003579942\right) + 0.0008327945\right)\right)\right)\right)\right)\right)}\right)} - 1\right)} \]

   14. Simplified 66.2%

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

   16. Applied egg-rr 66.3%

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

   18. Simplified 66.3%

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

   if 1e8 < x

   1. Initial program 1.5%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 100000000:\\ \;\;\;\;\left(-1 - \left(x \cdot x\right) \cdot \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + x \cdot \left(x \cdot \left(0.0072644182 + \left(x \cdot x\right) \cdot \left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right)\right)\right)\right)\right)\right) \cdot \frac{x}{-1 - x \cdot \left(x \cdot \left(0.7715471019 + x \cdot \left(x \cdot \left(0.2909738639 + \left(x \cdot x\right) \cdot \left(0.0694555761 + \left(x \cdot x\right) \cdot \left(0.0140005442 + x \cdot \left(x \cdot \left(0.0008327945 + x \cdot \left(x \cdot 0.0003579942\right)\right)\right)\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.3:\\ \;\;\;\;x\_m \cdot \frac{-1 - \left(x\_m \cdot x\_m\right) \cdot \left(0.1049934947 + x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(0.0005064034 + \left(x\_m \cdot x\_m\right) \cdot 0.0001789971\right)\right) + \left(0.0424060604 + x\_m \cdot \left(x\_m \cdot 0.0072644182\right)\right)\right)\right)\right)}{-1 - \left(x\_m \cdot x\_m\right) \cdot \left(0.7715471019 + x\_m \cdot \left(x\_m \cdot \left(0.2909738639 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0694555761 + x\_m \cdot \left(x\_m \cdot \left(0.0140005442 + \left(x\_m \cdot x\_m\right) \cdot 0.0008327945\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.3)
    (*
     x_m
     (/
      (-
       -1.0
       (*
        (* x_m x_m)
        (+
         0.1049934947
         (*
          x_m
          (*
           x_m
           (+
            (*
             x_m
             (*
              (* x_m (* x_m x_m))
              (+ 0.0005064034 (* (* x_m x_m) 0.0001789971))))
            (+ 0.0424060604 (* x_m (* x_m 0.0072644182)))))))))
      (-
       -1.0
       (*
        (* x_m x_m)
        (+
         0.7715471019
         (*
          x_m
          (*
           x_m
           (+
            0.2909738639
            (*
             (* x_m x_m)
             (+
              0.0694555761
              (*
               x_m
               (* x_m (+ 0.0140005442 (* (* x_m x_m) 0.0008327945))))))))))))))
    (/ 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 <= 2.3) {
		tmp = x_m * ((-1.0 - ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * ((x_m * ((x_m * (x_m * x_m)) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))) + (0.0424060604 + (x_m * (x_m * 0.0072644182))))))))) / (-1.0 - ((x_m * x_m) * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + (x_m * (x_m * (0.0140005442 + ((x_m * x_m) * 0.0008327945)))))))))))));
	} 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 <= 2.3d0) then
        tmp = x_m * (((-1.0d0) - ((x_m * x_m) * (0.1049934947d0 + (x_m * (x_m * ((x_m * ((x_m * (x_m * x_m)) * (0.0005064034d0 + ((x_m * x_m) * 0.0001789971d0)))) + (0.0424060604d0 + (x_m * (x_m * 0.0072644182d0))))))))) / ((-1.0d0) - ((x_m * x_m) * (0.7715471019d0 + (x_m * (x_m * (0.2909738639d0 + ((x_m * x_m) * (0.0694555761d0 + (x_m * (x_m * (0.0140005442d0 + ((x_m * x_m) * 0.0008327945d0)))))))))))))
    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 <= 2.3) {
		tmp = x_m * ((-1.0 - ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * ((x_m * ((x_m * (x_m * x_m)) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))) + (0.0424060604 + (x_m * (x_m * 0.0072644182))))))))) / (-1.0 - ((x_m * x_m) * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + (x_m * (x_m * (0.0140005442 + ((x_m * x_m) * 0.0008327945)))))))))))));
	} 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 <= 2.3:
		tmp = x_m * ((-1.0 - ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * ((x_m * ((x_m * (x_m * x_m)) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))) + (0.0424060604 + (x_m * (x_m * 0.0072644182))))))))) / (-1.0 - ((x_m * x_m) * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + (x_m * (x_m * (0.0140005442 + ((x_m * x_m) * 0.0008327945)))))))))))))
	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 <= 2.3)
		tmp = Float64(x_m * Float64(Float64(-1.0 - Float64(Float64(x_m * x_m) * Float64(0.1049934947 + Float64(x_m * Float64(x_m * Float64(Float64(x_m * Float64(Float64(x_m * Float64(x_m * x_m)) * Float64(0.0005064034 + Float64(Float64(x_m * x_m) * 0.0001789971)))) + Float64(0.0424060604 + Float64(x_m * Float64(x_m * 0.0072644182))))))))) / Float64(-1.0 - Float64(Float64(x_m * x_m) * Float64(0.7715471019 + Float64(x_m * Float64(x_m * Float64(0.2909738639 + Float64(Float64(x_m * x_m) * Float64(0.0694555761 + Float64(x_m * Float64(x_m * Float64(0.0140005442 + Float64(Float64(x_m * x_m) * 0.0008327945))))))))))))));
	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 <= 2.3)
		tmp = x_m * ((-1.0 - ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * ((x_m * ((x_m * (x_m * x_m)) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))) + (0.0424060604 + (x_m * (x_m * 0.0072644182))))))))) / (-1.0 - ((x_m * x_m) * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + (x_m * (x_m * (0.0140005442 + ((x_m * x_m) * 0.0008327945)))))))))))));
	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, 2.3], N[(x$95$m * N[(N[(-1.0 - N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.1049934947 + N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * N[(N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.0005064034 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0001789971), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 + N[(x$95$m * N[(x$95$m * 0.0072644182), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.7715471019 + N[(x$95$m * N[(x$95$m * N[(0.2909738639 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0694555761 + N[(x$95$m * N[(x$95$m * N[(0.0140005442 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0008327945), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   1. Initial program 65.8%

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

   4. Applied egg-rr 5.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0001789971 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{\left(\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0694555761\right) + 0.0140005442 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)} - 1} \]

   6. Simplified 66.3%

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

   8. Applied egg-rr 5.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot \left(1 + \left(x \cdot \left(x \cdot \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right)\right)\right) + \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)\right)}{\left(1 + \left(x \cdot x\right) \cdot \left(0.7715471019 + x \cdot \left(x \cdot \left(0.2909738639 + x \cdot \left(x \cdot 0.0694555761\right)\right)\right)\right)\right) + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.0003579942\right) + \left(0.0140005442 + \left(x \cdot x\right) \cdot 0.0008327945\right)\right)}\right)} - 1} \]

   10. Simplified 66.3%

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

   12. Applied egg-rr 62.7%

       \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1 + x \cdot \left(x \cdot \left(\left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + \left(x \cdot x\right) \cdot 0.0072644182\right)\right) + \left(x \cdot x\right) \cdot \left(\left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}{\left(x \cdot x\right) \cdot 0.7715471019 + \left(1 + x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(0.2909738639 + \left(x \cdot x\right) \cdot 0.0694555761\right) + x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0140005442 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.0003579942\right) + 0.0008327945\right)\right)\right)\right)\right)\right)}\right)} - 1\right)} \]

   14. Simplified 66.2%

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

   15. Taylor expanded in x around 0 62.5%

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

       1. *-commutative 62.5%

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

       2. unpow2 62.5%

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

   17. Simplified 62.5%

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

   if 2.2999999999999998 < x

   1. Initial program 1.5%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;x \cdot \frac{-1 - \left(x \cdot x\right) \cdot \left(0.1049934947 + x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right) + \left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right)\right)\right)\right)}{-1 - \left(x \cdot x\right) \cdot \left(0.7715471019 + x \cdot \left(x \cdot \left(0.2909738639 + \left(x \cdot x\right) \cdot \left(0.0694555761 + x \cdot \left(x \cdot \left(0.0140005442 + \left(x \cdot x\right) \cdot 0.0008327945\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5:\\ \;\;\;\;x\_m \cdot \frac{-1 - \left(x\_m \cdot x\_m\right) \cdot \left(0.1049934947 + x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot \left(\left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \left(0.0005064034 + \left(x\_m \cdot x\_m\right) \cdot 0.0001789971\right)\right) + \left(0.0424060604 + x\_m \cdot \left(x\_m \cdot 0.0072644182\right)\right)\right)\right)\right)}{-1 - \left(x\_m \cdot x\_m\right) \cdot \left(0.7715471019 + x\_m \cdot \left(x\_m \cdot \left(0.2909738639 + \left(x\_m \cdot x\_m\right) \cdot \left(0.0694555761 + x\_m \cdot \left(x\_m \cdot 0.0140005442\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot 2 + \frac{-1.0056716002661497}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.5)
    (*
     x_m
     (/
      (-
       -1.0
       (*
        (* x_m x_m)
        (+
         0.1049934947
         (*
          x_m
          (*
           x_m
           (+
            (*
             x_m
             (*
              (* x_m (* x_m x_m))
              (+ 0.0005064034 (* (* x_m x_m) 0.0001789971))))
            (+ 0.0424060604 (* x_m (* x_m 0.0072644182)))))))))
      (-
       -1.0
       (*
        (* x_m x_m)
        (+
         0.7715471019
         (*
          x_m
          (*
           x_m
           (+
            0.2909738639
            (*
             (* x_m x_m)
             (+ 0.0694555761 (* x_m (* x_m 0.0140005442))))))))))))
    (/ 1.0 (+ (* x_m 2.0) (/ -1.0056716002661497 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.5) {
		tmp = x_m * ((-1.0 - ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * ((x_m * ((x_m * (x_m * x_m)) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))) + (0.0424060604 + (x_m * (x_m * 0.0072644182))))))))) / (-1.0 - ((x_m * x_m) * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + (x_m * (x_m * 0.0140005442)))))))))));
	} else {
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.5d0) then
        tmp = x_m * (((-1.0d0) - ((x_m * x_m) * (0.1049934947d0 + (x_m * (x_m * ((x_m * ((x_m * (x_m * x_m)) * (0.0005064034d0 + ((x_m * x_m) * 0.0001789971d0)))) + (0.0424060604d0 + (x_m * (x_m * 0.0072644182d0))))))))) / ((-1.0d0) - ((x_m * x_m) * (0.7715471019d0 + (x_m * (x_m * (0.2909738639d0 + ((x_m * x_m) * (0.0694555761d0 + (x_m * (x_m * 0.0140005442d0)))))))))))
    else
        tmp = 1.0d0 / ((x_m * 2.0d0) + ((-1.0056716002661497d0) / x_m))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.5) {
		tmp = x_m * ((-1.0 - ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * ((x_m * ((x_m * (x_m * x_m)) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))) + (0.0424060604 + (x_m * (x_m * 0.0072644182))))))))) / (-1.0 - ((x_m * x_m) * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + (x_m * (x_m * 0.0140005442)))))))))));
	} else {
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.5:
		tmp = x_m * ((-1.0 - ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * ((x_m * ((x_m * (x_m * x_m)) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))) + (0.0424060604 + (x_m * (x_m * 0.0072644182))))))))) / (-1.0 - ((x_m * x_m) * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + (x_m * (x_m * 0.0140005442)))))))))))
	else:
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.5)
		tmp = Float64(x_m * Float64(Float64(-1.0 - Float64(Float64(x_m * x_m) * Float64(0.1049934947 + Float64(x_m * Float64(x_m * Float64(Float64(x_m * Float64(Float64(x_m * Float64(x_m * x_m)) * Float64(0.0005064034 + Float64(Float64(x_m * x_m) * 0.0001789971)))) + Float64(0.0424060604 + Float64(x_m * Float64(x_m * 0.0072644182))))))))) / Float64(-1.0 - Float64(Float64(x_m * x_m) * Float64(0.7715471019 + Float64(x_m * Float64(x_m * Float64(0.2909738639 + Float64(Float64(x_m * x_m) * Float64(0.0694555761 + Float64(x_m * Float64(x_m * 0.0140005442))))))))))));
	else
		tmp = Float64(1.0 / Float64(Float64(x_m * 2.0) + Float64(-1.0056716002661497 / x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.5)
		tmp = x_m * ((-1.0 - ((x_m * x_m) * (0.1049934947 + (x_m * (x_m * ((x_m * ((x_m * (x_m * x_m)) * (0.0005064034 + ((x_m * x_m) * 0.0001789971)))) + (0.0424060604 + (x_m * (x_m * 0.0072644182))))))))) / (-1.0 - ((x_m * x_m) * (0.7715471019 + (x_m * (x_m * (0.2909738639 + ((x_m * x_m) * (0.0694555761 + (x_m * (x_m * 0.0140005442)))))))))));
	else
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.5], N[(x$95$m * N[(N[(-1.0 - N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.1049934947 + N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * N[(N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.0005064034 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0001789971), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 + N[(x$95$m * N[(x$95$m * 0.0072644182), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.7715471019 + N[(x$95$m * N[(x$95$m * N[(0.2909738639 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.0694555761 + N[(x$95$m * N[(x$95$m * 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(-1.0056716002661497 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 65.8%

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

   4. Applied egg-rr 5.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + \left(0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0001789971 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{\left(\left(\left(1 + \left(\left(x \cdot x\right) \cdot 0.7715471019 + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0694555761\right) + 0.0140005442 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0008327945 + 0.0003579942 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)} - 1} \]

   6. Simplified 66.3%

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

   8. Applied egg-rr 5.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot \left(1 + \left(x \cdot \left(x \cdot \left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right)\right)\right) + \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right)\right)\right)}{\left(1 + \left(x \cdot x\right) \cdot \left(0.7715471019 + x \cdot \left(x \cdot \left(0.2909738639 + x \cdot \left(x \cdot 0.0694555761\right)\right)\right)\right)\right) + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.0003579942\right) + \left(0.0140005442 + \left(x \cdot x\right) \cdot 0.0008327945\right)\right)}\right)} - 1} \]

   10. Simplified 66.3%

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

   12. Applied egg-rr 62.7%

       \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1 + x \cdot \left(x \cdot \left(\left(0.1049934947 + \left(x \cdot x\right) \cdot \left(0.0424060604 + \left(x \cdot x\right) \cdot 0.0072644182\right)\right) + \left(x \cdot x\right) \cdot \left(\left(0.0005064034 + x \cdot \left(x \cdot 0.0001789971\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}{\left(x \cdot x\right) \cdot 0.7715471019 + \left(1 + x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(0.2909738639 + \left(x \cdot x\right) \cdot 0.0694555761\right) + x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0140005442 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.0003579942\right) + 0.0008327945\right)\right)\right)\right)\right)\right)}\right)} - 1\right)} \]

   14. Simplified 66.2%

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

   15. Taylor expanded in x around 0 62.5%

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

       1. *-commutative 62.5%

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

   17. Simplified 62.5%

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

   if 1.5 < x

   1. Initial program 1.5%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

         \[\leadsto \frac{0.5 + \frac{\color{blue}{0.2514179000665374}}{{x}^{2}}}{x} \]

      3. unpow2 100.0%

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

   5. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 + \left(-1.0056716002661497 \cdot \frac{1}{{x}^{2}}\right)\right)}} \]

      2. associate-*r/ 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \left(-\color{blue}{\frac{1.0056716002661497 \cdot 1}{{x}^{2}}}\right)\right)} \]

      3. metadata-eval 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \left(-\frac{\color{blue}{1.0056716002661497}}{{x}^{2}}\right)\right)} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \color{blue}{\frac{-1.0056716002661497}{{x}^{2}}}\right)} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \frac{\color{blue}{-1.0056716002661497}}{{x}^{2}}\right)} \]

      6. unpow2 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \frac{-1.0056716002661497}{\color{blue}{x \cdot x}}\right)} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(2 + \frac{-1.0056716002661497}{x \cdot x}\right)}} \]

   11. Taylor expanded in x around inf 100.0%

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

       1. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 + \left(-1.0056716002661497\right) \cdot \frac{1}{{x}^{2}}\right)}} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \color{blue}{-1.0056716002661497} \cdot \frac{1}{{x}^{2}}\right)} \]

       3. associate-*r/ 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \color{blue}{\frac{-1.0056716002661497 \cdot 1}{{x}^{2}}}\right)} \]

       4. metadata-eval 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \frac{\color{blue}{-1.0056716002661497}}{{x}^{2}}\right)} \]

       5. unpow2 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \frac{-1.0056716002661497}{\color{blue}{x \cdot x}}\right)} \]

       6. distribute-lft-in 100.0%

          \[\leadsto \frac{1}{\color{blue}{x \cdot 2 + x \cdot \frac{-1.0056716002661497}{x \cdot x}}} \]

       7. *-commutative 100.0%

          \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497}{x \cdot x} \cdot x}} \]

       8. unpow2 100.0%

          \[\leadsto \frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{\color{blue}{{x}^{2}}} \cdot x} \]

       9. associate-*l/ 100.0%

          \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497 \cdot x}{{x}^{2}}}} \]

       10. unpow2 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \frac{-1.0056716002661497 \cdot x}{\color{blue}{x \cdot x}}} \]

       11. times-frac 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497}{x} \cdot \frac{x}{x}}} \]

       12. *-inverses 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{x} \cdot \color{blue}{1}} \]

       13. *-rgt-identity 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497}{x}}} \]

   13. Simplified 100.0%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;x \cdot \frac{-1 - \left(x \cdot x\right) \cdot \left(0.1049934947 + x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.0005064034 + \left(x \cdot x\right) \cdot 0.0001789971\right)\right) + \left(0.0424060604 + x \cdot \left(x \cdot 0.0072644182\right)\right)\right)\right)\right)}{-1 - \left(x \cdot x\right) \cdot \left(0.7715471019 + x \cdot \left(x \cdot \left(0.2909738639 + \left(x \cdot x\right) \cdot \left(0.0694555761 + x \cdot \left(x \cdot 0.0140005442\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2:\\ \;\;\;\;x\_m \cdot \left(1 + x\_m \cdot \left(x\_m \cdot \left(-0.6665536072 + \left(x\_m \cdot x\_m\right) \cdot \left(0.265709700396151 + x\_m \cdot \left(x\_m \cdot -0.0732490286039007\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot 2 + \frac{-1.0056716002661497}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.2)
    (*
     x_m
     (+
      1.0
      (*
       x_m
       (*
        x_m
        (+
         -0.6665536072
         (*
          (* x_m x_m)
          (+ 0.265709700396151 (* x_m (* x_m -0.0732490286039007)))))))))
    (/ 1.0 (+ (* x_m 2.0) (/ -1.0056716002661497 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.2) {
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + ((x_m * x_m) * (0.265709700396151 + (x_m * (x_m * -0.0732490286039007))))))));
	} else {
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.2d0) then
        tmp = x_m * (1.0d0 + (x_m * (x_m * ((-0.6665536072d0) + ((x_m * x_m) * (0.265709700396151d0 + (x_m * (x_m * (-0.0732490286039007d0)))))))))
    else
        tmp = 1.0d0 / ((x_m * 2.0d0) + ((-1.0056716002661497d0) / x_m))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.2) {
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + ((x_m * x_m) * (0.265709700396151 + (x_m * (x_m * -0.0732490286039007))))))));
	} else {
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.2:
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + ((x_m * x_m) * (0.265709700396151 + (x_m * (x_m * -0.0732490286039007))))))))
	else:
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.2)
		tmp = Float64(x_m * Float64(1.0 + Float64(x_m * Float64(x_m * Float64(-0.6665536072 + Float64(Float64(x_m * x_m) * Float64(0.265709700396151 + Float64(x_m * Float64(x_m * -0.0732490286039007)))))))));
	else
		tmp = Float64(1.0 / Float64(Float64(x_m * 2.0) + Float64(-1.0056716002661497 / x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.2)
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + ((x_m * x_m) * (0.265709700396151 + (x_m * (x_m * -0.0732490286039007))))))));
	else
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.2], N[(x$95$m * N[(1.0 + N[(x$95$m * N[(x$95$m * N[(-0.6665536072 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.265709700396151 + N[(x$95$m * N[(x$95$m * -0.0732490286039007), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(-1.0056716002661497 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 65.8%

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

   3. Taylor expanded in x around 0 62.4%

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

      1. unpow2 62.4%

         \[\leadsto \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right) - 0.6665536072\right)\right) \cdot x \]

      2. associate-*l* 62.4%

         \[\leadsto \left(1 + \color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right) - 0.6665536072\right)\right)}\right) \cdot x \]

      3. sub-neg 62.4%

         \[\leadsto \left(1 + x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right) + \left(-0.6665536072\right)\right)}\right)\right) \cdot x \]

      4. metadata-eval 62.4%

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left({x}^{2} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right) + \color{blue}{-0.6665536072}\right)\right)\right) \cdot x \]

      5. +-commutative 62.4%

         \[\leadsto \left(1 + x \cdot \left(x \cdot \color{blue}{\left(-0.6665536072 + {x}^{2} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right)\right)}\right)\right) \cdot x \]

      6. unpow2 62.4%

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(-0.6665536072 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.265709700396151 + -0.0732490286039007 \cdot {x}^{2}\right)\right)\right)\right) \cdot x \]

      7. *-commutative 62.4%

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(-0.6665536072 + \left(x \cdot x\right) \cdot \left(0.265709700396151 + \color{blue}{{x}^{2} \cdot -0.0732490286039007}\right)\right)\right)\right) \cdot x \]

      8. unpow2 62.4%

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

      9. associate-*l* 62.4%

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

   5. Simplified 62.4%

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

   if 1.19999999999999996 < x

   1. Initial program 1.5%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

         \[\leadsto \frac{0.5 + \frac{\color{blue}{0.2514179000665374}}{{x}^{2}}}{x} \]

      3. unpow2 100.0%

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

   5. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 + \left(-1.0056716002661497 \cdot \frac{1}{{x}^{2}}\right)\right)}} \]

      2. associate-*r/ 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \left(-\color{blue}{\frac{1.0056716002661497 \cdot 1}{{x}^{2}}}\right)\right)} \]

      3. metadata-eval 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \left(-\frac{\color{blue}{1.0056716002661497}}{{x}^{2}}\right)\right)} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \color{blue}{\frac{-1.0056716002661497}{{x}^{2}}}\right)} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \frac{\color{blue}{-1.0056716002661497}}{{x}^{2}}\right)} \]

      6. unpow2 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \frac{-1.0056716002661497}{\color{blue}{x \cdot x}}\right)} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(2 + \frac{-1.0056716002661497}{x \cdot x}\right)}} \]

   11. Taylor expanded in x around inf 100.0%

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

       1. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 + \left(-1.0056716002661497\right) \cdot \frac{1}{{x}^{2}}\right)}} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \color{blue}{-1.0056716002661497} \cdot \frac{1}{{x}^{2}}\right)} \]

       3. associate-*r/ 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \color{blue}{\frac{-1.0056716002661497 \cdot 1}{{x}^{2}}}\right)} \]

       4. metadata-eval 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \frac{\color{blue}{-1.0056716002661497}}{{x}^{2}}\right)} \]

       5. unpow2 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \frac{-1.0056716002661497}{\color{blue}{x \cdot x}}\right)} \]

       6. distribute-lft-in 100.0%

          \[\leadsto \frac{1}{\color{blue}{x \cdot 2 + x \cdot \frac{-1.0056716002661497}{x \cdot x}}} \]

       7. *-commutative 100.0%

          \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497}{x \cdot x} \cdot x}} \]

       8. unpow2 100.0%

          \[\leadsto \frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{\color{blue}{{x}^{2}}} \cdot x} \]

       9. associate-*l/ 100.0%

          \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497 \cdot x}{{x}^{2}}}} \]

       10. unpow2 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \frac{-1.0056716002661497 \cdot x}{\color{blue}{x \cdot x}}} \]

       11. times-frac 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497}{x} \cdot \frac{x}{x}}} \]

       12. *-inverses 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{x} \cdot \color{blue}{1}} \]

       13. *-rgt-identity 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497}{x}}} \]

   13. Simplified 100.0%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.6665536072 + \left(x \cdot x\right) \cdot \left(0.265709700396151 + x \cdot \left(x \cdot -0.0732490286039007\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;x\_m \cdot \left(1 + x\_m \cdot \left(x\_m \cdot \left(-0.6665536072 + x\_m \cdot \left(x\_m \cdot 0.265709700396151\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot 2 + \frac{-1.0056716002661497}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.15)
    (*
     x_m
     (+
      1.0
      (* x_m (* x_m (+ -0.6665536072 (* x_m (* x_m 0.265709700396151)))))))
    (/ 1.0 (+ (* x_m 2.0) (/ -1.0056716002661497 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.15) {
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + (x_m * (x_m * 0.265709700396151))))));
	} else {
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.15d0) then
        tmp = x_m * (1.0d0 + (x_m * (x_m * ((-0.6665536072d0) + (x_m * (x_m * 0.265709700396151d0))))))
    else
        tmp = 1.0d0 / ((x_m * 2.0d0) + ((-1.0056716002661497d0) / x_m))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.15) {
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + (x_m * (x_m * 0.265709700396151))))));
	} else {
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.15:
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + (x_m * (x_m * 0.265709700396151))))))
	else:
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.15)
		tmp = Float64(x_m * Float64(1.0 + Float64(x_m * Float64(x_m * Float64(-0.6665536072 + Float64(x_m * Float64(x_m * 0.265709700396151)))))));
	else
		tmp = Float64(1.0 / Float64(Float64(x_m * 2.0) + Float64(-1.0056716002661497 / x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.15)
		tmp = x_m * (1.0 + (x_m * (x_m * (-0.6665536072 + (x_m * (x_m * 0.265709700396151))))));
	else
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.15], N[(x$95$m * N[(1.0 + N[(x$95$m * N[(x$95$m * N[(-0.6665536072 + N[(x$95$m * N[(x$95$m * 0.265709700396151), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(-1.0056716002661497 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 65.8%

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

   3. Taylor expanded in x around 0 63.0%

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

      1. unpow2 63.0%

         \[\leadsto \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.265709700396151 \cdot {x}^{2} - 0.6665536072\right)\right) \cdot x \]

      2. associate-*l* 63.0%

         \[\leadsto \left(1 + \color{blue}{x \cdot \left(x \cdot \left(0.265709700396151 \cdot {x}^{2} - 0.6665536072\right)\right)}\right) \cdot x \]

      3. sub-neg 63.0%

         \[\leadsto \left(1 + x \cdot \left(x \cdot \color{blue}{\left(0.265709700396151 \cdot {x}^{2} + \left(-0.6665536072\right)\right)}\right)\right) \cdot x \]

      4. metadata-eval 63.0%

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(0.265709700396151 \cdot {x}^{2} + \color{blue}{-0.6665536072}\right)\right)\right) \cdot x \]

      5. +-commutative 63.0%

         \[\leadsto \left(1 + x \cdot \left(x \cdot \color{blue}{\left(-0.6665536072 + 0.265709700396151 \cdot {x}^{2}\right)}\right)\right) \cdot x \]

      6. *-commutative 63.0%

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(-0.6665536072 + \color{blue}{{x}^{2} \cdot 0.265709700396151}\right)\right)\right) \cdot x \]

      7. unpow2 63.0%

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(-0.6665536072 + \color{blue}{\left(x \cdot x\right)} \cdot 0.265709700396151\right)\right)\right) \cdot x \]

      8. associate-*l* 63.0%

         \[\leadsto \left(1 + x \cdot \left(x \cdot \left(-0.6665536072 + \color{blue}{x \cdot \left(x \cdot 0.265709700396151\right)}\right)\right)\right) \cdot x \]

   5. Simplified 63.0%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.6665536072 + x \cdot \left(x \cdot 0.265709700396151\right)\right)\right)\right)} \cdot x \]

   if 1.1499999999999999 < x

   1. Initial program 1.5%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

         \[\leadsto \frac{0.5 + \frac{\color{blue}{0.2514179000665374}}{{x}^{2}}}{x} \]

      3. unpow2 100.0%

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

   5. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 + \left(-1.0056716002661497 \cdot \frac{1}{{x}^{2}}\right)\right)}} \]

      2. associate-*r/ 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \left(-\color{blue}{\frac{1.0056716002661497 \cdot 1}{{x}^{2}}}\right)\right)} \]

      3. metadata-eval 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \left(-\frac{\color{blue}{1.0056716002661497}}{{x}^{2}}\right)\right)} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \color{blue}{\frac{-1.0056716002661497}{{x}^{2}}}\right)} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \frac{\color{blue}{-1.0056716002661497}}{{x}^{2}}\right)} \]

      6. unpow2 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \frac{-1.0056716002661497}{\color{blue}{x \cdot x}}\right)} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(2 + \frac{-1.0056716002661497}{x \cdot x}\right)}} \]

   11. Taylor expanded in x around inf 100.0%

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

       1. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 + \left(-1.0056716002661497\right) \cdot \frac{1}{{x}^{2}}\right)}} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \color{blue}{-1.0056716002661497} \cdot \frac{1}{{x}^{2}}\right)} \]

       3. associate-*r/ 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \color{blue}{\frac{-1.0056716002661497 \cdot 1}{{x}^{2}}}\right)} \]

       4. metadata-eval 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \frac{\color{blue}{-1.0056716002661497}}{{x}^{2}}\right)} \]

       5. unpow2 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \frac{-1.0056716002661497}{\color{blue}{x \cdot x}}\right)} \]

       6. distribute-lft-in 100.0%

          \[\leadsto \frac{1}{\color{blue}{x \cdot 2 + x \cdot \frac{-1.0056716002661497}{x \cdot x}}} \]

       7. *-commutative 100.0%

          \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497}{x \cdot x} \cdot x}} \]

       8. unpow2 100.0%

          \[\leadsto \frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{\color{blue}{{x}^{2}}} \cdot x} \]

       9. associate-*l/ 100.0%

          \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497 \cdot x}{{x}^{2}}}} \]

       10. unpow2 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \frac{-1.0056716002661497 \cdot x}{\color{blue}{x \cdot x}}} \]

       11. times-frac 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497}{x} \cdot \frac{x}{x}}} \]

       12. *-inverses 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{x} \cdot \color{blue}{1}} \]

       13. *-rgt-identity 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497}{x}}} \]

   13. Simplified 100.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.6665536072 + x \cdot \left(x \cdot 0.265709700396151\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.0d0) then
        tmp = x_m * (1.0d0 + ((x_m * x_m) * (-0.6665536072d0)))
    else
        tmp = 1.0d0 / ((x_m * 2.0d0) + ((-1.0056716002661497d0) / x_m))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	} else {
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072))
	else:
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(x_m * Float64(1.0 + Float64(Float64(x_m * x_m) * -0.6665536072)));
	else
		tmp = Float64(1.0 / Float64(Float64(x_m * 2.0) + Float64(-1.0056716002661497 / x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = x_m * (1.0 + ((x_m * x_m) * -0.6665536072));
	else
		tmp = 1.0 / ((x_m * 2.0) + (-1.0056716002661497 / x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.0], N[(x$95$m * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(-1.0056716002661497 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 65.8%

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

   3. Taylor expanded in x around 0 62.3%

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

      1. unpow2 62.3%

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

   5. Simplified 62.3%

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

   if 1 < x

   1. Initial program 1.5%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

         \[\leadsto \frac{0.5 + \frac{\color{blue}{0.2514179000665374}}{{x}^{2}}}{x} \]

      3. unpow2 100.0%

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

   5. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 + \left(-1.0056716002661497 \cdot \frac{1}{{x}^{2}}\right)\right)}} \]

      2. associate-*r/ 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \left(-\color{blue}{\frac{1.0056716002661497 \cdot 1}{{x}^{2}}}\right)\right)} \]

      3. metadata-eval 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \left(-\frac{\color{blue}{1.0056716002661497}}{{x}^{2}}\right)\right)} \]

      4. distribute-neg-frac 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \color{blue}{\frac{-1.0056716002661497}{{x}^{2}}}\right)} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \frac{\color{blue}{-1.0056716002661497}}{{x}^{2}}\right)} \]

      6. unpow2 100.0%

         \[\leadsto \frac{1}{x \cdot \left(2 + \frac{-1.0056716002661497}{\color{blue}{x \cdot x}}\right)} \]

   10. Simplified 100.0%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(2 + \frac{-1.0056716002661497}{x \cdot x}\right)}} \]

   11. Taylor expanded in x around inf 100.0%

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

       1. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{1}{x \cdot \color{blue}{\left(2 + \left(-1.0056716002661497\right) \cdot \frac{1}{{x}^{2}}\right)}} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \color{blue}{-1.0056716002661497} \cdot \frac{1}{{x}^{2}}\right)} \]

       3. associate-*r/ 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \color{blue}{\frac{-1.0056716002661497 \cdot 1}{{x}^{2}}}\right)} \]

       4. metadata-eval 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \frac{\color{blue}{-1.0056716002661497}}{{x}^{2}}\right)} \]

       5. unpow2 100.0%

          \[\leadsto \frac{1}{x \cdot \left(2 + \frac{-1.0056716002661497}{\color{blue}{x \cdot x}}\right)} \]

       6. distribute-lft-in 100.0%

          \[\leadsto \frac{1}{\color{blue}{x \cdot 2 + x \cdot \frac{-1.0056716002661497}{x \cdot x}}} \]

       7. *-commutative 100.0%

          \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497}{x \cdot x} \cdot x}} \]

       8. unpow2 100.0%

          \[\leadsto \frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{\color{blue}{{x}^{2}}} \cdot x} \]

       9. associate-*l/ 100.0%

          \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497 \cdot x}{{x}^{2}}}} \]

       10. unpow2 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \frac{-1.0056716002661497 \cdot x}{\color{blue}{x \cdot x}}} \]

       11. times-frac 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497}{x} \cdot \frac{x}{x}}} \]

       12. *-inverses 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \frac{-1.0056716002661497}{x} \cdot \color{blue}{1}} \]

       13. *-rgt-identity 100.0%

           \[\leadsto \frac{1}{x \cdot 2 + \color{blue}{\frac{-1.0056716002661497}{x}}} \]

   13. Simplified 100.0%

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

4. Final simplification 72.4%

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

Alternative 8: 99.2% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.78000000000000003

   1. Initial program 65.8%

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

   3. Taylor expanded in x around 0 62.3%

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

      1. unpow2 62.3%

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

   5. Simplified 62.3%

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

   if 0.78000000000000003 < x

   1. Initial program 1.5%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.78:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 65.8%

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

   3. Taylor expanded in x around 0 63.2%

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

   if 0.69999999999999996 < x

   1. Initial program 1.5%

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

   3. Taylor expanded in x around inf 100.0%

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

Alternative 10: 50.9% accurate, 173.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.4%

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

3. Taylor expanded in x around 0 47.1%

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

Reproduce

herbie shell --seed 2024107 
(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))