Jmat.Real.dawson

Percentage Accurate: 54.1% → 99.2%

Time: 5.4s

Alternatives: 4

Speedup: 24.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% 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: 99.2% accurate, 13.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.78) (not (<= x 0.78)))
   (/ 0.5 x)
   (* x (+ 1.0 (* (* x x) -0.6665536072)))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.78) || !(x <= 0.78)) {
		tmp = 0.5 / x;
	} else {
		tmp = x * (1.0 + ((x * x) * -0.6665536072));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.78d0)) .or. (.not. (x <= 0.78d0))) then
        tmp = 0.5d0 / x
    else
        tmp = x * (1.0d0 + ((x * x) * (-0.6665536072d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.78) || !(x <= 0.78)) {
		tmp = 0.5 / x;
	} else {
		tmp = x * (1.0 + ((x * x) * -0.6665536072));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.78) or not (x <= 0.78):
		tmp = 0.5 / x
	else:
		tmp = x * (1.0 + ((x * x) * -0.6665536072))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.78) || !(x <= 0.78))
		tmp = Float64(0.5 / x);
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.6665536072)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.78) || ~((x <= 0.78)))
		tmp = 0.5 / x;
	else
		tmp = x * (1.0 + ((x * x) * -0.6665536072));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.78], N[Not[LessEqual[x, 0.78]], $MachinePrecision]], N[(0.5 / x), $MachinePrecision], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.78000000000000003 or 0.78000000000000003 < x

   1. Initial program 3.6%

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

   3. Simplified 3.6%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -0.78000000000000003 < x < 0.78000000000000003

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. unpow2 99.3%

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

   6. Simplified 99.3%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.6665536072 \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

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

Alternative 2: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	t_0 = 1.0 + ((x * x) * 0.7715471019)
	t_1 = (x * x) * (x * x)
	t_2 = (x * x) * t_1
	t_3 = x * (x * (x * x))
	t_4 = (x * x) * t_2
	t_5 = (x * x) * t_3
	t_6 = t_3 * t_3
	t_7 = (x * x) * t_6
	t_8 = 1.0 + (0.1049934947 * (x * x))
	t_9 = (x * x) * t_4
	tmp = 0
	if (x * (((((t_8 + (0.0424060604 * t_1)) + (0.0072644182 * t_2)) + (0.0005064034 * t_4)) + (0.0001789971 * t_9)) / (((((t_0 + (t_1 * 0.2909738639)) + (t_2 * 0.0694555761)) + (t_4 * 0.0140005442)) + (t_9 * 0.0008327945)) + (0.0003579942 * ((x * x) * t_9))))) <= 0.5:
		tmp = x * ((((t_8 + (0.0424060604 * t_3)) + (0.0072644182 * t_5)) + ((0.0005064034 * t_6) + (0.0001789971 * t_7))) / (((t_0 + (0.2909738639 * t_3)) + ((0.0694555761 * t_5) + (0.0140005442 * t_6))) + ((0.0008327945 * t_7) + (0.0003579942 * (t_3 * t_6)))))
	else:
		tmp = 0.5 / x
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 1 (*.f64 1049934947/10000000000 (*.f64 x x))) (*.f64 106015151/2500000000 (*.f64 (*.f64 x x) (*.f64 x x)))) (*.f64 36322091/5000000000 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)))) (*.f64 2532017/5000000000 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (*.f64 1789971/10000000000 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 1 (*.f64 7715471019/10000000000 (*.f64 x x))) (*.f64 2909738639/10000000000 (*.f64 (*.f64 x x) (*.f64 x x)))) (*.f64 694555761/10000000000 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)))) (*.f64 70002721/5000000000 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (*.f64 1665589/2000000000 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (*.f64 (*.f64 2 1789971/10000000000) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x))))) x) < 0.5

   1. Initial program 96.8%

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

   3. Simplified 96.8%

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

   if 0.5 < (*.f64 (/.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 1 (*.f64 1049934947/10000000000 (*.f64 x x))) (*.f64 106015151/2500000000 (*.f64 (*.f64 x x) (*.f64 x x)))) (*.f64 36322091/5000000000 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)))) (*.f64 2532017/5000000000 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (*.f64 1789971/10000000000 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 1 (*.f64 7715471019/10000000000 (*.f64 x x))) (*.f64 2909738639/10000000000 (*.f64 (*.f64 x x) (*.f64 x x)))) (*.f64 694555761/10000000000 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)))) (*.f64 70002721/5000000000 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (*.f64 1665589/2000000000 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)))) (*.f64 (*.f64 2 1789971/10000000000) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 x x) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x)) (*.f64 x x))))) x)

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. 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 98.5%

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

Alternative 3: 99.0% accurate, 24.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.7) (/ 0.5 x) (if (<= x 0.7) x (/ 0.5 x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.7) {
		tmp = 0.5 / x;
	} else if (x <= 0.7) {
		tmp = x;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.7d0)) then
        tmp = 0.5d0 / x
    else if (x <= 0.7d0) then
        tmp = x
    else
        tmp = 0.5d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.7) {
		tmp = 0.5 / x;
	} else if (x <= 0.7) {
		tmp = x;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.7:
		tmp = 0.5 / x
	elif x <= 0.7:
		tmp = x
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.7)
		tmp = Float64(0.5 / x);
	elseif (x <= 0.7)
		tmp = x;
	else
		tmp = Float64(0.5 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.7)
		tmp = 0.5 / x;
	elseif (x <= 0.7)
		tmp = x;
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.7], N[(0.5 / x), $MachinePrecision], If[LessEqual[x, 0.7], x, N[(0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.69999999999999996 or 0.69999999999999996 < x

   1. Initial program 3.6%

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

   3. Simplified 3.6%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -0.69999999999999996 < x < 0.69999999999999996

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.5%

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

4. Final simplification 99.3%

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

Alternative 4: 51.7% accurate, 173.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 47.3%

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

3. Simplified 47.3%

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

4. Taylor expanded in x around 0 46.6%

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

5. Final simplification 46.6%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023189 
(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))