Jmat.Real.dawson

Percentage Accurate: 97.4% → 99.4%

Time: 40.3s

Alternatives: 11

Speedup: 24.3×

• could not determine a ground truth

  1. x = -6.345523484422262e+49

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: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;\frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \frac{\frac{0.10624017004622396}{x}}{x}\right)}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{x}{1 + \left({x}^{4} \cdot 0.17858401087518092 + x \cdot \left(x \cdot 0.6665536072\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-1.0056716002661497 + \left(x \cdot x\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.3)
   (/
    x
    (+
     (* x (* x 2.0))
     (- -1.0056716002661497 (/ (/ 0.10624017004622396 x) x))))
   (if (<= x 1.42)
     (/
      x
      (+ 1.0 (+ (* (pow x 4.0) 0.17858401087518092) (* x (* x 0.6665536072)))))
     (/ x (+ -1.0056716002661497 (* (* x x) 2.0))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.3) {
		tmp = x / ((x * (x * 2.0)) + (-1.0056716002661497 - ((0.10624017004622396 / x) / x)));
	} else if (x <= 1.42) {
		tmp = x / (1.0 + ((pow(x, 4.0) * 0.17858401087518092) + (x * (x * 0.6665536072))));
	} else {
		tmp = x / (-1.0056716002661497 + ((x * x) * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.3d0)) then
        tmp = x / ((x * (x * 2.0d0)) + ((-1.0056716002661497d0) - ((0.10624017004622396d0 / x) / x)))
    else if (x <= 1.42d0) then
        tmp = x / (1.0d0 + (((x ** 4.0d0) * 0.17858401087518092d0) + (x * (x * 0.6665536072d0))))
    else
        tmp = x / ((-1.0056716002661497d0) + ((x * x) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.3) {
		tmp = x / ((x * (x * 2.0)) + (-1.0056716002661497 - ((0.10624017004622396 / x) / x)));
	} else if (x <= 1.42) {
		tmp = x / (1.0 + ((Math.pow(x, 4.0) * 0.17858401087518092) + (x * (x * 0.6665536072))));
	} else {
		tmp = x / (-1.0056716002661497 + ((x * x) * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.3:
		tmp = x / ((x * (x * 2.0)) + (-1.0056716002661497 - ((0.10624017004622396 / x) / x)))
	elif x <= 1.42:
		tmp = x / (1.0 + ((math.pow(x, 4.0) * 0.17858401087518092) + (x * (x * 0.6665536072))))
	else:
		tmp = x / (-1.0056716002661497 + ((x * x) * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.3)
		tmp = Float64(x / Float64(Float64(x * Float64(x * 2.0)) + Float64(-1.0056716002661497 - Float64(Float64(0.10624017004622396 / x) / x))));
	elseif (x <= 1.42)
		tmp = Float64(x / Float64(1.0 + Float64(Float64((x ^ 4.0) * 0.17858401087518092) + Float64(x * Float64(x * 0.6665536072)))));
	else
		tmp = Float64(x / Float64(-1.0056716002661497 + Float64(Float64(x * x) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.3)
		tmp = x / ((x * (x * 2.0)) + (-1.0056716002661497 - ((0.10624017004622396 / x) / x)));
	elseif (x <= 1.42)
		tmp = x / (1.0 + (((x ^ 4.0) * 0.17858401087518092) + (x * (x * 0.6665536072))));
	else
		tmp = x / (-1.0056716002661497 + ((x * x) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.3], N[(x / N[(N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0056716002661497 - N[(N[(0.10624017004622396 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.42], N[(x / N[(1.0 + N[(N[(N[Power[x, 4.0], $MachinePrecision] * 0.17858401087518092), $MachinePrecision] + N[(x * N[(x * 0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(-1.0056716002661497 + N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2999999999999998

   1. Initial program 90.3%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in x around inf 83.7%

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

      1. associate--r+ 83.7%

         \[\leadsto \frac{x}{\color{blue}{\left(2 \cdot {x}^{2} - 1.0056716002661497\right) - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}}} \]

      2. unpow2 83.7%

         \[\leadsto \frac{x}{\left(2 \cdot \color{blue}{\left(x \cdot x\right)} - 1.0056716002661497\right) - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}} \]

      3. *-commutative 83.7%

         \[\leadsto \frac{x}{\left(\color{blue}{\left(x \cdot x\right) \cdot 2} - 1.0056716002661497\right) - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}} \]

      4. associate-*l* 83.7%

         \[\leadsto \frac{x}{\left(\color{blue}{x \cdot \left(x \cdot 2\right)} - 1.0056716002661497\right) - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}} \]

      5. fma-neg 83.7%

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, x \cdot 2, -1.0056716002661497\right)} - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}} \]

      6. metadata-eval 83.7%

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, x \cdot 2, \color{blue}{-1.0056716002661497}\right) - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}} \]

      7. unpow2 83.7%

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, x \cdot 2, -1.0056716002661497\right) - 0.10624017004622396 \cdot \frac{1}{\color{blue}{x \cdot x}}} \]

      8. associate-*r/ 83.7%

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, x \cdot 2, -1.0056716002661497\right) - \color{blue}{\frac{0.10624017004622396 \cdot 1}{x \cdot x}}} \]

      9. metadata-eval 83.7%

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, x \cdot 2, -1.0056716002661497\right) - \frac{\color{blue}{0.10624017004622396}}{x \cdot x}} \]

   6. Simplified 83.7%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, x \cdot 2, -1.0056716002661497\right) - \frac{0.10624017004622396}{x \cdot x}}} \]
   7. Step-by-step derivation

      1. fma-udef 83.7%

         \[\leadsto \frac{x}{\color{blue}{\left(x \cdot \left(x \cdot 2\right) + -1.0056716002661497\right)} - \frac{0.10624017004622396}{x \cdot x}} \]

   8. Applied egg-rr 83.7%

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

      1. associate--l+ 83.7%

         \[\leadsto \frac{x}{\color{blue}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \frac{0.10624017004622396}{x \cdot x}\right)}} \]

      2. div-inv 83.7%

         \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \color{blue}{0.10624017004622396 \cdot \frac{1}{x \cdot x}}\right)} \]

      3. cancel-sign-sub-inv 83.7%

         \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \color{blue}{\left(-1.0056716002661497 + \left(-0.10624017004622396\right) \cdot \frac{1}{x \cdot x}\right)}} \]

      4. metadata-eval 83.7%

         \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 + \color{blue}{-0.10624017004622396} \cdot \frac{1}{x \cdot x}\right)} \]

      5. pow2 83.7%

         \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 + -0.10624017004622396 \cdot \frac{1}{\color{blue}{{x}^{2}}}\right)} \]

      6. pow-flip 83.7%

         \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 + -0.10624017004622396 \cdot \color{blue}{{x}^{\left(-2\right)}}\right)} \]

      7. metadata-eval 83.7%

         \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 + -0.10624017004622396 \cdot {x}^{\color{blue}{-2}}\right)} \]

   10. Applied egg-rr 83.7%

       \[\leadsto \frac{x}{\color{blue}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 + -0.10624017004622396 \cdot {x}^{-2}\right)}} \]

   11. Taylor expanded in x around 0 83.7%

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

       1. distribute-neg-in 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \color{blue}{\left(\left(-1.0056716002661497\right) + \left(-0.10624017004622396 \cdot \frac{1}{{x}^{2}}\right)\right)}} \]

       2. metadata-eval 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(\color{blue}{-1.0056716002661497} + \left(-0.10624017004622396 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

       3. unsub-neg 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \color{blue}{\left(-1.0056716002661497 - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}\right)}} \]

       4. associate-*r/ 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \color{blue}{\frac{0.10624017004622396 \cdot 1}{{x}^{2}}}\right)} \]

       5. metadata-eval 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \frac{\color{blue}{0.10624017004622396}}{{x}^{2}}\right)} \]

       6. unpow2 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \frac{0.10624017004622396}{\color{blue}{x \cdot x}}\right)} \]

       7. associate-/r* 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \color{blue}{\frac{\frac{0.10624017004622396}{x}}{x}}\right)} \]

   13. Simplified 83.7%

       \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \color{blue}{\left(-1.0056716002661497 - \frac{\frac{0.10624017004622396}{x}}{x}\right)}} \]

   if -2.2999999999999998 < x < 1.4199999999999999

   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%

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

   4. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{x}{\color{blue}{1 + \left(0.17858401087518092 \cdot {x}^{4} + 0.6665536072 \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{x}{1 + \left(\color{blue}{{x}^{4} \cdot 0.17858401087518092} + 0.6665536072 \cdot {x}^{2}\right)} \]

      2. fma-def 99.6%

         \[\leadsto \frac{x}{1 + \color{blue}{\mathsf{fma}\left({x}^{4}, 0.17858401087518092, 0.6665536072 \cdot {x}^{2}\right)}} \]

      3. unpow2 99.6%

         \[\leadsto \frac{x}{1 + \mathsf{fma}\left({x}^{4}, 0.17858401087518092, 0.6665536072 \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

      4. *-commutative 99.6%

         \[\leadsto \frac{x}{1 + \mathsf{fma}\left({x}^{4}, 0.17858401087518092, \color{blue}{\left(x \cdot x\right) \cdot 0.6665536072}\right)} \]

      5. associate-*l* 99.6%

         \[\leadsto \frac{x}{1 + \mathsf{fma}\left({x}^{4}, 0.17858401087518092, \color{blue}{x \cdot \left(x \cdot 0.6665536072\right)}\right)} \]

   6. Simplified 99.6%

      \[\leadsto \frac{x}{\color{blue}{1 + \mathsf{fma}\left({x}^{4}, 0.17858401087518092, x \cdot \left(x \cdot 0.6665536072\right)\right)}} \]
   7. Step-by-step derivation

      1. fma-udef 99.6%

         \[\leadsto \frac{x}{1 + \color{blue}{\left({x}^{4} \cdot 0.17858401087518092 + x \cdot \left(x \cdot 0.6665536072\right)\right)}} \]

   8. Applied egg-rr 99.6%

      \[\leadsto \frac{x}{1 + \color{blue}{\left({x}^{4} \cdot 0.17858401087518092 + x \cdot \left(x \cdot 0.6665536072\right)\right)}} \]

   if 1.4199999999999999 < x

   1. Initial program 79.7%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in x around inf 99.5%

      \[\leadsto \frac{x}{\color{blue}{2 \cdot {x}^{2} - 1.0056716002661497}} \]
   5. Step-by-step derivation

      1. fma-neg 99.5%

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(2, {x}^{2}, -1.0056716002661497\right)}} \]

      2. unpow2 99.5%

         \[\leadsto \frac{x}{\mathsf{fma}\left(2, \color{blue}{x \cdot x}, -1.0056716002661497\right)} \]

      3. metadata-eval 99.5%

         \[\leadsto \frac{x}{\mathsf{fma}\left(2, x \cdot x, \color{blue}{-1.0056716002661497}\right)} \]

   6. Simplified 99.5%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(2, x \cdot x, -1.0056716002661497\right)}} \]
   7. Step-by-step derivation

      1. fma-udef 99.5%

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

   8. Applied egg-rr 99.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;\frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \frac{\frac{0.10624017004622396}{x}}{x}\right)}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{x}{1 + \left({x}^{4} \cdot 0.17858401087518092 + x \cdot \left(x \cdot 0.6665536072\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-1.0056716002661497 + \left(x \cdot x\right) \cdot 2}\\ \end{array} \]

Alternative 2: 97.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.0072644182, {x}^{6}, 0.1049934947 \cdot \left(x \cdot x\right) + 1\right)\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, 1\right)\right)\right)\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   0.0005064034
   (pow x 8.0)
   (fma
    0.0001789971
    (pow x 10.0)
    (fma
     0.0424060604
     (pow x 4.0)
     (fma 0.0072644182 (pow x 6.0) (+ (* 0.1049934947 (* x x)) 1.0)))))
  (/
   x
   (fma
    (pow x 10.0)
    0.0008327945
    (fma
     0.0003579942
     (pow x 12.0)
     (fma
      (pow x 6.0)
      0.0694555761
      (fma
       (pow x 8.0)
       0.0140005442
       (fma x (* x 0.7715471019) (fma (pow x 4.0) 0.2909738639 1.0)))))))))

C

double code(double x) {
	return fma(0.0005064034, pow(x, 8.0), fma(0.0001789971, pow(x, 10.0), fma(0.0424060604, pow(x, 4.0), fma(0.0072644182, pow(x, 6.0), ((0.1049934947 * (x * x)) + 1.0))))) * (x / fma(pow(x, 10.0), 0.0008327945, fma(0.0003579942, pow(x, 12.0), fma(pow(x, 6.0), 0.0694555761, fma(pow(x, 8.0), 0.0140005442, fma(x, (x * 0.7715471019), fma(pow(x, 4.0), 0.2909738639, 1.0)))))));
}

Julia

function code(x)
	return Float64(fma(0.0005064034, (x ^ 8.0), fma(0.0001789971, (x ^ 10.0), fma(0.0424060604, (x ^ 4.0), fma(0.0072644182, (x ^ 6.0), Float64(Float64(0.1049934947 * Float64(x * x)) + 1.0))))) * Float64(x / fma((x ^ 10.0), 0.0008327945, fma(0.0003579942, (x ^ 12.0), fma((x ^ 6.0), 0.0694555761, fma((x ^ 8.0), 0.0140005442, fma(x, Float64(x * 0.7715471019), fma((x ^ 4.0), 0.2909738639, 1.0))))))))
end

Wolfram

code[x_] := N[(N[(0.0005064034 * N[Power[x, 8.0], $MachinePrecision] + N[(0.0001789971 * N[Power[x, 10.0], $MachinePrecision] + N[(0.0424060604 * N[Power[x, 4.0], $MachinePrecision] + N[(0.0072644182 * N[Power[x, 6.0], $MachinePrecision] + N[(N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[Power[x, 10.0], $MachinePrecision] * 0.0008327945 + N[(0.0003579942 * N[Power[x, 12.0], $MachinePrecision] + N[(N[Power[x, 6.0], $MachinePrecision] * 0.0694555761 + N[(N[Power[x, 8.0], $MachinePrecision] * 0.0140005442 + N[(x * N[(x * 0.7715471019), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * 0.2909738639 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, 1\right)\right)\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. fma-udef 98.4%

      \[\leadsto \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \color{blue}{0.1049934947 \cdot \left(x \cdot x\right) + 1}\right)\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, 1\right)\right)\right)\right)\right)\right)} \]

5. Applied egg-rr 98.4%

   \[\leadsto \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \color{blue}{0.1049934947 \cdot \left(x \cdot x\right) + 1}\right)\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, 1\right)\right)\right)\right)\right)\right)} \]

6. Final simplification 98.4%

   \[\leadsto \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.0072644182, {x}^{6}, 0.1049934947 \cdot \left(x \cdot x\right) + 1\right)\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, 1\right)\right)\right)\right)\right)\right)} \]

Alternative 3: 97.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x \cdot x\right)}^{4}\\ \frac{x}{\frac{t_0 \cdot \left(0.0140005442 + x \cdot \left(x \cdot 0.0008327945\right)\right) + \mathsf{fma}\left(0.0003579942, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left(x, x \cdot 0.7715471019, 1\right)\right)\right)\right)}{\mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right) + t_0 \cdot \left(0.0005064034 + 0.0001789971 \cdot \left(x \cdot x\right)\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (* x x) 4.0)))
   (/
    x
    (/
     (+
      (* t_0 (+ 0.0140005442 (* x (* x 0.0008327945))))
      (fma
       0.0003579942
       (pow (* x x) 6.0)
       (fma
        (pow x 4.0)
        0.2909738639
        (fma (pow x 6.0) 0.0694555761 (fma x (* x 0.7715471019) 1.0)))))
     (+
      (fma
       0.0424060604
       (pow x 4.0)
       (fma 0.0072644182 (pow x 6.0) (fma 0.1049934947 (* x x) 1.0)))
      (* t_0 (+ 0.0005064034 (* 0.0001789971 (* x x)))))))))

C

double code(double x) {
	double t_0 = pow((x * x), 4.0);
	return x / (((t_0 * (0.0140005442 + (x * (x * 0.0008327945)))) + fma(0.0003579942, pow((x * x), 6.0), fma(pow(x, 4.0), 0.2909738639, fma(pow(x, 6.0), 0.0694555761, fma(x, (x * 0.7715471019), 1.0))))) / (fma(0.0424060604, pow(x, 4.0), fma(0.0072644182, pow(x, 6.0), fma(0.1049934947, (x * x), 1.0))) + (t_0 * (0.0005064034 + (0.0001789971 * (x * x))))));
}

Julia

function code(x)
	t_0 = Float64(x * x) ^ 4.0
	return Float64(x / Float64(Float64(Float64(t_0 * Float64(0.0140005442 + Float64(x * Float64(x * 0.0008327945)))) + fma(0.0003579942, (Float64(x * x) ^ 6.0), fma((x ^ 4.0), 0.2909738639, fma((x ^ 6.0), 0.0694555761, fma(x, Float64(x * 0.7715471019), 1.0))))) / Float64(fma(0.0424060604, (x ^ 4.0), fma(0.0072644182, (x ^ 6.0), fma(0.1049934947, Float64(x * x), 1.0))) + Float64(t_0 * Float64(0.0005064034 + Float64(0.0001789971 * Float64(x * x)))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x * x), $MachinePrecision], 4.0], $MachinePrecision]}, N[(x / N[(N[(N[(t$95$0 * N[(0.0140005442 + N[(x * N[(x * 0.0008327945), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0003579942 * N[Power[N[(x * x), $MachinePrecision], 6.0], $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * 0.2909738639 + N[(N[Power[x, 6.0], $MachinePrecision] * 0.0694555761 + N[(x * N[(x * 0.7715471019), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.0424060604 * N[Power[x, 4.0], $MachinePrecision] + N[(0.0072644182 * N[Power[x, 6.0], $MachinePrecision] + N[(0.1049934947 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(0.0005064034 + N[(0.0001789971 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ t_2 := t_0 \cdot t_0\\ t_3 := \left(x \cdot x\right) \cdot t_2\\ x \cdot \frac{\left(\left(\left(0.1049934947 \cdot \left(x \cdot x\right) + 1\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot t_1\right) + \left(0.0005064034 \cdot t_2 + 0.0001789971 \cdot t_3\right)}{\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + 0.2909738639 \cdot t_0\right) + \left(0.0694555761 \cdot t_1 + 0.0140005442 \cdot t_2\right)\right) + \left(0.0008327945 \cdot t_3 + 0.0003579942 \cdot \left(t_0 \cdot t_2\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x))))
        (t_1 (* (* x x) t_0))
        (t_2 (* t_0 t_0))
        (t_3 (* (* x x) t_2)))
   (*
    x
    (/
     (+
      (+
       (+ (+ (* 0.1049934947 (* x x)) 1.0) (* 0.0424060604 t_0))
       (* 0.0072644182 t_1))
      (+ (* 0.0005064034 t_2) (* 0.0001789971 t_3)))
     (+
      (+
       (+ (+ 1.0 (* (* x x) 0.7715471019)) (* 0.2909738639 t_0))
       (+ (* 0.0694555761 t_1) (* 0.0140005442 t_2)))
      (+ (* 0.0008327945 t_3) (* 0.0003579942 (* t_0 t_2))))))))

C

double code(double x) {
	double t_0 = x * (x * (x * x));
	double t_1 = (x * x) * t_0;
	double t_2 = t_0 * t_0;
	double t_3 = (x * x) * t_2;
	return x * ((((((0.1049934947 * (x * x)) + 1.0) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + ((0.0005064034 * t_2) + (0.0001789971 * t_3))) / ((((1.0 + ((x * x) * 0.7715471019)) + (0.2909738639 * t_0)) + ((0.0694555761 * t_1) + (0.0140005442 * t_2))) + ((0.0008327945 * t_3) + (0.0003579942 * (t_0 * t_2)))));
}

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 = (x * x) * t_0
    t_2 = t_0 * t_0
    t_3 = (x * x) * t_2
    code = x * ((((((0.1049934947d0 * (x * x)) + 1.0d0) + (0.0424060604d0 * t_0)) + (0.0072644182d0 * t_1)) + ((0.0005064034d0 * t_2) + (0.0001789971d0 * t_3))) / ((((1.0d0 + ((x * x) * 0.7715471019d0)) + (0.2909738639d0 * t_0)) + ((0.0694555761d0 * t_1) + (0.0140005442d0 * t_2))) + ((0.0008327945d0 * t_3) + (0.0003579942d0 * (t_0 * t_2)))))
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (x * x));
	double t_1 = (x * x) * t_0;
	double t_2 = t_0 * t_0;
	double t_3 = (x * x) * t_2;
	return x * ((((((0.1049934947 * (x * x)) + 1.0) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + ((0.0005064034 * t_2) + (0.0001789971 * t_3))) / ((((1.0 + ((x * x) * 0.7715471019)) + (0.2909738639 * t_0)) + ((0.0694555761 * t_1) + (0.0140005442 * t_2))) + ((0.0008327945 * t_3) + (0.0003579942 * (t_0 * t_2)))));
}

Python

def code(x):
	t_0 = x * (x * (x * x))
	t_1 = (x * x) * t_0
	t_2 = t_0 * t_0
	t_3 = (x * x) * t_2
	return x * ((((((0.1049934947 * (x * x)) + 1.0) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + ((0.0005064034 * t_2) + (0.0001789971 * t_3))) / ((((1.0 + ((x * x) * 0.7715471019)) + (0.2909738639 * t_0)) + ((0.0694555761 * t_1) + (0.0140005442 * t_2))) + ((0.0008327945 * t_3) + (0.0003579942 * (t_0 * t_2)))))

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	t_1 = Float64(Float64(x * x) * t_0)
	t_2 = Float64(t_0 * t_0)
	t_3 = Float64(Float64(x * x) * t_2)
	return Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(0.1049934947 * Float64(x * x)) + 1.0) + Float64(0.0424060604 * t_0)) + Float64(0.0072644182 * t_1)) + Float64(Float64(0.0005064034 * t_2) + Float64(0.0001789971 * t_3))) / Float64(Float64(Float64(Float64(1.0 + Float64(Float64(x * x) * 0.7715471019)) + Float64(0.2909738639 * t_0)) + Float64(Float64(0.0694555761 * t_1) + Float64(0.0140005442 * t_2))) + Float64(Float64(0.0008327945 * t_3) + Float64(0.0003579942 * Float64(t_0 * t_2))))))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * (x * x));
	t_1 = (x * x) * t_0;
	t_2 = t_0 * t_0;
	t_3 = (x * x) * t_2;
	tmp = x * ((((((0.1049934947 * (x * x)) + 1.0) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + ((0.0005064034 * t_2) + (0.0001789971 * t_3))) / ((((1.0 + ((x * x) * 0.7715471019)) + (0.2909738639 * t_0)) + ((0.0694555761 * t_1) + (0.0140005442 * t_2))) + ((0.0008327945 * t_3) + (0.0003579942 * (t_0 * t_2)))));
end

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 5: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ t_2 := \left(x \cdot x\right) \cdot t_1\\ t_3 := \left(x \cdot x\right) \cdot t_2\\ x \cdot \frac{\left(\left(\left(\left(0.1049934947 \cdot \left(x \cdot x\right) + 1\right) + 0.0424060604 \cdot t_0\right) + 0.0072644182 \cdot t_1\right) + 0.0005064034 \cdot t_2\right) + 0.0001789971 \cdot t_3}{\left(\left(\left(\left(0.2909738639 \cdot t_0 + \left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right)\right) + 0.0694555761 \cdot t_1\right) + 0.0140005442 \cdot t_2\right) + 0.0008327945 \cdot t_3\right) + 0.0003579942 \cdot \left(\left(x \cdot x\right) \cdot t_3\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x)))
        (t_1 (* (* x x) t_0))
        (t_2 (* (* x x) t_1))
        (t_3 (* (* x x) t_2)))
   (*
    x
    (/
     (+
      (+
       (+
        (+ (+ (* 0.1049934947 (* x x)) 1.0) (* 0.0424060604 t_0))
        (* 0.0072644182 t_1))
       (* 0.0005064034 t_2))
      (* 0.0001789971 t_3))
     (+
      (+
       (+
        (+
         (+ (* 0.2909738639 t_0) (+ 1.0 (* (* x x) 0.7715471019)))
         (* 0.0694555761 t_1))
        (* 0.0140005442 t_2))
       (* 0.0008327945 t_3))
      (* 0.0003579942 (* (* x x) t_3)))))))

C

double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = (x * x) * t_0;
	double t_2 = (x * x) * t_1;
	double t_3 = (x * x) * t_2;
	return x * (((((((0.1049934947 * (x * x)) + 1.0) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((0.2909738639 * t_0) + (1.0 + ((x * x) * 0.7715471019))) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + (0.0003579942 * ((x * x) * t_3))));
}

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 = (x * x) * t_0
    t_2 = (x * x) * t_1
    t_3 = (x * x) * t_2
    code = x * (((((((0.1049934947d0 * (x * x)) + 1.0d0) + (0.0424060604d0 * t_0)) + (0.0072644182d0 * t_1)) + (0.0005064034d0 * t_2)) + (0.0001789971d0 * t_3)) / ((((((0.2909738639d0 * t_0) + (1.0d0 + ((x * x) * 0.7715471019d0))) + (0.0694555761d0 * t_1)) + (0.0140005442d0 * t_2)) + (0.0008327945d0 * t_3)) + (0.0003579942d0 * ((x * x) * t_3))))
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = (x * x) * t_0;
	double t_2 = (x * x) * t_1;
	double t_3 = (x * x) * t_2;
	return x * (((((((0.1049934947 * (x * x)) + 1.0) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((0.2909738639 * t_0) + (1.0 + ((x * x) * 0.7715471019))) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + (0.0003579942 * ((x * x) * t_3))));
}

Python

def code(x):
	t_0 = (x * x) * (x * x)
	t_1 = (x * x) * t_0
	t_2 = (x * x) * t_1
	t_3 = (x * x) * t_2
	return x * (((((((0.1049934947 * (x * x)) + 1.0) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((0.2909738639 * t_0) + (1.0 + ((x * x) * 0.7715471019))) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + (0.0003579942 * ((x * x) * t_3))))

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	t_1 = Float64(Float64(x * x) * t_0)
	t_2 = Float64(Float64(x * x) * t_1)
	t_3 = Float64(Float64(x * x) * t_2)
	return Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.1049934947 * Float64(x * x)) + 1.0) + 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(0.2909738639 * t_0) + Float64(1.0 + Float64(Float64(x * x) * 0.7715471019))) + Float64(0.0694555761 * t_1)) + Float64(0.0140005442 * t_2)) + Float64(0.0008327945 * t_3)) + Float64(0.0003579942 * Float64(Float64(x * x) * t_3)))))
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * (x * x);
	t_1 = (x * x) * t_0;
	t_2 = (x * x) * t_1;
	t_3 = (x * x) * t_2;
	tmp = x * (((((((0.1049934947 * (x * x)) + 1.0) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((0.2909738639 * t_0) + (1.0 + ((x * x) * 0.7715471019))) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + (0.0003579942 * ((x * x) * t_3))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision]}, N[(x * N[(N[(N[(N[(N[(N[(N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $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[(0.2909738639 * t$95$0), $MachinePrecision] + N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.7715471019), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.0003579942 * N[(N[(x * x), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 98.4%

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

2. Final simplification 98.4%

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

Alternative 6: 99.2% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \frac{\frac{0.10624017004622396}{x}}{x}\right)}\\ \mathbf{elif}\;x \leq 1.22:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot 0.6665536072}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-1.0056716002661497 + \left(x \cdot x\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (/
    x
    (+
     (* x (* x 2.0))
     (- -1.0056716002661497 (/ (/ 0.10624017004622396 x) x))))
   (if (<= x 1.22)
     (/ x (+ 1.0 (* (* x x) 0.6665536072)))
     (/ x (+ -1.0056716002661497 (* (* x x) 2.0))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = x / ((x * (x * 2.0)) + (-1.0056716002661497 - ((0.10624017004622396 / x) / x)));
	} else if (x <= 1.22) {
		tmp = x / (1.0 + ((x * x) * 0.6665536072));
	} else {
		tmp = x / (-1.0056716002661497 + ((x * x) * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.25d0)) then
        tmp = x / ((x * (x * 2.0d0)) + ((-1.0056716002661497d0) - ((0.10624017004622396d0 / x) / x)))
    else if (x <= 1.22d0) then
        tmp = x / (1.0d0 + ((x * x) * 0.6665536072d0))
    else
        tmp = x / ((-1.0056716002661497d0) + ((x * x) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = x / ((x * (x * 2.0)) + (-1.0056716002661497 - ((0.10624017004622396 / x) / x)));
	} else if (x <= 1.22) {
		tmp = x / (1.0 + ((x * x) * 0.6665536072));
	} else {
		tmp = x / (-1.0056716002661497 + ((x * x) * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = x / ((x * (x * 2.0)) + (-1.0056716002661497 - ((0.10624017004622396 / x) / x)))
	elif x <= 1.22:
		tmp = x / (1.0 + ((x * x) * 0.6665536072))
	else:
		tmp = x / (-1.0056716002661497 + ((x * x) * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = Float64(x / Float64(Float64(x * Float64(x * 2.0)) + Float64(-1.0056716002661497 - Float64(Float64(0.10624017004622396 / x) / x))));
	elseif (x <= 1.22)
		tmp = Float64(x / Float64(1.0 + Float64(Float64(x * x) * 0.6665536072)));
	else
		tmp = Float64(x / Float64(-1.0056716002661497 + Float64(Float64(x * x) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = x / ((x * (x * 2.0)) + (-1.0056716002661497 - ((0.10624017004622396 / x) / x)));
	elseif (x <= 1.22)
		tmp = x / (1.0 + ((x * x) * 0.6665536072));
	else
		tmp = x / (-1.0056716002661497 + ((x * x) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[(x / N[(N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0056716002661497 - N[(N[(0.10624017004622396 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22], N[(x / N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(-1.0056716002661497 + N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 90.3%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in x around inf 83.7%

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

      1. associate--r+ 83.7%

         \[\leadsto \frac{x}{\color{blue}{\left(2 \cdot {x}^{2} - 1.0056716002661497\right) - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}}} \]

      2. unpow2 83.7%

         \[\leadsto \frac{x}{\left(2 \cdot \color{blue}{\left(x \cdot x\right)} - 1.0056716002661497\right) - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}} \]

      3. *-commutative 83.7%

         \[\leadsto \frac{x}{\left(\color{blue}{\left(x \cdot x\right) \cdot 2} - 1.0056716002661497\right) - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}} \]

      4. associate-*l* 83.7%

         \[\leadsto \frac{x}{\left(\color{blue}{x \cdot \left(x \cdot 2\right)} - 1.0056716002661497\right) - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}} \]

      5. fma-neg 83.7%

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, x \cdot 2, -1.0056716002661497\right)} - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}} \]

      6. metadata-eval 83.7%

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, x \cdot 2, \color{blue}{-1.0056716002661497}\right) - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}} \]

      7. unpow2 83.7%

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, x \cdot 2, -1.0056716002661497\right) - 0.10624017004622396 \cdot \frac{1}{\color{blue}{x \cdot x}}} \]

      8. associate-*r/ 83.7%

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, x \cdot 2, -1.0056716002661497\right) - \color{blue}{\frac{0.10624017004622396 \cdot 1}{x \cdot x}}} \]

      9. metadata-eval 83.7%

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, x \cdot 2, -1.0056716002661497\right) - \frac{\color{blue}{0.10624017004622396}}{x \cdot x}} \]

   6. Simplified 83.7%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, x \cdot 2, -1.0056716002661497\right) - \frac{0.10624017004622396}{x \cdot x}}} \]
   7. Step-by-step derivation

      1. fma-udef 83.7%

         \[\leadsto \frac{x}{\color{blue}{\left(x \cdot \left(x \cdot 2\right) + -1.0056716002661497\right)} - \frac{0.10624017004622396}{x \cdot x}} \]

   8. Applied egg-rr 83.7%

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

      1. associate--l+ 83.7%

         \[\leadsto \frac{x}{\color{blue}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \frac{0.10624017004622396}{x \cdot x}\right)}} \]

      2. div-inv 83.7%

         \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \color{blue}{0.10624017004622396 \cdot \frac{1}{x \cdot x}}\right)} \]

      3. cancel-sign-sub-inv 83.7%

         \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \color{blue}{\left(-1.0056716002661497 + \left(-0.10624017004622396\right) \cdot \frac{1}{x \cdot x}\right)}} \]

      4. metadata-eval 83.7%

         \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 + \color{blue}{-0.10624017004622396} \cdot \frac{1}{x \cdot x}\right)} \]

      5. pow2 83.7%

         \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 + -0.10624017004622396 \cdot \frac{1}{\color{blue}{{x}^{2}}}\right)} \]

      6. pow-flip 83.7%

         \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 + -0.10624017004622396 \cdot \color{blue}{{x}^{\left(-2\right)}}\right)} \]

      7. metadata-eval 83.7%

         \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 + -0.10624017004622396 \cdot {x}^{\color{blue}{-2}}\right)} \]

   10. Applied egg-rr 83.7%

       \[\leadsto \frac{x}{\color{blue}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 + -0.10624017004622396 \cdot {x}^{-2}\right)}} \]

   11. Taylor expanded in x around 0 83.7%

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

       1. distribute-neg-in 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \color{blue}{\left(\left(-1.0056716002661497\right) + \left(-0.10624017004622396 \cdot \frac{1}{{x}^{2}}\right)\right)}} \]

       2. metadata-eval 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(\color{blue}{-1.0056716002661497} + \left(-0.10624017004622396 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

       3. unsub-neg 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \color{blue}{\left(-1.0056716002661497 - 0.10624017004622396 \cdot \frac{1}{{x}^{2}}\right)}} \]

       4. associate-*r/ 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \color{blue}{\frac{0.10624017004622396 \cdot 1}{{x}^{2}}}\right)} \]

       5. metadata-eval 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \frac{\color{blue}{0.10624017004622396}}{{x}^{2}}\right)} \]

       6. unpow2 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \frac{0.10624017004622396}{\color{blue}{x \cdot x}}\right)} \]

       7. associate-/r* 83.7%

          \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \color{blue}{\frac{\frac{0.10624017004622396}{x}}{x}}\right)} \]

   13. Simplified 83.7%

       \[\leadsto \frac{x}{x \cdot \left(x \cdot 2\right) + \color{blue}{\left(-1.0056716002661497 - \frac{\frac{0.10624017004622396}{x}}{x}\right)}} \]

   if -1.25 < x < 1.21999999999999997

   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%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. unpow2 99.3%

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

   6. Simplified 99.3%

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

   if 1.21999999999999997 < x

   1. Initial program 79.7%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in x around inf 99.5%

      \[\leadsto \frac{x}{\color{blue}{2 \cdot {x}^{2} - 1.0056716002661497}} \]
   5. Step-by-step derivation

      1. fma-neg 99.5%

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(2, {x}^{2}, -1.0056716002661497\right)}} \]

      2. unpow2 99.5%

         \[\leadsto \frac{x}{\mathsf{fma}\left(2, \color{blue}{x \cdot x}, -1.0056716002661497\right)} \]

      3. metadata-eval 99.5%

         \[\leadsto \frac{x}{\mathsf{fma}\left(2, x \cdot x, \color{blue}{-1.0056716002661497}\right)} \]

   6. Simplified 99.5%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(2, x \cdot x, -1.0056716002661497\right)}} \]
   7. Step-by-step derivation

      1. fma-udef 99.5%

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

   8. Applied egg-rr 99.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\frac{x}{x \cdot \left(x \cdot 2\right) + \left(-1.0056716002661497 - \frac{\frac{0.10624017004622396}{x}}{x}\right)}\\ \mathbf{elif}\;x \leq 1.22:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot 0.6665536072}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-1.0056716002661497 + \left(x \cdot x\right) \cdot 2}\\ \end{array} \]

Alternative 7: 98.7% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 0.88\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot 0.6665536072}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.4) (not (<= x 0.88)))
   (/ 0.5 x)
   (/ x (+ 1.0 (* (* x x) 0.6665536072)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.4) || !(x <= 0.88)) {
		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 <= (-1.4d0)) .or. (.not. (x <= 0.88d0))) 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 <= -1.4) || !(x <= 0.88)) {
		tmp = 0.5 / x;
	} else {
		tmp = x / (1.0 + ((x * x) * 0.6665536072));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.4) or not (x <= 0.88):
		tmp = 0.5 / x
	else:
		tmp = x / (1.0 + ((x * x) * 0.6665536072))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.4) || !(x <= 0.88))
		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 <= -1.4) || ~((x <= 0.88)))
		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, -1.4], N[Not[LessEqual[x, 0.88]], $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 < -1.3999999999999999 or 0.880000000000000004 < x

   1. Initial program 84.2%

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

   3. Simplified 84.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, 1\right)\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in x around inf 88.7%

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

   if -1.3999999999999999 < x < 0.880000000000000004

   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%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. unpow2 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 0.88\right):\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot 0.6665536072}\\ \end{array} \]

Alternative 8: 99.2% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \lor \neg \left(x \leq 1.22\right):\\ \;\;\;\;\frac{x}{-1.0056716002661497 + \left(x \cdot x\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot 0.6665536072}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.25) (not (<= x 1.22)))
   (/ x (+ -1.0056716002661497 (* (* x x) 2.0)))
   (/ x (+ 1.0 (* (* x x) 0.6665536072)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.25) || !(x <= 1.22)) {
		tmp = x / (-1.0056716002661497 + ((x * x) * 2.0));
	} 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 <= (-1.25d0)) .or. (.not. (x <= 1.22d0))) then
        tmp = x / ((-1.0056716002661497d0) + ((x * x) * 2.0d0))
    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 <= -1.25) || !(x <= 1.22)) {
		tmp = x / (-1.0056716002661497 + ((x * x) * 2.0));
	} else {
		tmp = x / (1.0 + ((x * x) * 0.6665536072));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.25) or not (x <= 1.22):
		tmp = x / (-1.0056716002661497 + ((x * x) * 2.0))
	else:
		tmp = x / (1.0 + ((x * x) * 0.6665536072))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.25) || !(x <= 1.22))
		tmp = Float64(x / Float64(-1.0056716002661497 + Float64(Float64(x * x) * 2.0)));
	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 <= -1.25) || ~((x <= 1.22)))
		tmp = x / (-1.0056716002661497 + ((x * x) * 2.0));
	else
		tmp = x / (1.0 + ((x * x) * 0.6665536072));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.25], N[Not[LessEqual[x, 1.22]], $MachinePrecision]], N[(x / N[(-1.0056716002661497 + N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $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 < -1.25 or 1.21999999999999997 < x

   1. Initial program 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in x around inf 92.2%

      \[\leadsto \frac{x}{\color{blue}{2 \cdot {x}^{2} - 1.0056716002661497}} \]
   5. Step-by-step derivation

      1. fma-neg 92.2%

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(2, {x}^{2}, -1.0056716002661497\right)}} \]

      2. unpow2 92.2%

         \[\leadsto \frac{x}{\mathsf{fma}\left(2, \color{blue}{x \cdot x}, -1.0056716002661497\right)} \]

      3. metadata-eval 92.2%

         \[\leadsto \frac{x}{\mathsf{fma}\left(2, x \cdot x, \color{blue}{-1.0056716002661497}\right)} \]

   6. Simplified 92.2%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(2, x \cdot x, -1.0056716002661497\right)}} \]
   7. Step-by-step derivation

      1. fma-udef 92.2%

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

   8. Applied egg-rr 92.2%

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

   if -1.25 < x < 1.21999999999999997

   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%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. unpow2 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \lor \neg \left(x \leq 1.22\right):\\ \;\;\;\;\frac{x}{-1.0056716002661497 + \left(x \cdot x\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot 0.6665536072}\\ \end{array} \]

Alternative 9: 98.7% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.78)
   (/ 0.5 x)
   (if (<= x 0.8) (* x (+ 1.0 (* (* x x) -0.6665536072))) (/ 0.5 x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.78) {
		tmp = 0.5 / x;
	} else if (x <= 0.8) {
		tmp = x * (1.0 + ((x * x) * -0.6665536072));
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.78d0)) then
        tmp = 0.5d0 / x
    else if (x <= 0.8d0) then
        tmp = x * (1.0d0 + ((x * x) * (-0.6665536072d0)))
    else
        tmp = 0.5d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.78) {
		tmp = 0.5 / x;
	} else if (x <= 0.8) {
		tmp = x * (1.0 + ((x * x) * -0.6665536072));
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.78:
		tmp = 0.5 / x
	elif x <= 0.8:
		tmp = x * (1.0 + ((x * x) * -0.6665536072))
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.78)
		tmp = Float64(0.5 / x);
	elseif (x <= 0.8)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.6665536072)));
	else
		tmp = Float64(0.5 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.78)
		tmp = 0.5 / x;
	elseif (x <= 0.8)
		tmp = x * (1.0 + ((x * x) * -0.6665536072));
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.78], N[(0.5 / x), $MachinePrecision], If[LessEqual[x, 0.8], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.78000000000000003 or 0.80000000000000004 < x

   1. Initial program 84.2%

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

   3. Simplified 84.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, 1\right)\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in x around inf 88.7%

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

   if -0.78000000000000003 < x < 0.80000000000000004

   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 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.6665536072\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 10: 98.2% 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 84.2%

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

   3. Simplified 84.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, 1\right)\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in x around inf 88.7%

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

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, 1\right)\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in x around 0 98.6%

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

4. Final simplification 97.6%

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

Alternative 11: 89.9% 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 98.4%

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

3. Simplified 98.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left({x}^{4}, 0.2909738639, 1\right)\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in x around 0 89.5%

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

5. Final simplification 89.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023278 
(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))