Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Percentage Accurate: 51.6% → 79.7%

Time: 5.6s

Alternatives: 6

Speedup: 2.6×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t_0}{x \cdot x + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 51.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t_0}{x \cdot x + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 79.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := x \cdot x - t_0\\ t_2 := -1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-161}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 100000000000:\\ \;\;\;\;\frac{t_1}{x \cdot x + t_0}\\ \mathbf{elif}\;x \cdot x \leq 10^{+119}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(y \cdot 4, y, {x}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (- (* x x) t_0))
        (t_2 (+ -1.0 (/ 0.5 (/ (/ y x) (/ x y))))))
   (if (<= (* x x) 2e-161)
     t_2
     (if (<= (* x x) 100000000000.0)
       (/ t_1 (+ (* x x) t_0))
       (if (<= (* x x) 1e+119)
         t_2
         (if (<= (* x x) 5e+295)
           (/ t_1 (fma (* y 4.0) y (pow x 2.0)))
           (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = (x * x) - t_0;
	double t_2 = -1.0 + (0.5 / ((y / x) / (x / y)));
	double tmp;
	if ((x * x) <= 2e-161) {
		tmp = t_2;
	} else if ((x * x) <= 100000000000.0) {
		tmp = t_1 / ((x * x) + t_0);
	} else if ((x * x) <= 1e+119) {
		tmp = t_2;
	} else if ((x * x) <= 5e+295) {
		tmp = t_1 / fma((y * 4.0), y, pow(x, 2.0));
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(x * x) - t_0)
	t_2 = Float64(-1.0 + Float64(0.5 / Float64(Float64(y / x) / Float64(x / y))))
	tmp = 0.0
	if (Float64(x * x) <= 2e-161)
		tmp = t_2;
	elseif (Float64(x * x) <= 100000000000.0)
		tmp = Float64(t_1 / Float64(Float64(x * x) + t_0));
	elseif (Float64(x * x) <= 1e+119)
		tmp = t_2;
	elseif (Float64(x * x) <= 5e+295)
		tmp = Float64(t_1 / fma(Float64(y * 4.0), y, (x ^ 2.0)));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 + N[(0.5 / N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e-161], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 100000000000.0], N[(t$95$1 / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+119], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 5e+295], N[(t$95$1 / N[(N[(y * 4.0), $MachinePrecision] * y + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 2.00000000000000006e-161 or 1e11 < (*.f64 x x) < 9.99999999999999944e118

   1. Initial program 54.1%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. +-commutative 54.1%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right) \cdot y + x \cdot x}} \]

      2. fma-def 54.1%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(y \cdot 4, y, x \cdot x\right)}} \]

      3. pow2 54.1%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(y \cdot 4, y, \color{blue}{{x}^{2}}\right)} \]

   3. Applied egg-rr 54.1%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(y \cdot 4, y, {x}^{2}\right)}} \]

   4. Taylor expanded in x around 0 81.1%

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

      1. sub-neg 81.1%

         \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} + \left(-1\right)} \]

      2. metadata-eval 81.1%

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

      3. +-commutative 81.1%

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

      4. associate-*r/ 81.1%

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

      5. associate-/l* 81.1%

         \[\leadsto -1 + \color{blue}{\frac{0.5}{\frac{{y}^{2}}{{x}^{2}}}} \]

      6. unpow2 81.1%

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

      7. unpow2 81.1%

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

      8. times-frac 87.8%

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

      9. unpow2 87.8%

         \[\leadsto -1 + \frac{0.5}{\color{blue}{{\left(\frac{y}{x}\right)}^{2}}} \]

   6. Simplified 87.8%

      \[\leadsto \color{blue}{-1 + \frac{0.5}{{\left(\frac{y}{x}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. pow2 87.8%

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

      2. clear-num 87.8%

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

      3. un-div-inv 87.8%

         \[\leadsto -1 + \frac{0.5}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   8. Applied egg-rr 87.8%

      \[\leadsto -1 + \frac{0.5}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   if 2.00000000000000006e-161 < (*.f64 x x) < 1e11

   1. Initial program 78.4%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 9.99999999999999944e118 < (*.f64 x x) < 4.99999999999999991e295

   1. Initial program 86.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. +-commutative 86.0%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right) \cdot y + x \cdot x}} \]

      2. fma-def 86.0%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(y \cdot 4, y, x \cdot x\right)}} \]

      3. pow2 86.0%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(y \cdot 4, y, \color{blue}{{x}^{2}}\right)} \]

   3. Applied egg-rr 86.0%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(y \cdot 4, y, {x}^{2}\right)}} \]

   if 4.99999999999999991e295 < (*.f64 x x)

   1. Initial program 1.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in y around 0 71.7%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. pow2 71.7%

         \[\leadsto 1 + -8 \cdot \frac{{y}^{2}}{\color{blue}{x \cdot x}} \]

      2. unpow2 71.7%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{x \cdot x} \]

      3. times-frac 86.0%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   4. Applied egg-rr 86.0%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-161}:\\ \;\;\;\;-1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{elif}\;x \cdot x \leq 100000000000:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 10^{+119}:\\ \;\;\;\;-1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(y \cdot 4, y, {x}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 2: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t_0}{x \cdot x + t_0}\\ t_2 := -1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-161}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 100000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 10^{+119}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+295}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0)))
        (t_2 (+ -1.0 (/ 0.5 (/ (/ y x) (/ x y))))))
   (if (<= (* x x) 2e-161)
     t_2
     (if (<= (* x x) 100000000000.0)
       t_1
       (if (<= (* x x) 1e+119)
         t_2
         (if (<= (* x x) 5e+295) t_1 (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double t_2 = -1.0 + (0.5 / ((y / x) / (x / y)));
	double tmp;
	if ((x * x) <= 2e-161) {
		tmp = t_2;
	} else if ((x * x) <= 100000000000.0) {
		tmp = t_1;
	} else if ((x * x) <= 1e+119) {
		tmp = t_2;
	} else if ((x * x) <= 5e+295) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    t_2 = (-1.0d0) + (0.5d0 / ((y / x) / (x / y)))
    if ((x * x) <= 2d-161) then
        tmp = t_2
    else if ((x * x) <= 100000000000.0d0) then
        tmp = t_1
    else if ((x * x) <= 1d+119) then
        tmp = t_2
    else if ((x * x) <= 5d+295) then
        tmp = t_1
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double t_2 = -1.0 + (0.5 / ((y / x) / (x / y)));
	double tmp;
	if ((x * x) <= 2e-161) {
		tmp = t_2;
	} else if ((x * x) <= 100000000000.0) {
		tmp = t_1;
	} else if ((x * x) <= 1e+119) {
		tmp = t_2;
	} else if ((x * x) <= 5e+295) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	t_2 = -1.0 + (0.5 / ((y / x) / (x / y)))
	tmp = 0
	if (x * x) <= 2e-161:
		tmp = t_2
	elif (x * x) <= 100000000000.0:
		tmp = t_1
	elif (x * x) <= 1e+119:
		tmp = t_2
	elif (x * x) <= 5e+295:
		tmp = t_1
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	t_2 = Float64(-1.0 + Float64(0.5 / Float64(Float64(y / x) / Float64(x / y))))
	tmp = 0.0
	if (Float64(x * x) <= 2e-161)
		tmp = t_2;
	elseif (Float64(x * x) <= 100000000000.0)
		tmp = t_1;
	elseif (Float64(x * x) <= 1e+119)
		tmp = t_2;
	elseif (Float64(x * x) <= 5e+295)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	t_2 = -1.0 + (0.5 / ((y / x) / (x / y)));
	tmp = 0.0;
	if ((x * x) <= 2e-161)
		tmp = t_2;
	elseif ((x * x) <= 100000000000.0)
		tmp = t_1;
	elseif ((x * x) <= 1e+119)
		tmp = t_2;
	elseif ((x * x) <= 5e+295)
		tmp = t_1;
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 + N[(0.5 / N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e-161], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 100000000000.0], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 1e+119], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 5e+295], t$95$1, N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.00000000000000006e-161 or 1e11 < (*.f64 x x) < 9.99999999999999944e118

   1. Initial program 54.1%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. +-commutative 54.1%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right) \cdot y + x \cdot x}} \]

      2. fma-def 54.1%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(y \cdot 4, y, x \cdot x\right)}} \]

      3. pow2 54.1%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(y \cdot 4, y, \color{blue}{{x}^{2}}\right)} \]

   3. Applied egg-rr 54.1%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(y \cdot 4, y, {x}^{2}\right)}} \]

   4. Taylor expanded in x around 0 81.1%

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

      1. sub-neg 81.1%

         \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} + \left(-1\right)} \]

      2. metadata-eval 81.1%

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

      3. +-commutative 81.1%

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

      4. associate-*r/ 81.1%

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

      5. associate-/l* 81.1%

         \[\leadsto -1 + \color{blue}{\frac{0.5}{\frac{{y}^{2}}{{x}^{2}}}} \]

      6. unpow2 81.1%

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

      7. unpow2 81.1%

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

      8. times-frac 87.8%

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

      9. unpow2 87.8%

         \[\leadsto -1 + \frac{0.5}{\color{blue}{{\left(\frac{y}{x}\right)}^{2}}} \]

   6. Simplified 87.8%

      \[\leadsto \color{blue}{-1 + \frac{0.5}{{\left(\frac{y}{x}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. pow2 87.8%

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

      2. clear-num 87.8%

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

      3. un-div-inv 87.8%

         \[\leadsto -1 + \frac{0.5}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   8. Applied egg-rr 87.8%

      \[\leadsto -1 + \frac{0.5}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   if 2.00000000000000006e-161 < (*.f64 x x) < 1e11 or 9.99999999999999944e118 < (*.f64 x x) < 4.99999999999999991e295

   1. Initial program 82.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 4.99999999999999991e295 < (*.f64 x x)

   1. Initial program 1.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in y around 0 71.7%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. pow2 71.7%

         \[\leadsto 1 + -8 \cdot \frac{{y}^{2}}{\color{blue}{x \cdot x}} \]

      2. unpow2 71.7%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{x \cdot x} \]

      3. times-frac 86.0%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   4. Applied egg-rr 86.0%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-161}:\\ \;\;\;\;-1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{elif}\;x \cdot x \leq 100000000000:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 10^{+119}:\\ \;\;\;\;-1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 3: 64.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-18} \lor \neg \left(x \leq 1.5 \cdot 10^{-11}\right) \land x \leq 2.8 \cdot 10^{+59}:\\ \;\;\;\;-1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 1.55e-18) (and (not (<= x 1.5e-11)) (<= x 2.8e+59)))
   (+ -1.0 (/ 0.5 (/ (/ y x) (/ x y))))
   1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 1.55e-18) || (!(x <= 1.5e-11) && (x <= 2.8e+59))) {
		tmp = -1.0 + (0.5 / ((y / x) / (x / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= 1.55d-18) .or. (.not. (x <= 1.5d-11)) .and. (x <= 2.8d+59)) then
        tmp = (-1.0d0) + (0.5d0 / ((y / x) / (x / y)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= 1.55e-18) || (!(x <= 1.5e-11) && (x <= 2.8e+59))) {
		tmp = -1.0 + (0.5 / ((y / x) / (x / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 1.55e-18) or (not (x <= 1.5e-11) and (x <= 2.8e+59)):
		tmp = -1.0 + (0.5 / ((y / x) / (x / y)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 1.55e-18) || (!(x <= 1.5e-11) && (x <= 2.8e+59)))
		tmp = Float64(-1.0 + Float64(0.5 / Float64(Float64(y / x) / Float64(x / y))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 1.55e-18) || (~((x <= 1.5e-11)) && (x <= 2.8e+59)))
		tmp = -1.0 + (0.5 / ((y / x) / (x / y)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 1.55e-18], And[N[Not[LessEqual[x, 1.5e-11]], $MachinePrecision], LessEqual[x, 2.8e+59]]], N[(-1.0 + N[(0.5 / N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000003e-18 or 1.5e-11 < x < 2.7999999999999998e59

   1. Initial program 54.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. +-commutative 54.2%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right) \cdot y + x \cdot x}} \]

      2. fma-def 54.2%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(y \cdot 4, y, x \cdot x\right)}} \]

      3. pow2 54.2%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(y \cdot 4, y, \color{blue}{{x}^{2}}\right)} \]

   3. Applied egg-rr 54.2%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(y \cdot 4, y, {x}^{2}\right)}} \]

   4. Taylor expanded in x around 0 59.1%

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

      1. sub-neg 59.1%

         \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} + \left(-1\right)} \]

      2. metadata-eval 59.1%

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

      3. +-commutative 59.1%

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

      4. associate-*r/ 59.1%

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

      5. associate-/l* 59.1%

         \[\leadsto -1 + \color{blue}{\frac{0.5}{\frac{{y}^{2}}{{x}^{2}}}} \]

      6. unpow2 59.1%

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

      7. unpow2 59.1%

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

      8. times-frac 64.4%

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

      9. unpow2 64.4%

         \[\leadsto -1 + \frac{0.5}{\color{blue}{{\left(\frac{y}{x}\right)}^{2}}} \]

   6. Simplified 64.4%

      \[\leadsto \color{blue}{-1 + \frac{0.5}{{\left(\frac{y}{x}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. pow2 64.4%

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

      2. clear-num 64.4%

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

      3. un-div-inv 64.4%

         \[\leadsto -1 + \frac{0.5}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   8. Applied egg-rr 64.4%

      \[\leadsto -1 + \frac{0.5}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   if 1.55000000000000003e-18 < x < 1.5e-11 or 2.7999999999999998e59 < x

   1. Initial program 30.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around inf 81.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-18} \lor \neg \left(x \leq 1.5 \cdot 10^{-11}\right) \land x \leq 2.8 \cdot 10^{+59}:\\ \;\;\;\;-1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 64.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-22} \lor \neg \left(x \leq 1.5 \cdot 10^{-11}\right) \land x \leq 7.8 \cdot 10^{+61}:\\ \;\;\;\;-1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 3.6e-22) (and (not (<= x 1.5e-11)) (<= x 7.8e+61)))
   (+ -1.0 (/ 0.5 (/ (/ y x) (/ x y))))
   (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 3.6e-22) || (!(x <= 1.5e-11) && (x <= 7.8e+61))) {
		tmp = -1.0 + (0.5 / ((y / x) / (x / y)));
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= 3.6d-22) .or. (.not. (x <= 1.5d-11)) .and. (x <= 7.8d+61)) then
        tmp = (-1.0d0) + (0.5d0 / ((y / x) / (x / y)))
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= 3.6e-22) || (!(x <= 1.5e-11) && (x <= 7.8e+61))) {
		tmp = -1.0 + (0.5 / ((y / x) / (x / y)));
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 3.6e-22) or (not (x <= 1.5e-11) and (x <= 7.8e+61)):
		tmp = -1.0 + (0.5 / ((y / x) / (x / y)))
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 3.6e-22) || (!(x <= 1.5e-11) && (x <= 7.8e+61)))
		tmp = Float64(-1.0 + Float64(0.5 / Float64(Float64(y / x) / Float64(x / y))));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 3.6e-22) || (~((x <= 1.5e-11)) && (x <= 7.8e+61)))
		tmp = -1.0 + (0.5 / ((y / x) / (x / y)));
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 3.6e-22], And[N[Not[LessEqual[x, 1.5e-11]], $MachinePrecision], LessEqual[x, 7.8e+61]]], N[(-1.0 + N[(0.5 / N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5999999999999998e-22 or 1.5e-11 < x < 7.79999999999999975e61

   1. Initial program 54.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. +-commutative 54.0%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right) \cdot y + x \cdot x}} \]

      2. fma-def 54.0%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(y \cdot 4, y, x \cdot x\right)}} \]

      3. pow2 54.0%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\mathsf{fma}\left(y \cdot 4, y, \color{blue}{{x}^{2}}\right)} \]

   3. Applied egg-rr 54.0%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\mathsf{fma}\left(y \cdot 4, y, {x}^{2}\right)}} \]

   4. Taylor expanded in x around 0 59.4%

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

      1. sub-neg 59.4%

         \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} + \left(-1\right)} \]

      2. metadata-eval 59.4%

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

      3. +-commutative 59.4%

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

      4. associate-*r/ 59.4%

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

      5. associate-/l* 59.4%

         \[\leadsto -1 + \color{blue}{\frac{0.5}{\frac{{y}^{2}}{{x}^{2}}}} \]

      6. unpow2 59.4%

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

      7. unpow2 59.4%

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

      8. times-frac 64.7%

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

      9. unpow2 64.7%

         \[\leadsto -1 + \frac{0.5}{\color{blue}{{\left(\frac{y}{x}\right)}^{2}}} \]

   6. Simplified 64.7%

      \[\leadsto \color{blue}{-1 + \frac{0.5}{{\left(\frac{y}{x}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. pow2 64.7%

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

      2. clear-num 64.7%

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

      3. un-div-inv 64.7%

         \[\leadsto -1 + \frac{0.5}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   8. Applied egg-rr 64.7%

      \[\leadsto -1 + \frac{0.5}{\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}} \]

   if 3.5999999999999998e-22 < x < 1.5e-11 or 7.79999999999999975e61 < x

   1. Initial program 32.1%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in y around 0 73.7%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. pow2 73.7%

         \[\leadsto 1 + -8 \cdot \frac{{y}^{2}}{\color{blue}{x \cdot x}} \]

      2. unpow2 73.7%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{x \cdot x} \]

      3. times-frac 83.3%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   4. Applied egg-rr 83.3%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-22} \lor \neg \left(x \leq 1.5 \cdot 10^{-11}\right) \land x \leq 7.8 \cdot 10^{+61}:\\ \;\;\;\;-1 + \frac{0.5}{\frac{\frac{y}{x}}{\frac{x}{y}}}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 5: 62.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-18}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 10^{-11}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+60}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.55e-18) -1.0 (if (<= x 1e-11) 1.0 (if (<= x 1.9e+60) -1.0 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.55e-18) {
		tmp = -1.0;
	} else if (x <= 1e-11) {
		tmp = 1.0;
	} else if (x <= 1.9e+60) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.55d-18) then
        tmp = -1.0d0
    else if (x <= 1d-11) then
        tmp = 1.0d0
    else if (x <= 1.9d+60) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.55e-18) {
		tmp = -1.0;
	} else if (x <= 1e-11) {
		tmp = 1.0;
	} else if (x <= 1.9e+60) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.55e-18:
		tmp = -1.0
	elif x <= 1e-11:
		tmp = 1.0
	elif x <= 1.9e+60:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.55e-18)
		tmp = -1.0;
	elseif (x <= 1e-11)
		tmp = 1.0;
	elseif (x <= 1.9e+60)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.55e-18)
		tmp = -1.0;
	elseif (x <= 1e-11)
		tmp = 1.0;
	elseif (x <= 1.9e+60)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.55e-18], -1.0, If[LessEqual[x, 1e-11], 1.0, If[LessEqual[x, 1.9e+60], -1.0, 1.0]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000003e-18 or 9.99999999999999939e-12 < x < 1.90000000000000005e60

   1. Initial program 54.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 63.2%

      \[\leadsto \color{blue}{-1} \]

   if 1.55000000000000003e-18 < x < 9.99999999999999939e-12 or 1.90000000000000005e60 < x

   1. Initial program 30.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around inf 81.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-18}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 10^{-11}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+60}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 50.0% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 49.2%

   \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

2. Taylor expanded in x around 0 53.4%

   \[\leadsto \color{blue}{-1} \]

3. Final simplification 53.4%

   \[\leadsto -1 \]

Developer target: 52.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 4\\ t_1 := x \cdot x + t_0\\ t_2 := \frac{t_0}{t_1}\\ t_3 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x \cdot x - t_3}{x \cdot x + t_3} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{t_1} - t_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{t_1}}\right)}^{2} - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 4.0))
        (t_1 (+ (* x x) t_0))
        (t_2 (/ t_0 t_1))
        (t_3 (* (* y 4.0) y)))
   (if (< (/ (- (* x x) t_3) (+ (* x x) t_3)) 0.9743233849626781)
     (- (/ (* x x) t_1) t_2)
     (- (pow (/ x (sqrt t_1)) 2.0) t_2))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = pow((x / sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (y * y) * 4.0d0
    t_1 = (x * x) + t_0
    t_2 = t_0 / t_1
    t_3 = (y * 4.0d0) * y
    if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781d0) then
        tmp = ((x * x) / t_1) - t_2
    else
        tmp = ((x / sqrt(t_1)) ** 2.0d0) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = Math.pow((x / Math.sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 4.0
	t_1 = (x * x) + t_0
	t_2 = t_0 / t_1
	t_3 = (y * 4.0) * y
	tmp = 0
	if (((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781:
		tmp = ((x * x) / t_1) - t_2
	else:
		tmp = math.pow((x / math.sqrt(t_1)), 2.0) - t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 4.0)
	t_1 = Float64(Float64(x * x) + t_0)
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_3) / Float64(Float64(x * x) + t_3)) < 0.9743233849626781)
		tmp = Float64(Float64(Float64(x * x) / t_1) - t_2);
	else
		tmp = Float64((Float64(x / sqrt(t_1)) ^ 2.0) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 4.0;
	t_1 = (x * x) + t_0;
	t_2 = t_0 / t_1;
	t_3 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781)
		tmp = ((x * x) / t_1) - t_2;
	else
		tmp = ((x / sqrt(t_1)) ^ 2.0) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[Less[N[(N[(N[(x * x), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 0.9743233849626781], N[(N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Power[N[(x / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2023311 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))