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

Percentage Accurate: 50.5% → 81.4%

Time: 8.2s

Alternatives: 8

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: 50.5% 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: 81.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(x, x, t_0\right)}{\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 0.0)
     1.0
     (if (<= t_0 2e+260)
       (pow (/ (fma x x t_0) (fma y (* y -4.0) (* x x))) -1.0)
       (fma 0.5 (* (/ x y) (/ x y)) -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0;
	} else if (t_0 <= 2e+260) {
		tmp = pow((fma(x, x, t_0) / fma(y, (y * -4.0), (x * x))), -1.0);
	} else {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = 1.0;
	elseif (t_0 <= 2e+260)
		tmp = Float64(fma(x, x, t_0) / fma(y, Float64(y * -4.0), Float64(x * x))) ^ -1.0;
	else
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], 1.0, If[LessEqual[t$95$0, 2e+260], N[Power[N[(N[(x * x + t$95$0), $MachinePrecision] / N[(y * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 0.0

   1. Initial program 48.4%

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

   2. Taylor expanded in x around inf 85.7%

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

   if 0.0 < (*.f64 (*.f64 y 4) y) < 2.00000000000000013e260

   1. Initial program 78.8%

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

      1. clear-num 78.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x - \left(y \cdot 4\right) \cdot y}}} \]

      2. inv-pow 78.8%

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

      3. fma-def 78.8%

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

      4. *-commutative 78.8%

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

      5. sub-neg 78.8%

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

      6. +-commutative 78.8%

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

      7. *-commutative 78.8%

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

      8. distribute-rgt-neg-in 78.8%

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

      9. fma-def 78.8%

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

      10. distribute-rgt-neg-in 78.8%

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

      11. metadata-eval 78.8%

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

   3. Applied egg-rr 78.8%

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

   if 2.00000000000000013e260 < (*.f64 (*.f64 y 4) y)

   1. Initial program 5.6%

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

   2. Taylor expanded in x around 0 76.1%

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

      1. fma-neg 76.1%

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

      2. unpow2 76.1%

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

      3. unpow2 76.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]

      4. times-frac 93.4%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]

      5. metadata-eval 93.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, \color{blue}{-1}\right) \]

   4. Simplified 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+260}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternative 2: 81.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x \cdot x - t_0}{\mathsf{fma}\left(y \cdot 4, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 0.0)
     1.0
     (if (<= t_0 2e+260)
       (/ (- (* x x) t_0) (fma (* y 4.0) y (* x x)))
       (fma 0.5 (* (/ x y) (/ x y)) -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0;
	} else if (t_0 <= 2e+260) {
		tmp = ((x * x) - t_0) / fma((y * 4.0), y, (x * x));
	} else {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = 1.0;
	elseif (t_0 <= 2e+260)
		tmp = Float64(Float64(Float64(x * x) - t_0) / fma(Float64(y * 4.0), y, Float64(x * x)));
	else
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], 1.0, If[LessEqual[t$95$0, 2e+260], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(y * 4.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 0.0

   1. Initial program 48.4%

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

   2. Taylor expanded in x around inf 85.7%

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

   if 0.0 < (*.f64 (*.f64 y 4) y) < 2.00000000000000013e260

   1. Initial program 78.8%

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

      1. +-commutative 78.8%

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

      2. fma-def 78.8%

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

   3. Applied egg-rr 78.8%

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

   if 2.00000000000000013e260 < (*.f64 (*.f64 y 4) y)

   1. Initial program 5.6%

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

   2. Taylor expanded in x around 0 76.1%

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

      1. fma-neg 76.1%

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

      2. unpow2 76.1%

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

      3. unpow2 76.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]

      4. times-frac 93.4%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]

      5. metadata-eval 93.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, \color{blue}{-1}\right) \]

   4. Simplified 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(y \cdot 4, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternative 3: 81.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x \cdot x - t_0}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 0.0)
     1.0
     (if (<= t_0 2e+260)
       (/ (- (* x x) t_0) (+ t_0 (* x x)))
       (fma 0.5 (* (/ x y) (/ x y)) -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0;
	} else if (t_0 <= 2e+260) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = 1.0;
	elseif (t_0 <= 2e+260)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	else
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], 1.0, If[LessEqual[t$95$0, 2e+260], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 0.0

   1. Initial program 48.4%

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

   2. Taylor expanded in x around inf 85.7%

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

   if 0.0 < (*.f64 (*.f64 y 4) y) < 2.00000000000000013e260

   1. Initial program 78.8%

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

   if 2.00000000000000013e260 < (*.f64 (*.f64 y 4) y)

   1. Initial program 5.6%

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

   2. Taylor expanded in x around 0 76.1%

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

      1. fma-neg 76.1%

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

      2. unpow2 76.1%

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

      3. unpow2 76.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]

      4. times-frac 93.4%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]

      5. metadata-eval 93.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, \color{blue}{-1}\right) \]

   4. Simplified 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternative 4: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x \cdot x - t_0}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y \cdot 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 0.0)
     1.0
     (if (<= t_0 2e+260)
       (/ (- (* x x) t_0) (+ t_0 (* x x)))
       (+ -1.0 (* (/ x y) (/ x (* y 4.0))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0;
	} else if (t_0 <= 2e+260) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + ((x / y) * (x / (y * 4.0)));
	}
	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) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 0.0d0) then
        tmp = 1.0d0
    else if (t_0 <= 2d+260) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    else
        tmp = (-1.0d0) + ((x / y) * (x / (y * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0;
	} else if (t_0 <= 2e+260) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + ((x / y) * (x / (y * 4.0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = 1.0
	elif t_0 <= 2e+260:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = -1.0 + ((x / y) * (x / (y * 4.0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = 1.0;
	elseif (t_0 <= 2e+260)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / Float64(y * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 1.0;
	elseif (t_0 <= 2e+260)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	else
		tmp = -1.0 + ((x / y) * (x / (y * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], 1.0, If[LessEqual[t$95$0, 2e+260], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 0.0

   1. Initial program 48.4%

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

   2. Taylor expanded in x around inf 85.7%

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

   if 0.0 < (*.f64 (*.f64 y 4) y) < 2.00000000000000013e260

   1. Initial program 78.8%

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

   if 2.00000000000000013e260 < (*.f64 (*.f64 y 4) y)

   1. Initial program 5.6%

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

   2. Taylor expanded in x around 0 5.7%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{4 \cdot {y}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 5.7%

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

      2. unpow2 5.7%

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

      3. associate-*r* 5.7%

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

   4. Simplified 5.7%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{y \cdot \left(y \cdot 4\right)}} \]
   5. Step-by-step derivation

      1. div-sub 5.7%

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

      2. times-frac 5.7%

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

      3. *-commutative 5.7%

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

      4. pow1 5.7%

         \[\leadsto \frac{x}{y} \cdot \frac{x}{y \cdot 4} - \frac{\color{blue}{{\left(y \cdot \left(y \cdot 4\right)\right)}^{1}}}{y \cdot \left(y \cdot 4\right)} \]

      5. pow1 5.7%

         \[\leadsto \frac{x}{y} \cdot \frac{x}{y \cdot 4} - \frac{{\left(y \cdot \left(y \cdot 4\right)\right)}^{1}}{\color{blue}{{\left(y \cdot \left(y \cdot 4\right)\right)}^{1}}} \]

      6. pow-div 93.1%

         \[\leadsto \frac{x}{y} \cdot \frac{x}{y \cdot 4} - \color{blue}{{\left(y \cdot \left(y \cdot 4\right)\right)}^{\left(1 - 1\right)}} \]

      7. metadata-eval 93.1%

         \[\leadsto \frac{x}{y} \cdot \frac{x}{y \cdot 4} - {\left(y \cdot \left(y \cdot 4\right)\right)}^{\color{blue}{0}} \]

      8. metadata-eval 93.1%

         \[\leadsto \frac{x}{y} \cdot \frac{x}{y \cdot 4} - \color{blue}{1} \]

   6. Applied egg-rr 93.1%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y \cdot 4} - 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y \cdot 4}\\ \end{array} \]

Alternative 5: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-31} \lor \neg \left(t_0 \leq 2 \cdot 10^{+58}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \left(y \cdot \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 1e-147)
     1.0
     (if (or (<= t_0 5e-31) (not (<= t_0 2e+58)))
       (+ -1.0 (* (/ x y) (/ x (* y 4.0))))
       (+ 1.0 (* -4.0 (* y (/ y (* x x)))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-147) {
		tmp = 1.0;
	} else if ((t_0 <= 5e-31) || !(t_0 <= 2e+58)) {
		tmp = -1.0 + ((x / y) * (x / (y * 4.0)));
	} else {
		tmp = 1.0 + (-4.0 * (y * (y / (x * 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) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 1d-147) then
        tmp = 1.0d0
    else if ((t_0 <= 5d-31) .or. (.not. (t_0 <= 2d+58))) then
        tmp = (-1.0d0) + ((x / y) * (x / (y * 4.0d0)))
    else
        tmp = 1.0d0 + ((-4.0d0) * (y * (y / (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-147) {
		tmp = 1.0;
	} else if ((t_0 <= 5e-31) || !(t_0 <= 2e+58)) {
		tmp = -1.0 + ((x / y) * (x / (y * 4.0)));
	} else {
		tmp = 1.0 + (-4.0 * (y * (y / (x * x))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 1e-147:
		tmp = 1.0
	elif (t_0 <= 5e-31) or not (t_0 <= 2e+58):
		tmp = -1.0 + ((x / y) * (x / (y * 4.0)))
	else:
		tmp = 1.0 + (-4.0 * (y * (y / (x * x))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 1e-147)
		tmp = 1.0;
	elseif ((t_0 <= 5e-31) || !(t_0 <= 2e+58))
		tmp = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / Float64(y * 4.0))));
	else
		tmp = Float64(1.0 + Float64(-4.0 * Float64(y * Float64(y / Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 1e-147)
		tmp = 1.0;
	elseif ((t_0 <= 5e-31) || ~((t_0 <= 2e+58)))
		tmp = -1.0 + ((x / y) * (x / (y * 4.0)));
	else
		tmp = 1.0 + (-4.0 * (y * (y / (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-147], 1.0, If[Or[LessEqual[t$95$0, 5e-31], N[Not[LessEqual[t$95$0, 2e+58]], $MachinePrecision]], N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-4.0 * N[(y * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 9.9999999999999997e-148

   1. Initial program 62.1%

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

   2. Taylor expanded in x around inf 77.7%

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

   if 9.9999999999999997e-148 < (*.f64 (*.f64 y 4) y) < 5e-31 or 1.99999999999999989e58 < (*.f64 (*.f64 y 4) y)

   1. Initial program 41.3%

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

   2. Taylor expanded in x around 0 35.6%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{4 \cdot {y}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 35.6%

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

      2. unpow2 35.6%

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

      3. associate-*r* 35.6%

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

   4. Simplified 35.6%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{y \cdot \left(y \cdot 4\right)}} \]
   5. Step-by-step derivation

      1. div-sub 35.6%

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

      2. times-frac 35.7%

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

      3. *-commutative 35.7%

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

      4. pow1 35.7%

         \[\leadsto \frac{x}{y} \cdot \frac{x}{y \cdot 4} - \frac{\color{blue}{{\left(y \cdot \left(y \cdot 4\right)\right)}^{1}}}{y \cdot \left(y \cdot 4\right)} \]

      5. pow1 35.7%

         \[\leadsto \frac{x}{y} \cdot \frac{x}{y \cdot 4} - \frac{{\left(y \cdot \left(y \cdot 4\right)\right)}^{1}}{\color{blue}{{\left(y \cdot \left(y \cdot 4\right)\right)}^{1}}} \]

      6. pow-div 82.3%

         \[\leadsto \frac{x}{y} \cdot \frac{x}{y \cdot 4} - \color{blue}{{\left(y \cdot \left(y \cdot 4\right)\right)}^{\left(1 - 1\right)}} \]

      7. metadata-eval 82.3%

         \[\leadsto \frac{x}{y} \cdot \frac{x}{y \cdot 4} - {\left(y \cdot \left(y \cdot 4\right)\right)}^{\color{blue}{0}} \]

      8. metadata-eval 82.3%

         \[\leadsto \frac{x}{y} \cdot \frac{x}{y \cdot 4} - \color{blue}{1} \]

   6. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y \cdot 4} - 1} \]

   if 5e-31 < (*.f64 (*.f64 y 4) y) < 1.99999999999999989e58

   1. Initial program 59.8%

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

   2. Taylor expanded in x around inf 23.7%

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

      1. unpow2 23.7%

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

   4. Simplified 23.7%

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

   5. Taylor expanded in x around inf 64.7%

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

      1. unpow2 64.7%

         \[\leadsto 1 + -4 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 64.7%

         \[\leadsto 1 + -4 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. associate-*r/ 64.7%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(y \cdot \frac{y}{x \cdot x}\right)} \]

   7. Simplified 64.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{-31} \lor \neg \left(y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+58}\right):\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \left(y \cdot \frac{y}{x \cdot x}\right)\\ \end{array} \]

Alternative 6: 62.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-51}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-16}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+29}:\\ \;\;\;\;1 + -4 \cdot \left(y \cdot \frac{y}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7e-51)
   1.0
   (if (<= y 2.9e-16)
     -1.0
     (if (<= y 1.2e+29) (+ 1.0 (* -4.0 (* y (/ y (* x x))))) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7e-51) {
		tmp = 1.0;
	} else if (y <= 2.9e-16) {
		tmp = -1.0;
	} else if (y <= 1.2e+29) {
		tmp = 1.0 + (-4.0 * (y * (y / (x * x))));
	} 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 (y <= 7d-51) then
        tmp = 1.0d0
    else if (y <= 2.9d-16) then
        tmp = -1.0d0
    else if (y <= 1.2d+29) then
        tmp = 1.0d0 + ((-4.0d0) * (y * (y / (x * x))))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7e-51) {
		tmp = 1.0;
	} else if (y <= 2.9e-16) {
		tmp = -1.0;
	} else if (y <= 1.2e+29) {
		tmp = 1.0 + (-4.0 * (y * (y / (x * x))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7e-51:
		tmp = 1.0
	elif y <= 2.9e-16:
		tmp = -1.0
	elif y <= 1.2e+29:
		tmp = 1.0 + (-4.0 * (y * (y / (x * x))))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7e-51)
		tmp = 1.0;
	elseif (y <= 2.9e-16)
		tmp = -1.0;
	elseif (y <= 1.2e+29)
		tmp = Float64(1.0 + Float64(-4.0 * Float64(y * Float64(y / Float64(x * x)))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e-51)
		tmp = 1.0;
	elseif (y <= 2.9e-16)
		tmp = -1.0;
	elseif (y <= 1.2e+29)
		tmp = 1.0 + (-4.0 * (y * (y / (x * x))));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7e-51], 1.0, If[LessEqual[y, 2.9e-16], -1.0, If[LessEqual[y, 1.2e+29], N[(1.0 + N[(-4.0 * N[(y * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.9999999999999995e-51

   1. Initial program 51.1%

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

   2. Taylor expanded in x around inf 56.3%

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

   if 6.9999999999999995e-51 < y < 2.8999999999999998e-16 or 1.2e29 < y

   1. Initial program 50.0%

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

   2. Taylor expanded in x around 0 86.9%

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

   if 2.8999999999999998e-16 < y < 1.2e29

   1. Initial program 58.2%

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

   2. Taylor expanded in x around inf 17.7%

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

      1. unpow2 17.7%

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

   4. Simplified 17.7%

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

   5. Taylor expanded in x around inf 60.7%

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

      1. unpow2 60.7%

         \[\leadsto 1 + -4 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 60.7%

         \[\leadsto 1 + -4 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. associate-*r/ 60.7%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(y \cdot \frac{y}{x \cdot x}\right)} \]

   7. Simplified 60.7%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-51}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-16}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+29}:\\ \;\;\;\;1 + -4 \cdot \left(y \cdot \frac{y}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7: 62.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-51}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-16}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.5e-51) 1.0 (if (<= y 3.8e-16) -1.0 (if (<= y 7.6e+28) 1.0 -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-51) {
		tmp = 1.0;
	} else if (y <= 3.8e-16) {
		tmp = -1.0;
	} else if (y <= 7.6e+28) {
		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 (y <= 8.5d-51) then
        tmp = 1.0d0
    else if (y <= 3.8d-16) then
        tmp = -1.0d0
    else if (y <= 7.6d+28) 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 (y <= 8.5e-51) {
		tmp = 1.0;
	} else if (y <= 3.8e-16) {
		tmp = -1.0;
	} else if (y <= 7.6e+28) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e-51:
		tmp = 1.0
	elif y <= 3.8e-16:
		tmp = -1.0
	elif y <= 7.6e+28:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-51)
		tmp = 1.0;
	elseif (y <= 3.8e-16)
		tmp = -1.0;
	elseif (y <= 7.6e+28)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.5e-51)
		tmp = 1.0;
	elseif (y <= 3.8e-16)
		tmp = -1.0;
	elseif (y <= 7.6e+28)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e-51], 1.0, If[LessEqual[y, 3.8e-16], -1.0, If[LessEqual[y, 7.6e+28], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 8.50000000000000036e-51 or 3.80000000000000012e-16 < y < 7.5999999999999998e28

   1. Initial program 51.5%

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

   2. Taylor expanded in x around inf 56.5%

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

   if 8.50000000000000036e-51 < y < 3.80000000000000012e-16 or 7.5999999999999998e28 < y

   1. Initial program 50.0%

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

   2. Taylor expanded in x around 0 86.9%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-51}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-16}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8: 50.1% 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 51.1%

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

2. Taylor expanded in x around 0 54.3%

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

3. Final simplification 54.3%

   \[\leadsto -1 \]

Developer target: 51.0% 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 2023224 
(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))))