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

Percentage Accurate: 50.7% → 78.9%

Time: 25.5s

Alternatives: 5

Speedup: 2.4×

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: 50.7% 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: 78.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.3333333333333333\\ t_1 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-167}:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_0 + \left(e^{\mathsf{log1p}\left(2 \cdot t\_0\right)} + -1\right), -1\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+46}:\\ \;\;\;\;\frac{x \cdot x - t\_1}{x \cdot x + t\_1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (/ x y) (/ x y)) 0.3333333333333333)) (t_1 (* y (* y 4.0))))
   (if (<= (* x x) 1e-167)
     (fma 0.5 (+ t_0 (+ (exp (log1p (* 2.0 t_0))) -1.0)) -1.0)
     (if (<= (* x x) 1e+46) (/ (- (* x x) t_1) (+ (* x x) t_1)) 1.0))))

C

double code(double x, double y) {
	double t_0 = ((x / y) * (x / y)) * 0.3333333333333333;
	double t_1 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-167) {
		tmp = fma(0.5, (t_0 + (exp(log1p((2.0 * t_0))) + -1.0)), -1.0);
	} else if ((x * x) <= 1e+46) {
		tmp = ((x * x) - t_1) / ((x * x) + t_1);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x / y) * Float64(x / y)) * 0.3333333333333333)
	t_1 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 1e-167)
		tmp = fma(0.5, Float64(t_0 + Float64(exp(log1p(Float64(2.0 * t_0))) + -1.0)), -1.0);
	elseif (Float64(x * x) <= 1e+46)
		tmp = Float64(Float64(Float64(x * x) - t_1) / Float64(Float64(x * x) + t_1));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e-167], N[(0.5 * N[(t$95$0 + N[(N[Exp[N[Log[1 + N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+46], N[(N[(N[(x * x), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1e-167

   1. Initial program 55.2%

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

   3. Taylor expanded in x around 0 80.3%

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

      1. fma-neg 80.3%

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

      2. unpow2 80.3%

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

      3. unpow2 80.3%

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

      4. times-frac 91.9%

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

      5. metadata-eval 91.9%

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

   5. Simplified 91.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
   6. Step-by-step derivation

      1. add-log-exp 91.7%

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

      2. add-cube-cbrt 91.7%

         \[\leadsto \mathsf{fma}\left(0.5, \log \color{blue}{\left(\left(\sqrt[3]{e^{\frac{x}{y} \cdot \frac{x}{y}}} \cdot \sqrt[3]{e^{\frac{x}{y} \cdot \frac{x}{y}}}\right) \cdot \sqrt[3]{e^{\frac{x}{y} \cdot \frac{x}{y}}}\right)}, -1\right) \]

      3. log-prod 91.7%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\log \left(\sqrt[3]{e^{\frac{x}{y} \cdot \frac{x}{y}}} \cdot \sqrt[3]{e^{\frac{x}{y} \cdot \frac{x}{y}}}\right) + \log \left(\sqrt[3]{e^{\frac{x}{y} \cdot \frac{x}{y}}}\right)}, -1\right) \]

      4. frac-times 80.3%

         \[\leadsto \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{e^{\color{blue}{\frac{x \cdot x}{y \cdot y}}}} \cdot \sqrt[3]{e^{\frac{x}{y} \cdot \frac{x}{y}}}\right) + \log \left(\sqrt[3]{e^{\frac{x}{y} \cdot \frac{x}{y}}}\right), -1\right) \]

      5. frac-times 80.3%

         \[\leadsto \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{e^{\frac{x \cdot x}{y \cdot y}}} \cdot \sqrt[3]{e^{\color{blue}{\frac{x \cdot x}{y \cdot y}}}}\right) + \log \left(\sqrt[3]{e^{\frac{x}{y} \cdot \frac{x}{y}}}\right), -1\right) \]

      6. frac-times 80.3%

         \[\leadsto \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{e^{\frac{x \cdot x}{y \cdot y}}} \cdot \sqrt[3]{e^{\frac{x \cdot x}{y \cdot y}}}\right) + \log \left(\sqrt[3]{e^{\color{blue}{\frac{x \cdot x}{y \cdot y}}}}\right), -1\right) \]

   7. Applied egg-rr 80.3%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\log \left(\sqrt[3]{e^{\frac{x \cdot x}{y \cdot y}}} \cdot \sqrt[3]{e^{\frac{x \cdot x}{y \cdot y}}}\right) + \log \left(\sqrt[3]{e^{\frac{x \cdot x}{y \cdot y}}}\right)}, -1\right) \]
   8. Step-by-step derivation

   9. Simplified 91.7%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right) + \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right)}, -1\right) \]
   10. Step-by-step derivation

       1. log1p-expm1-u 91.7%

          \[\leadsto \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right) + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right)\right)\right)}, -1\right) \]

       2. expm1-undefine 91.7%

          \[\leadsto \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right) + \mathsf{log1p}\left(\color{blue}{e^{\log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right)} - 1}\right), -1\right) \]

       3. add-exp-log 91.7%

          \[\leadsto \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right) + \mathsf{log1p}\left(\color{blue}{\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}} - 1\right), -1\right) \]

       4. pow1/3 91.7%

          \[\leadsto \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right) + \mathsf{log1p}\left(\color{blue}{{\left(e^{{\left(\frac{x}{y}\right)}^{2}}\right)}^{0.3333333333333333}} - 1\right), -1\right) \]

       5. pow-to-exp 91.7%

          \[\leadsto \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right) + \mathsf{log1p}\left(\color{blue}{e^{\log \left(e^{{\left(\frac{x}{y}\right)}^{2}}\right) \cdot 0.3333333333333333}} - 1\right), -1\right) \]

       6. expm1-define 91.7%

          \[\leadsto \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right) + \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(e^{{\left(\frac{x}{y}\right)}^{2}}\right) \cdot 0.3333333333333333\right)}\right), -1\right) \]

       7. add-log-exp 91.7%

          \[\leadsto \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right) + \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} \cdot 0.3333333333333333\right)\right), -1\right) \]

       8. log1p-expm1-u 91.7%

          \[\leadsto \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right) + \color{blue}{{\left(\frac{x}{y}\right)}^{2} \cdot 0.3333333333333333}, -1\right) \]

       9. unpow2 91.7%

          \[\leadsto \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right) + \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot 0.3333333333333333, -1\right) \]

   11. Applied egg-rr 91.7%

       \[\leadsto \mathsf{fma}\left(0.5, \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right) + \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.3333333333333333}, -1\right) \]
   12. Step-by-step derivation

       1. expm1-log1p-u 91.7%

          \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right)\right)\right)} + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.3333333333333333, -1\right) \]

       2. expm1-undefine 91.7%

          \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right)\right)} - 1\right)} + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.3333333333333333, -1\right) \]

   13. Applied egg-rr 91.9%

       \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \left(\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.3333333333333333\right)\right)} - 1\right)} + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.3333333333333333, -1\right) \]

   if 1e-167 < (*.f64 x x) < 9.9999999999999999e45

   1. Initial program 83.0%

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

   if 9.9999999999999999e45 < (*.f64 x x)

   1. Initial program 39.8%

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

   3. Taylor expanded in x around inf 82.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-167}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.3333333333333333 + \left(e^{\mathsf{log1p}\left(2 \cdot \left(\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.3333333333333333\right)\right)} + -1\right), -1\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+46}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-167}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+46}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-167)
     (fma 0.5 (* (/ x y) (/ x y)) -1.0)
     (if (<= (* x x) 1e+46) (/ (- (* x x) t_0) (+ (* x x) t_0)) 1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-167) {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	} else if ((x * x) <= 1e+46) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 1e-167)
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	elseif (Float64(x * x) <= 1e+46)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e-167], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+46], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1e-167

   1. Initial program 55.2%

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

   3. Taylor expanded in x around 0 80.3%

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

      1. fma-neg 80.3%

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

      2. unpow2 80.3%

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

      3. unpow2 80.3%

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

      4. times-frac 91.9%

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

      5. metadata-eval 91.9%

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

   5. Simplified 91.9%

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

   if 1e-167 < (*.f64 x x) < 9.9999999999999999e45

   1. Initial program 83.0%

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

   if 9.9999999999999999e45 < (*.f64 x x)

   1. Initial program 39.8%

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

   3. Taylor expanded in x around inf 82.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-167}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+46}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-167}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 3.6 \cdot 10^{+50}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-167)
     -1.0
     (if (<= (* x x) 3.6e+50) (/ (- (* x x) t_0) (+ (* x x) t_0)) 1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-167) {
		tmp = -1.0;
	} else if ((x * x) <= 3.6e+50) {
		tmp = ((x * x) - t_0) / ((x * x) + t_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) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 1d-167) then
        tmp = -1.0d0
    else if ((x * x) <= 3.6d+50) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0
    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 ((x * x) <= 1e-167) {
		tmp = -1.0;
	} else if ((x * x) <= 3.6e+50) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 1e-167:
		tmp = -1.0
	elif (x * x) <= 3.6e+50:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 1e-167)
		tmp = -1.0;
	elseif (Float64(x * x) <= 3.6e+50)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 1e-167)
		tmp = -1.0;
	elseif ((x * x) <= 3.6e+50)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e-167], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 3.6e+50], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1e-167

   1. Initial program 55.2%

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

   3. Taylor expanded in x around 0 91.1%

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

   if 1e-167 < (*.f64 x x) < 3.59999999999999986e50

   1. Initial program 83.0%

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

   if 3.59999999999999986e50 < (*.f64 x x)

   1. Initial program 39.8%

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

   3. Taylor expanded in x around inf 82.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-167}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 3.6 \cdot 10^{+50}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-110}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* x x) 5e-110) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e-110) {
		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 * x) <= 5d-110) 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 * x) <= 5e-110) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 5e-110:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5e-110)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 5e-110)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-110], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5e-110

   1. Initial program 60.4%

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

   3. Taylor expanded in x around 0 87.9%

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

   if 5e-110 < (*.f64 x x)

   1. Initial program 48.3%

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

   3. Taylor expanded in x around inf 76.9%

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

Alternative 5: 50.6% 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 53.5%

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

3. Taylor expanded in x around 0 51.8%

   \[\leadsto \color{blue}{-1} \]
4. Add Preprocessing

Developer target: 51.2% 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 2024097 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (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))))