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

Percentage Accurate: 50.5% → 80.3%

Time: 12.5s

Alternatives: 7

Speedup: 19.0×

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: 80.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-317}:\\ \;\;\;\;\log \left({\left(\mathsf{fma}\left(0.25, {\left(\frac{x}{y}\right)}^{2}, 1\right)\right)}^{2}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+210}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{\mathsf{fma}\left(x, x, t\_0\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+229}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-317)
     (+ (log (pow (fma 0.25 (pow (/ x y) 2.0) 1.0) 2.0)) -1.0)
     (if (<= (* x x) 1e+210)
       (/ (- (* x x) t_0) (fma x x t_0))
       (if (<= (* x x) 2e+229) -1.0 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-317) {
		tmp = log(pow(fma(0.25, pow((x / y), 2.0), 1.0), 2.0)) + -1.0;
	} else if ((x * x) <= 1e+210) {
		tmp = ((x * x) - t_0) / fma(x, x, t_0);
	} else if ((x * x) <= 2e+229) {
		tmp = -1.0;
	} 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-317)
		tmp = Float64(log((fma(0.25, (Float64(x / y) ^ 2.0), 1.0) ^ 2.0)) + -1.0);
	elseif (Float64(x * x) <= 1e+210)
		tmp = Float64(Float64(Float64(x * x) - t_0) / fma(x, x, t_0));
	elseif (Float64(x * x) <= 2e+229)
		tmp = -1.0;
	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-317], N[(N[Log[N[Power[N[(0.25 * N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+210], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(x * x + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+229], -1.0, 1.0]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 1.00000023e-317

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. fma-define 43.9%

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

      3. *-commutative 43.9%

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

   3. Simplified 43.9%

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

   5. Taylor expanded in x around 0 87.7%

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

      1. add-log-exp 87.7%

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

      2. add-sqr-sqrt 87.7%

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

      3. log-prod 87.7%

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

   7. Applied egg-rr 92.3%

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

      1. count-2 92.3%

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

      2. unpow2 92.3%

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

      3. associate-*r/ 92.3%

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

      4. associate-*l/ 91.9%

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

      5. unpow2 91.9%

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

      6. *-lft-identity 91.9%

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

      7. associate-*l/ 91.9%

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

      8. associate-*r/ 91.9%

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

      9. associate-*l/ 87.7%

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

      10. *-rgt-identity 87.7%

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

      11. associate-/l* 87.7%

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

      12. unpow2 87.7%

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

      13. swap-sqr 92.3%

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

      14. associate-/r/ 92.3%

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

      15. associate-/r/ 92.3%

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

      16. unpow-1 92.3%

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

      17. unpow-1 92.3%

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

      18. pow-sqr 92.3%

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

      19. metadata-eval 92.3%

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

   9. Simplified 92.3%

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

   10. Taylor expanded in y around inf 87.7%

       \[\leadsto 2 \cdot \log \color{blue}{\left(1 + 0.25 \cdot \frac{{x}^{2}}{{y}^{2}}\right)} - 1 \]
   11. Step-by-step derivation

       1. +-commutative 87.7%

          \[\leadsto 2 \cdot \log \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right)} - 1 \]

       2. *-commutative 87.7%

          \[\leadsto 2 \cdot \log \left(\color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot 0.25} + 1\right) - 1 \]

       3. unpow2 87.7%

          \[\leadsto 2 \cdot \log \left(\frac{{x}^{2}}{\color{blue}{y \cdot y}} \cdot 0.25 + 1\right) - 1 \]

       4. unpow2 87.7%

          \[\leadsto 2 \cdot \log \left(\frac{\color{blue}{x \cdot x}}{y \cdot y} \cdot 0.25 + 1\right) - 1 \]

       5. times-frac 93.1%

          \[\leadsto 2 \cdot \log \left(\color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot 0.25 + 1\right) - 1 \]

       6. unpow2 93.1%

          \[\leadsto 2 \cdot \log \left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} \cdot 0.25 + 1\right) - 1 \]

   12. Simplified 93.1%

       \[\leadsto 2 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{2} \cdot 0.25 + 1\right)} - 1 \]
   13. Step-by-step derivation

       1. add-log-exp 93.1%

          \[\leadsto \color{blue}{\log \left(e^{2 \cdot \log \left({\left(\frac{x}{y}\right)}^{2} \cdot 0.25 + 1\right)}\right)} - 1 \]

       2. *-commutative 93.1%

          \[\leadsto \log \left(e^{\color{blue}{\log \left({\left(\frac{x}{y}\right)}^{2} \cdot 0.25 + 1\right) \cdot 2}}\right) - 1 \]

       3. exp-to-pow 93.1%

          \[\leadsto \log \color{blue}{\left({\left({\left(\frac{x}{y}\right)}^{2} \cdot 0.25 + 1\right)}^{2}\right)} - 1 \]

       4. *-commutative 93.1%

          \[\leadsto \log \left({\left(\color{blue}{0.25 \cdot {\left(\frac{x}{y}\right)}^{2}} + 1\right)}^{2}\right) - 1 \]

       5. fma-define 93.1%

          \[\leadsto \log \left({\color{blue}{\left(\mathsf{fma}\left(0.25, {\left(\frac{x}{y}\right)}^{2}, 1\right)\right)}}^{2}\right) - 1 \]

   14. Applied egg-rr 93.1%

       \[\leadsto \color{blue}{\log \left({\left(\mathsf{fma}\left(0.25, {\left(\frac{x}{y}\right)}^{2}, 1\right)\right)}^{2}\right)} - 1 \]

   if 1.00000023e-317 < (*.f64 x x) < 9.99999999999999927e209

   1. Initial program 76.9%

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

      1. *-commutative 76.9%

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

      2. fma-define 77.0%

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

      3. *-commutative 77.0%

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

   3. Simplified 77.0%

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

   if 9.99999999999999927e209 < (*.f64 x x) < 2e229

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. fma-define 0.0%

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

      3. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 2e229 < (*.f64 x x)

   1. Initial program 19.1%

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

      1. *-commutative 19.1%

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

      2. fma-define 19.1%

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

      3. *-commutative 19.1%

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

   3. Simplified 19.1%

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

   5. Taylor expanded in x around inf 84.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-317}:\\ \;\;\;\;\log \left({\left(\mathsf{fma}\left(0.25, {\left(\frac{x}{y}\right)}^{2}, 1\right)\right)}^{2}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+210}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+229}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-317}:\\ \;\;\;\;-1 + 2 \cdot \log \left(1 + 0.25 \cdot \frac{\frac{x}{y}}{\frac{y}{x}}\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+210}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{\mathsf{fma}\left(x, x, t\_0\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+229}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-317)
     (+ -1.0 (* 2.0 (log (+ 1.0 (* 0.25 (/ (/ x y) (/ y x)))))))
     (if (<= (* x x) 1e+210)
       (/ (- (* x x) t_0) (fma x x t_0))
       (if (<= (* x x) 2e+229) -1.0 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-317) {
		tmp = -1.0 + (2.0 * log((1.0 + (0.25 * ((x / y) / (y / x))))));
	} else if ((x * x) <= 1e+210) {
		tmp = ((x * x) - t_0) / fma(x, x, t_0);
	} else if ((x * x) <= 2e+229) {
		tmp = -1.0;
	} 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-317)
		tmp = Float64(-1.0 + Float64(2.0 * log(Float64(1.0 + Float64(0.25 * Float64(Float64(x / y) / Float64(y / x)))))));
	elseif (Float64(x * x) <= 1e+210)
		tmp = Float64(Float64(Float64(x * x) - t_0) / fma(x, x, t_0));
	elseif (Float64(x * x) <= 2e+229)
		tmp = -1.0;
	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-317], N[(-1.0 + N[(2.0 * N[Log[N[(1.0 + N[(0.25 * N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+210], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(x * x + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+229], -1.0, 1.0]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 1.00000023e-317

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. fma-define 43.9%

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

      3. *-commutative 43.9%

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

   3. Simplified 43.9%

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

   5. Taylor expanded in x around 0 87.7%

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

      1. add-log-exp 87.7%

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

      2. add-sqr-sqrt 87.7%

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

      3. log-prod 87.7%

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

   7. Applied egg-rr 92.3%

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

      1. count-2 92.3%

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

      2. unpow2 92.3%

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

      3. associate-*r/ 92.3%

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

      4. associate-*l/ 91.9%

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

      5. unpow2 91.9%

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

      6. *-lft-identity 91.9%

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

      7. associate-*l/ 91.9%

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

      8. associate-*r/ 91.9%

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

      9. associate-*l/ 87.7%

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

      10. *-rgt-identity 87.7%

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

      11. associate-/l* 87.7%

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

      12. unpow2 87.7%

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

      13. swap-sqr 92.3%

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

      14. associate-/r/ 92.3%

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

      15. associate-/r/ 92.3%

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

      16. unpow-1 92.3%

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

      17. unpow-1 92.3%

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

      18. pow-sqr 92.3%

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

      19. metadata-eval 92.3%

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

   9. Simplified 92.3%

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

   10. Taylor expanded in y around inf 87.7%

       \[\leadsto 2 \cdot \log \color{blue}{\left(1 + 0.25 \cdot \frac{{x}^{2}}{{y}^{2}}\right)} - 1 \]
   11. Step-by-step derivation

       1. +-commutative 87.7%

          \[\leadsto 2 \cdot \log \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right)} - 1 \]

       2. *-commutative 87.7%

          \[\leadsto 2 \cdot \log \left(\color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot 0.25} + 1\right) - 1 \]

       3. unpow2 87.7%

          \[\leadsto 2 \cdot \log \left(\frac{{x}^{2}}{\color{blue}{y \cdot y}} \cdot 0.25 + 1\right) - 1 \]

       4. unpow2 87.7%

          \[\leadsto 2 \cdot \log \left(\frac{\color{blue}{x \cdot x}}{y \cdot y} \cdot 0.25 + 1\right) - 1 \]

       5. times-frac 93.1%

          \[\leadsto 2 \cdot \log \left(\color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot 0.25 + 1\right) - 1 \]

       6. unpow2 93.1%

          \[\leadsto 2 \cdot \log \left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} \cdot 0.25 + 1\right) - 1 \]

   12. Simplified 93.1%

       \[\leadsto 2 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{2} \cdot 0.25 + 1\right)} - 1 \]
   13. Step-by-step derivation

       1. unpow2 93.1%

          \[\leadsto 2 \cdot \log \left(\color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot 0.25 + 1\right) - 1 \]

       2. clear-num 93.1%

          \[\leadsto 2 \cdot \log \left(\left(\frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right) \cdot 0.25 + 1\right) - 1 \]

       3. un-div-inv 93.1%

          \[\leadsto 2 \cdot \log \left(\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \cdot 0.25 + 1\right) - 1 \]

   14. Applied egg-rr 93.1%

       \[\leadsto 2 \cdot \log \left(\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \cdot 0.25 + 1\right) - 1 \]

   if 1.00000023e-317 < (*.f64 x x) < 9.99999999999999927e209

   1. Initial program 76.9%

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

      1. *-commutative 76.9%

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

      2. fma-define 77.0%

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

      3. *-commutative 77.0%

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

   3. Simplified 77.0%

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

   if 9.99999999999999927e209 < (*.f64 x x) < 2e229

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. fma-define 0.0%

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

      3. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 2e229 < (*.f64 x x)

   1. Initial program 19.1%

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

      1. *-commutative 19.1%

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

      2. fma-define 19.1%

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

      3. *-commutative 19.1%

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

   3. Simplified 19.1%

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

   5. Taylor expanded in x around inf 84.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-317}:\\ \;\;\;\;-1 + 2 \cdot \log \left(1 + 0.25 \cdot \frac{\frac{x}{y}}{\frac{y}{x}}\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+210}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+229}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-317}:\\ \;\;\;\;-1 + 2 \cdot \log \left(1 + 0.25 \cdot \frac{\frac{x}{y}}{\frac{y}{x}}\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+210}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+229}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-317)
     (+ -1.0 (* 2.0 (log (+ 1.0 (* 0.25 (/ (/ x y) (/ y x)))))))
     (if (<= (* x x) 1e+210)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (if (<= (* x x) 2e+229) -1.0 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-317) {
		tmp = -1.0 + (2.0 * log((1.0 + (0.25 * ((x / y) / (y / x))))));
	} else if ((x * x) <= 1e+210) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 2e+229) {
		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) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 1d-317) then
        tmp = (-1.0d0) + (2.0d0 * log((1.0d0 + (0.25d0 * ((x / y) / (y / x))))))
    else if ((x * x) <= 1d+210) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else if ((x * x) <= 2d+229) then
        tmp = -1.0d0
    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-317) {
		tmp = -1.0 + (2.0 * Math.log((1.0 + (0.25 * ((x / y) / (y / x))))));
	} else if ((x * x) <= 1e+210) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 2e+229) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 1e-317:
		tmp = -1.0 + (2.0 * math.log((1.0 + (0.25 * ((x / y) / (y / x))))))
	elif (x * x) <= 1e+210:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	elif (x * x) <= 2e+229:
		tmp = -1.0
	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-317)
		tmp = Float64(-1.0 + Float64(2.0 * log(Float64(1.0 + Float64(0.25 * Float64(Float64(x / y) / Float64(y / x)))))));
	elseif (Float64(x * x) <= 1e+210)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	elseif (Float64(x * x) <= 2e+229)
		tmp = -1.0;
	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-317)
		tmp = -1.0 + (2.0 * log((1.0 + (0.25 * ((x / y) / (y / x))))));
	elseif ((x * x) <= 1e+210)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	elseif ((x * x) <= 2e+229)
		tmp = -1.0;
	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-317], N[(-1.0 + N[(2.0 * N[Log[N[(1.0 + N[(0.25 * N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+210], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+229], -1.0, 1.0]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 1.00000023e-317

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. fma-define 43.9%

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

      3. *-commutative 43.9%

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

   3. Simplified 43.9%

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

   5. Taylor expanded in x around 0 87.7%

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

      1. add-log-exp 87.7%

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

      2. add-sqr-sqrt 87.7%

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

      3. log-prod 87.7%

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

   7. Applied egg-rr 92.3%

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

      1. count-2 92.3%

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

      2. unpow2 92.3%

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

      3. associate-*r/ 92.3%

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

      4. associate-*l/ 91.9%

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

      5. unpow2 91.9%

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

      6. *-lft-identity 91.9%

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

      7. associate-*l/ 91.9%

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

      8. associate-*r/ 91.9%

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

      9. associate-*l/ 87.7%

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

      10. *-rgt-identity 87.7%

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

      11. associate-/l* 87.7%

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

      12. unpow2 87.7%

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

      13. swap-sqr 92.3%

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

      14. associate-/r/ 92.3%

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

      15. associate-/r/ 92.3%

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

      16. unpow-1 92.3%

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

      17. unpow-1 92.3%

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

      18. pow-sqr 92.3%

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

      19. metadata-eval 92.3%

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

   9. Simplified 92.3%

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

   10. Taylor expanded in y around inf 87.7%

       \[\leadsto 2 \cdot \log \color{blue}{\left(1 + 0.25 \cdot \frac{{x}^{2}}{{y}^{2}}\right)} - 1 \]
   11. Step-by-step derivation

       1. +-commutative 87.7%

          \[\leadsto 2 \cdot \log \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right)} - 1 \]

       2. *-commutative 87.7%

          \[\leadsto 2 \cdot \log \left(\color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot 0.25} + 1\right) - 1 \]

       3. unpow2 87.7%

          \[\leadsto 2 \cdot \log \left(\frac{{x}^{2}}{\color{blue}{y \cdot y}} \cdot 0.25 + 1\right) - 1 \]

       4. unpow2 87.7%

          \[\leadsto 2 \cdot \log \left(\frac{\color{blue}{x \cdot x}}{y \cdot y} \cdot 0.25 + 1\right) - 1 \]

       5. times-frac 93.1%

          \[\leadsto 2 \cdot \log \left(\color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot 0.25 + 1\right) - 1 \]

       6. unpow2 93.1%

          \[\leadsto 2 \cdot \log \left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} \cdot 0.25 + 1\right) - 1 \]

   12. Simplified 93.1%

       \[\leadsto 2 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{2} \cdot 0.25 + 1\right)} - 1 \]
   13. Step-by-step derivation

       1. unpow2 93.1%

          \[\leadsto 2 \cdot \log \left(\color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot 0.25 + 1\right) - 1 \]

       2. clear-num 93.1%

          \[\leadsto 2 \cdot \log \left(\left(\frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}}\right) \cdot 0.25 + 1\right) - 1 \]

       3. un-div-inv 93.1%

          \[\leadsto 2 \cdot \log \left(\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \cdot 0.25 + 1\right) - 1 \]

   14. Applied egg-rr 93.1%

       \[\leadsto 2 \cdot \log \left(\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \cdot 0.25 + 1\right) - 1 \]

   if 1.00000023e-317 < (*.f64 x x) < 9.99999999999999927e209

   1. Initial program 76.9%

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

   if 9.99999999999999927e209 < (*.f64 x x) < 2e229

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. fma-define 0.0%

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

      3. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 2e229 < (*.f64 x x)

   1. Initial program 19.1%

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

      1. *-commutative 19.1%

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

      2. fma-define 19.1%

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

      3. *-commutative 19.1%

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

   3. Simplified 19.1%

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

   5. Taylor expanded in x around inf 84.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-317}:\\ \;\;\;\;-1 + 2 \cdot \log \left(1 + 0.25 \cdot \frac{\frac{x}{y}}{\frac{y}{x}}\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+210}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+229}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-317}:\\ \;\;\;\;-1 + \left(-1 + \left(1 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+210}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+229}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-317)
     (+ -1.0 (+ -1.0 (+ 1.0 (* (* (/ x y) (/ x y)) 0.5))))
     (if (<= (* x x) 1e+210)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (if (<= (* x x) 2e+229) -1.0 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-317) {
		tmp = -1.0 + (-1.0 + (1.0 + (((x / y) * (x / y)) * 0.5)));
	} else if ((x * x) <= 1e+210) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 2e+229) {
		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) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 1d-317) then
        tmp = (-1.0d0) + ((-1.0d0) + (1.0d0 + (((x / y) * (x / y)) * 0.5d0)))
    else if ((x * x) <= 1d+210) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else if ((x * x) <= 2d+229) then
        tmp = -1.0d0
    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-317) {
		tmp = -1.0 + (-1.0 + (1.0 + (((x / y) * (x / y)) * 0.5)));
	} else if ((x * x) <= 1e+210) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else if ((x * x) <= 2e+229) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 1e-317:
		tmp = -1.0 + (-1.0 + (1.0 + (((x / y) * (x / y)) * 0.5)))
	elif (x * x) <= 1e+210:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	elif (x * x) <= 2e+229:
		tmp = -1.0
	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-317)
		tmp = Float64(-1.0 + Float64(-1.0 + Float64(1.0 + Float64(Float64(Float64(x / y) * Float64(x / y)) * 0.5))));
	elseif (Float64(x * x) <= 1e+210)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	elseif (Float64(x * x) <= 2e+229)
		tmp = -1.0;
	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-317)
		tmp = -1.0 + (-1.0 + (1.0 + (((x / y) * (x / y)) * 0.5)));
	elseif ((x * x) <= 1e+210)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	elseif ((x * x) <= 2e+229)
		tmp = -1.0;
	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-317], N[(-1.0 + N[(-1.0 + N[(1.0 + N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+210], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+229], -1.0, 1.0]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 1.00000023e-317

   1. Initial program 43.9%

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

      1. *-commutative 43.9%

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

      2. fma-define 43.9%

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

      3. *-commutative 43.9%

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

   3. Simplified 43.9%

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

   5. Taylor expanded in x around 0 87.7%

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

      1. unpow2 87.7%

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

      2. unpow2 87.7%

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

      3. times-frac 92.6%

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

   7. Applied egg-rr 92.6%

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

      1. *-commutative 92.6%

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

      2. pow2 92.6%

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

      3. expm1-log1p-u 92.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{x}{y}\right)}^{2} \cdot 0.5\right)\right)} - 1 \]

      4. expm1-define 92.6%

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

   9. Applied egg-rr 92.6%

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

       1. sqr-pow 92.6%

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

       2. metadata-eval 92.6%

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

       3. inv-pow 92.6%

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

       4. clear-num 92.6%

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

       5. metadata-eval 92.6%

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

       6. inv-pow 92.6%

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

       7. clear-num 92.6%

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

   11. Applied egg-rr 92.6%

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

   if 1.00000023e-317 < (*.f64 x x) < 9.99999999999999927e209

   1. Initial program 76.9%

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

   if 9.99999999999999927e209 < (*.f64 x x) < 2e229

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. fma-define 0.0%

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

      3. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 2e229 < (*.f64 x x)

   1. Initial program 19.1%

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

      1. *-commutative 19.1%

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

      2. fma-define 19.1%

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

      3. *-commutative 19.1%

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

   3. Simplified 19.1%

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

   5. Taylor expanded in x around inf 84.3%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-317}:\\ \;\;\;\;-1 + \left(-1 + \left(1 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+210}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+229}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+85}:\\ \;\;\;\;-1 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+107}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+114}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6e+85)
   (+ -1.0 (* (* (/ x y) (/ x y)) 0.5))
   (if (<= x 3.8e+107) 1.0 (if (<= x 8e+114) -1.0 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6e+85) {
		tmp = -1.0 + (((x / y) * (x / y)) * 0.5);
	} else if (x <= 3.8e+107) {
		tmp = 1.0;
	} else if (x <= 8e+114) {
		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 <= 6d+85) then
        tmp = (-1.0d0) + (((x / y) * (x / y)) * 0.5d0)
    else if (x <= 3.8d+107) then
        tmp = 1.0d0
    else if (x <= 8d+114) 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 <= 6e+85) {
		tmp = -1.0 + (((x / y) * (x / y)) * 0.5);
	} else if (x <= 3.8e+107) {
		tmp = 1.0;
	} else if (x <= 8e+114) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6e+85:
		tmp = -1.0 + (((x / y) * (x / y)) * 0.5)
	elif x <= 3.8e+107:
		tmp = 1.0
	elif x <= 8e+114:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6e+85)
		tmp = Float64(-1.0 + Float64(Float64(Float64(x / y) * Float64(x / y)) * 0.5));
	elseif (x <= 3.8e+107)
		tmp = 1.0;
	elseif (x <= 8e+114)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6e+85)
		tmp = -1.0 + (((x / y) * (x / y)) * 0.5);
	elseif (x <= 3.8e+107)
		tmp = 1.0;
	elseif (x <= 8e+114)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6e+85], N[(-1.0 + N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e+107], 1.0, If[LessEqual[x, 8e+114], -1.0, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.0000000000000001e85

   1. Initial program 55.3%

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

      1. *-commutative 55.3%

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

      2. fma-define 55.3%

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

      3. *-commutative 55.3%

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

   3. Simplified 55.3%

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

   5. Taylor expanded in x around 0 56.8%

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

      1. unpow2 56.8%

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

      2. unpow2 56.8%

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

      3. times-frac 60.0%

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

   7. Applied egg-rr 60.0%

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

   if 6.0000000000000001e85 < x < 3.7999999999999998e107 or 8e114 < x

   1. Initial program 19.6%

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

      1. *-commutative 19.6%

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

      2. fma-define 19.6%

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

      3. *-commutative 19.6%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in x around inf 84.2%

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

   if 3.7999999999999998e107 < x < 8e114

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. fma-define 0.0%

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

      3. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+85}:\\ \;\;\;\;-1 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+107}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+114}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-129}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.06 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+32}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.5e-129)
   -1.0
   (if (<= x 2.06e-122) 1.0 (if (<= x 1e+32) -1.0 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.5e-129) {
		tmp = -1.0;
	} else if (x <= 2.06e-122) {
		tmp = 1.0;
	} else if (x <= 1e+32) {
		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 <= 3.5d-129) then
        tmp = -1.0d0
    else if (x <= 2.06d-122) then
        tmp = 1.0d0
    else if (x <= 1d+32) 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 <= 3.5e-129) {
		tmp = -1.0;
	} else if (x <= 2.06e-122) {
		tmp = 1.0;
	} else if (x <= 1e+32) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.5e-129:
		tmp = -1.0
	elif x <= 2.06e-122:
		tmp = 1.0
	elif x <= 1e+32:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.5e-129)
		tmp = -1.0;
	elseif (x <= 2.06e-122)
		tmp = 1.0;
	elseif (x <= 1e+32)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.5e-129)
		tmp = -1.0;
	elseif (x <= 2.06e-122)
		tmp = 1.0;
	elseif (x <= 1e+32)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.5e-129], -1.0, If[LessEqual[x, 2.06e-122], 1.0, If[LessEqual[x, 1e+32], -1.0, 1.0]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4999999999999997e-129 or 2.06e-122 < x < 1.00000000000000005e32

   1. Initial program 53.8%

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

      1. *-commutative 53.8%

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

      2. fma-define 53.8%

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

      3. *-commutative 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in x around 0 60.0%

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

   if 3.4999999999999997e-129 < x < 2.06e-122 or 1.00000000000000005e32 < x

   1. Initial program 29.7%

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

      1. *-commutative 29.7%

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

      2. fma-define 29.7%

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

      3. *-commutative 29.7%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in x around inf 76.1%

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

Alternative 7: 51.5% 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 46.9%

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

   1. *-commutative 46.9%

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

   2. fma-define 46.9%

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

   3. *-commutative 46.9%

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

3. Simplified 46.9%

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

5. Taylor expanded in x around 0 49.7%

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

Developer target: 50.9% 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 2024100 
(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))))