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

Percentage Accurate: 50.4% → 80.5%

Time: 3.9s

Alternatives: 6

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: 50.4% 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: 80.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 10^{-224}:\\ \;\;\;\;1 - \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot 8\\ \mathbf{elif}\;t_0 \leq 10^{+177}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 1e-224)
     (- 1.0 (* (* (/ y x) (/ y x)) 8.0))
     (if (<= t_0 1e+177)
       (/ (* (+ x (* y 2.0)) (- x (* y 2.0))) (+ t_0 (* x x)))
       (fma 0.5 (* (/ x y) (/ x y)) -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-224) {
		tmp = 1.0 - (((y / x) * (y / x)) * 8.0);
	} else if (t_0 <= 1e+177) {
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 1e-224)
		tmp = Float64(1.0 - Float64(Float64(Float64(y / x) * Float64(y / x)) * 8.0));
	elseif (t_0 <= 1e+177)
		tmp = Float64(Float64(Float64(x + Float64(y * 2.0)) * Float64(x - Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)));
	else
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-224], N[(1.0 - N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+177], N[(N[(N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 1e-224

   1. Initial program 57.3%

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

      1. add-sqr-sqrt 57.3%

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

      2. difference-of-squares 57.3%

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

      3. *-commutative 57.3%

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

      4. associate-*r* 57.3%

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

      5. sqrt-prod 57.3%

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

      6. sqrt-unprod 35.3%

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

      7. add-sqr-sqrt 57.3%

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

      8. metadata-eval 57.3%

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

      9. *-commutative 57.3%

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

      10. associate-*r* 56.4%

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

      11. sqrt-prod 56.4%

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

      12. sqrt-unprod 35.3%

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

      13. add-sqr-sqrt 57.3%

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

      14. metadata-eval 57.3%

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

   3. Applied egg-rr 57.3%

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

   4. Taylor expanded in x around -inf 83.1%

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

      1. associate-+r+ 83.1%

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

      2. distribute-rgt-out 83.1%

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

      3. metadata-eval 83.1%

         \[\leadsto 1 + \left(\frac{y}{x} \cdot \color{blue}{0} + -1 \cdot \frac{4 \cdot {y}^{2} - -4 \cdot {y}^{2}}{{x}^{2}}\right) \]

      4. mul0-rgt 83.1%

         \[\leadsto 1 + \left(\color{blue}{0} + -1 \cdot \frac{4 \cdot {y}^{2} - -4 \cdot {y}^{2}}{{x}^{2}}\right) \]

      5. +-lft-identity 83.1%

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

      6. mul-1-neg 83.1%

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

      7. unsub-neg 83.1%

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

      8. unpow2 83.1%

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

      9. div-sub 83.1%

         \[\leadsto 1 - \color{blue}{\left(\frac{4 \cdot {y}^{2}}{x \cdot x} - \frac{-4 \cdot {y}^{2}}{x \cdot x}\right)} \]

      10. associate-*r/ 83.1%

          \[\leadsto 1 - \left(\color{blue}{4 \cdot \frac{{y}^{2}}{x \cdot x}} - \frac{-4 \cdot {y}^{2}}{x \cdot x}\right) \]

      11. unpow2 83.1%

          \[\leadsto 1 - \left(4 \cdot \frac{{y}^{2}}{\color{blue}{{x}^{2}}} - \frac{-4 \cdot {y}^{2}}{x \cdot x}\right) \]

      12. associate-*r/ 83.1%

          \[\leadsto 1 - \left(4 \cdot \frac{{y}^{2}}{{x}^{2}} - \color{blue}{-4 \cdot \frac{{y}^{2}}{x \cdot x}}\right) \]

      13. unpow2 83.1%

          \[\leadsto 1 - \left(4 \cdot \frac{{y}^{2}}{{x}^{2}} - -4 \cdot \frac{{y}^{2}}{\color{blue}{{x}^{2}}}\right) \]

   6. Simplified 93.1%

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

   if 1e-224 < (*.f64 (*.f64 y 4) y) < 1e177

   1. Initial program 83.1%

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

      1. add-sqr-sqrt 83.1%

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

      2. difference-of-squares 83.1%

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

      3. *-commutative 83.1%

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

      4. associate-*r* 83.1%

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

      5. sqrt-prod 83.1%

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

      6. sqrt-unprod 42.5%

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

      7. add-sqr-sqrt 52.0%

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

      8. metadata-eval 52.0%

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

      9. *-commutative 52.0%

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

      10. associate-*r* 52.0%

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

      11. sqrt-prod 52.0%

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

      12. sqrt-unprod 42.5%

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

      13. add-sqr-sqrt 83.1%

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

      14. metadata-eval 83.1%

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

   3. Applied egg-rr 83.1%

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

   if 1e177 < (*.f64 (*.f64 y 4) y)

   1. Initial program 27.1%

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

   2. Taylor expanded in x around 0 84.9%

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

      1. fma-neg 84.9%

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

      2. unpow2 84.9%

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

      3. unpow2 84.9%

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

      4. times-frac 91.1%

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

      5. metadata-eval 91.1%

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

   4. Simplified 91.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-224}:\\ \;\;\;\;1 - \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot 8\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+177}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternative 2: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-193} \lor \neg \left(t_0 \leq 10^{-163}\right) \land \left(t_0 \leq 2 \cdot 10^{-122} \lor \neg \left(t_0 \leq 5 \cdot 10^{-86}\right) \land t_0 \leq 5 \cdot 10^{-47}\right):\\ \;\;\;\;1 - \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (or (<= t_0 2e-193)
           (and (not (<= t_0 1e-163))
                (or (<= t_0 2e-122)
                    (and (not (<= t_0 5e-86)) (<= t_0 5e-47)))))
     (- 1.0 (* (* (/ y x) (/ y x)) 8.0))
     -1.0)))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((t_0 <= 2e-193) || (!(t_0 <= 1e-163) && ((t_0 <= 2e-122) || (!(t_0 <= 5e-86) && (t_0 <= 5e-47))))) {
		tmp = 1.0 - (((y / x) * (y / x)) * 8.0);
	} 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 ((t_0 <= 2d-193) .or. (.not. (t_0 <= 1d-163)) .and. (t_0 <= 2d-122) .or. (.not. (t_0 <= 5d-86)) .and. (t_0 <= 5d-47)) then
        tmp = 1.0d0 - (((y / x) * (y / x)) * 8.0d0)
    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 ((t_0 <= 2e-193) || (!(t_0 <= 1e-163) && ((t_0 <= 2e-122) || (!(t_0 <= 5e-86) && (t_0 <= 5e-47))))) {
		tmp = 1.0 - (((y / x) * (y / x)) * 8.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (t_0 <= 2e-193) or (not (t_0 <= 1e-163) and ((t_0 <= 2e-122) or (not (t_0 <= 5e-86) and (t_0 <= 5e-47)))):
		tmp = 1.0 - (((y / x) * (y / x)) * 8.0)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if ((t_0 <= 2e-193) || (!(t_0 <= 1e-163) && ((t_0 <= 2e-122) || (!(t_0 <= 5e-86) && (t_0 <= 5e-47)))))
		tmp = Float64(1.0 - Float64(Float64(Float64(y / x) * Float64(y / x)) * 8.0));
	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 ((t_0 <= 2e-193) || (~((t_0 <= 1e-163)) && ((t_0 <= 2e-122) || (~((t_0 <= 5e-86)) && (t_0 <= 5e-47)))))
		tmp = 1.0 - (((y / x) * (y / x)) * 8.0);
	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[Or[LessEqual[t$95$0, 2e-193], And[N[Not[LessEqual[t$95$0, 1e-163]], $MachinePrecision], Or[LessEqual[t$95$0, 2e-122], And[N[Not[LessEqual[t$95$0, 5e-86]], $MachinePrecision], LessEqual[t$95$0, 5e-47]]]]], N[(1.0 - N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y 4) y) < 2.0000000000000001e-193 or 9.99999999999999923e-164 < (*.f64 (*.f64 y 4) y) < 2.00000000000000012e-122 or 4.9999999999999999e-86 < (*.f64 (*.f64 y 4) y) < 5.00000000000000011e-47

   1. Initial program 58.7%

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

      1. add-sqr-sqrt 58.7%

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

      2. difference-of-squares 58.7%

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

      3. *-commutative 58.7%

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

      4. associate-*r* 58.7%

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

      5. sqrt-prod 58.7%

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

      6. sqrt-unprod 33.0%

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

      7. add-sqr-sqrt 54.1%

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

      8. metadata-eval 54.1%

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

      9. *-commutative 54.1%

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

      10. associate-*r* 53.4%

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

      11. sqrt-prod 53.4%

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

      12. sqrt-unprod 33.0%

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

      13. add-sqr-sqrt 58.7%

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

      14. metadata-eval 58.7%

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

   3. Applied egg-rr 58.7%

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

   4. Taylor expanded in x around -inf 80.2%

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

      1. associate-+r+ 80.2%

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

      2. distribute-rgt-out 80.2%

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

      3. metadata-eval 80.2%

         \[\leadsto 1 + \left(\frac{y}{x} \cdot \color{blue}{0} + -1 \cdot \frac{4 \cdot {y}^{2} - -4 \cdot {y}^{2}}{{x}^{2}}\right) \]

      4. mul0-rgt 80.2%

         \[\leadsto 1 + \left(\color{blue}{0} + -1 \cdot \frac{4 \cdot {y}^{2} - -4 \cdot {y}^{2}}{{x}^{2}}\right) \]

      5. +-lft-identity 80.2%

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

      6. mul-1-neg 80.2%

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

      7. unsub-neg 80.2%

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

      8. unpow2 80.2%

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

      9. div-sub 80.2%

         \[\leadsto 1 - \color{blue}{\left(\frac{4 \cdot {y}^{2}}{x \cdot x} - \frac{-4 \cdot {y}^{2}}{x \cdot x}\right)} \]

      10. associate-*r/ 80.2%

          \[\leadsto 1 - \left(\color{blue}{4 \cdot \frac{{y}^{2}}{x \cdot x}} - \frac{-4 \cdot {y}^{2}}{x \cdot x}\right) \]

      11. unpow2 80.2%

          \[\leadsto 1 - \left(4 \cdot \frac{{y}^{2}}{\color{blue}{{x}^{2}}} - \frac{-4 \cdot {y}^{2}}{x \cdot x}\right) \]

      12. associate-*r/ 80.2%

          \[\leadsto 1 - \left(4 \cdot \frac{{y}^{2}}{{x}^{2}} - \color{blue}{-4 \cdot \frac{{y}^{2}}{x \cdot x}}\right) \]

      13. unpow2 80.2%

          \[\leadsto 1 - \left(4 \cdot \frac{{y}^{2}}{{x}^{2}} - -4 \cdot \frac{{y}^{2}}{\color{blue}{{x}^{2}}}\right) \]

   6. Simplified 87.8%

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

   if 2.0000000000000001e-193 < (*.f64 (*.f64 y 4) y) < 9.99999999999999923e-164 or 2.00000000000000012e-122 < (*.f64 (*.f64 y 4) y) < 4.9999999999999999e-86 or 5.00000000000000011e-47 < (*.f64 (*.f64 y 4) y)

   1. Initial program 54.4%

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

   2. Taylor expanded in x around 0 84.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-193} \lor \neg \left(y \cdot \left(y \cdot 4\right) \leq 10^{-163}\right) \land \left(y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-122} \lor \neg \left(y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{-86}\right) \land y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{-47}\right):\\ \;\;\;\;1 - \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 10^{-224}:\\ \;\;\;\;1 - \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot 8\\ \mathbf{elif}\;t_0 \leq 10^{+177}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 1e-224)
     (- 1.0 (* (* (/ y x) (/ y x)) 8.0))
     (if (<= t_0 1e+177)
       (/ (* (+ x (* y 2.0)) (- x (* y 2.0))) (+ t_0 (* x x)))
       -1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-224) {
		tmp = 1.0 - (((y / x) * (y / x)) * 8.0);
	} else if (t_0 <= 1e+177) {
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 1d-224) then
        tmp = 1.0d0 - (((y / x) * (y / x)) * 8.0d0)
    else if (t_0 <= 1d+177) then
        tmp = ((x + (y * 2.0d0)) * (x - (y * 2.0d0))) / (t_0 + (x * x))
    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 (t_0 <= 1e-224) {
		tmp = 1.0 - (((y / x) * (y / x)) * 8.0);
	} else if (t_0 <= 1e+177) {
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 1e-224:
		tmp = 1.0 - (((y / x) * (y / x)) * 8.0)
	elif t_0 <= 1e+177:
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 1e-224)
		tmp = Float64(1.0 - Float64(Float64(Float64(y / x) * Float64(y / x)) * 8.0));
	elseif (t_0 <= 1e+177)
		tmp = Float64(Float64(Float64(x + Float64(y * 2.0)) * Float64(x - Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)));
	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 (t_0 <= 1e-224)
		tmp = 1.0 - (((y / x) * (y / x)) * 8.0);
	elseif (t_0 <= 1e+177)
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	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[t$95$0, 1e-224], N[(1.0 - N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+177], N[(N[(N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 1e-224

   1. Initial program 57.3%

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

      1. add-sqr-sqrt 57.3%

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

      2. difference-of-squares 57.3%

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

      3. *-commutative 57.3%

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

      4. associate-*r* 57.3%

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

      5. sqrt-prod 57.3%

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

      6. sqrt-unprod 35.3%

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

      7. add-sqr-sqrt 57.3%

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

      8. metadata-eval 57.3%

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

      9. *-commutative 57.3%

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

      10. associate-*r* 56.4%

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

      11. sqrt-prod 56.4%

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

      12. sqrt-unprod 35.3%

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

      13. add-sqr-sqrt 57.3%

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

      14. metadata-eval 57.3%

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

   3. Applied egg-rr 57.3%

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

   4. Taylor expanded in x around -inf 83.1%

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

      1. associate-+r+ 83.1%

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

      2. distribute-rgt-out 83.1%

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

      3. metadata-eval 83.1%

         \[\leadsto 1 + \left(\frac{y}{x} \cdot \color{blue}{0} + -1 \cdot \frac{4 \cdot {y}^{2} - -4 \cdot {y}^{2}}{{x}^{2}}\right) \]

      4. mul0-rgt 83.1%

         \[\leadsto 1 + \left(\color{blue}{0} + -1 \cdot \frac{4 \cdot {y}^{2} - -4 \cdot {y}^{2}}{{x}^{2}}\right) \]

      5. +-lft-identity 83.1%

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

      6. mul-1-neg 83.1%

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

      7. unsub-neg 83.1%

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

      8. unpow2 83.1%

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

      9. div-sub 83.1%

         \[\leadsto 1 - \color{blue}{\left(\frac{4 \cdot {y}^{2}}{x \cdot x} - \frac{-4 \cdot {y}^{2}}{x \cdot x}\right)} \]

      10. associate-*r/ 83.1%

          \[\leadsto 1 - \left(\color{blue}{4 \cdot \frac{{y}^{2}}{x \cdot x}} - \frac{-4 \cdot {y}^{2}}{x \cdot x}\right) \]

      11. unpow2 83.1%

          \[\leadsto 1 - \left(4 \cdot \frac{{y}^{2}}{\color{blue}{{x}^{2}}} - \frac{-4 \cdot {y}^{2}}{x \cdot x}\right) \]

      12. associate-*r/ 83.1%

          \[\leadsto 1 - \left(4 \cdot \frac{{y}^{2}}{{x}^{2}} - \color{blue}{-4 \cdot \frac{{y}^{2}}{x \cdot x}}\right) \]

      13. unpow2 83.1%

          \[\leadsto 1 - \left(4 \cdot \frac{{y}^{2}}{{x}^{2}} - -4 \cdot \frac{{y}^{2}}{\color{blue}{{x}^{2}}}\right) \]

   6. Simplified 93.1%

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

   if 1e-224 < (*.f64 (*.f64 y 4) y) < 1e177

   1. Initial program 83.1%

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

      1. add-sqr-sqrt 83.1%

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

      2. difference-of-squares 83.1%

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

      3. *-commutative 83.1%

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

      4. associate-*r* 83.1%

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

      5. sqrt-prod 83.1%

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

      6. sqrt-unprod 42.5%

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

      7. add-sqr-sqrt 52.0%

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

      8. metadata-eval 52.0%

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

      9. *-commutative 52.0%

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

      10. associate-*r* 52.0%

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

      11. sqrt-prod 52.0%

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

      12. sqrt-unprod 42.5%

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

      13. add-sqr-sqrt 83.1%

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

      14. metadata-eval 83.1%

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

   3. Applied egg-rr 83.1%

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

   if 1e177 < (*.f64 (*.f64 y 4) y)

   1. Initial program 27.1%

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

   2. Taylor expanded in x around 0 90.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-224}:\\ \;\;\;\;1 - \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot 8\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+177}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 10^{-224}:\\ \;\;\;\;1 - \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot 8\\ \mathbf{elif}\;t_0 \leq 10^{+177}:\\ \;\;\;\;\frac{x \cdot x - t_0}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 1e-224)
     (- 1.0 (* (* (/ y x) (/ y x)) 8.0))
     (if (<= t_0 1e+177) (/ (- (* x x) t_0) (+ t_0 (* x x))) -1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-224) {
		tmp = 1.0 - (((y / x) * (y / x)) * 8.0);
	} else if (t_0 <= 1e+177) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 1d-224) then
        tmp = 1.0d0 - (((y / x) * (y / x)) * 8.0d0)
    else if (t_0 <= 1d+177) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    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 (t_0 <= 1e-224) {
		tmp = 1.0 - (((y / x) * (y / x)) * 8.0);
	} else if (t_0 <= 1e+177) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 1e-224:
		tmp = 1.0 - (((y / x) * (y / x)) * 8.0)
	elif t_0 <= 1e+177:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 1e-224)
		tmp = Float64(1.0 - Float64(Float64(Float64(y / x) * Float64(y / x)) * 8.0));
	elseif (t_0 <= 1e+177)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	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 (t_0 <= 1e-224)
		tmp = 1.0 - (((y / x) * (y / x)) * 8.0);
	elseif (t_0 <= 1e+177)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	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[t$95$0, 1e-224], N[(1.0 - N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+177], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 1e-224

   1. Initial program 57.3%

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

      1. add-sqr-sqrt 57.3%

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

      2. difference-of-squares 57.3%

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

      3. *-commutative 57.3%

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

      4. associate-*r* 57.3%

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

      5. sqrt-prod 57.3%

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

      6. sqrt-unprod 35.3%

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

      7. add-sqr-sqrt 57.3%

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

      8. metadata-eval 57.3%

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

      9. *-commutative 57.3%

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

      10. associate-*r* 56.4%

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

      11. sqrt-prod 56.4%

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

      12. sqrt-unprod 35.3%

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

      13. add-sqr-sqrt 57.3%

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

      14. metadata-eval 57.3%

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

   3. Applied egg-rr 57.3%

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

   4. Taylor expanded in x around -inf 83.1%

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

      1. associate-+r+ 83.1%

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

      2. distribute-rgt-out 83.1%

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

      3. metadata-eval 83.1%

         \[\leadsto 1 + \left(\frac{y}{x} \cdot \color{blue}{0} + -1 \cdot \frac{4 \cdot {y}^{2} - -4 \cdot {y}^{2}}{{x}^{2}}\right) \]

      4. mul0-rgt 83.1%

         \[\leadsto 1 + \left(\color{blue}{0} + -1 \cdot \frac{4 \cdot {y}^{2} - -4 \cdot {y}^{2}}{{x}^{2}}\right) \]

      5. +-lft-identity 83.1%

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

      6. mul-1-neg 83.1%

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

      7. unsub-neg 83.1%

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

      8. unpow2 83.1%

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

      9. div-sub 83.1%

         \[\leadsto 1 - \color{blue}{\left(\frac{4 \cdot {y}^{2}}{x \cdot x} - \frac{-4 \cdot {y}^{2}}{x \cdot x}\right)} \]

      10. associate-*r/ 83.1%

          \[\leadsto 1 - \left(\color{blue}{4 \cdot \frac{{y}^{2}}{x \cdot x}} - \frac{-4 \cdot {y}^{2}}{x \cdot x}\right) \]

      11. unpow2 83.1%

          \[\leadsto 1 - \left(4 \cdot \frac{{y}^{2}}{\color{blue}{{x}^{2}}} - \frac{-4 \cdot {y}^{2}}{x \cdot x}\right) \]

      12. associate-*r/ 83.1%

          \[\leadsto 1 - \left(4 \cdot \frac{{y}^{2}}{{x}^{2}} - \color{blue}{-4 \cdot \frac{{y}^{2}}{x \cdot x}}\right) \]

      13. unpow2 83.1%

          \[\leadsto 1 - \left(4 \cdot \frac{{y}^{2}}{{x}^{2}} - -4 \cdot \frac{{y}^{2}}{\color{blue}{{x}^{2}}}\right) \]

   6. Simplified 93.1%

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

   if 1e-224 < (*.f64 (*.f64 y 4) y) < 1e177

   1. Initial program 83.1%

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

   if 1e177 < (*.f64 (*.f64 y 4) y)

   1. Initial program 27.1%

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

   2. Taylor expanded in x around 0 90.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-224}:\\ \;\;\;\;1 - \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot 8\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+177}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 72.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{-108}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-97}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-75}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{-33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.35e-108)
   -1.0
   (if (<= y 2.25e-97) 1.0 (if (<= y 2e-75) -1.0 (if (<= y 1e-33) 1.0 -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.35e-108) {
		tmp = -1.0;
	} else if (y <= 2.25e-97) {
		tmp = 1.0;
	} else if (y <= 2e-75) {
		tmp = -1.0;
	} else if (y <= 1e-33) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.35d-108)) then
        tmp = -1.0d0
    else if (y <= 2.25d-97) then
        tmp = 1.0d0
    else if (y <= 2d-75) then
        tmp = -1.0d0
    else if (y <= 1d-33) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.35e-108) {
		tmp = -1.0;
	} else if (y <= 2.25e-97) {
		tmp = 1.0;
	} else if (y <= 2e-75) {
		tmp = -1.0;
	} else if (y <= 1e-33) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.35e-108:
		tmp = -1.0
	elif y <= 2.25e-97:
		tmp = 1.0
	elif y <= 2e-75:
		tmp = -1.0
	elif y <= 1e-33:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.35e-108)
		tmp = -1.0;
	elseif (y <= 2.25e-97)
		tmp = 1.0;
	elseif (y <= 2e-75)
		tmp = -1.0;
	elseif (y <= 1e-33)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.35e-108)
		tmp = -1.0;
	elseif (y <= 2.25e-97)
		tmp = 1.0;
	elseif (y <= 2e-75)
		tmp = -1.0;
	elseif (y <= 1e-33)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.35e-108], -1.0, If[LessEqual[y, 2.25e-97], 1.0, If[LessEqual[y, 2e-75], -1.0, If[LessEqual[y, 1e-33], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.35000000000000006e-108 or 2.25000000000000005e-97 < y < 1.9999999999999999e-75 or 1.0000000000000001e-33 < y

   1. Initial program 54.4%

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

   2. Taylor expanded in x around 0 80.5%

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

   if -2.35000000000000006e-108 < y < 2.25000000000000005e-97 or 1.9999999999999999e-75 < y < 1.0000000000000001e-33

   1. Initial program 59.2%

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

   2. Taylor expanded in x around inf 89.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{-108}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-97}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-75}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{-33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 50.5% 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 56.2%

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

2. Taylor expanded in x around 0 54.5%

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

3. Final simplification 54.5%

   \[\leadsto -1 \]

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 2023175 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

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

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