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

Percentage Accurate: 50.2% → 99.9%

Time: 6.2s

Alternatives: 8

Speedup: 1.2×

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: 50.2% 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(x, 2 \cdot y\right)\\ \frac{\mathsf{fma}\left(2, y, x\right)}{t\_0} \cdot \frac{x + y \cdot -2}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot x (* 2.0 y))))
   (* (/ (fma 2.0 y x) t_0) (/ (+ x (* y -2.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = hypot(x, (2.0 * y));
	return (fma(2.0, y, x) / t_0) * ((x + (y * -2.0)) / t_0);
}

Julia

function code(x, y)
	t_0 = hypot(x, Float64(2.0 * y))
	return Float64(Float64(fma(2.0, y, x) / t_0) * Float64(Float64(x + Float64(y * -2.0)) / t_0))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(2.0 * y), $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[(N[(2.0 * y + x), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(x + N[(y * -2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 50.4%

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

   1. add-sqr-sqrt 50.4%

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

   2. difference-of-squares 50.4%

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

   3. *-commutative 50.4%

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

   4. associate-*r* 50.4%

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

   5. sqrt-prod 50.4%

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

   6. sqrt-unprod 26.9%

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

   7. add-sqr-sqrt 36.9%

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

   8. metadata-eval 36.9%

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

   9. *-commutative 36.9%

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

   10. associate-*r* 36.9%

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

   11. sqrt-prod 36.9%

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

   12. sqrt-unprod 26.9%

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

   13. add-sqr-sqrt 50.4%

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

   14. metadata-eval 50.4%

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

4. Applied egg-rr 50.4%

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

   1. add-sqr-sqrt 50.4%

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

   2. times-frac 51.9%

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

   3. +-commutative 51.9%

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

   4. *-commutative 51.9%

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

   5. fma-define 51.9%

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

   6. add-sqr-sqrt 51.9%

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

   7. hypot-define 51.9%

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

   8. sqrt-prod 27.7%

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

   9. *-commutative 27.7%

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

   10. sqrt-prod 27.7%

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

   11. metadata-eval 27.7%

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

   12. associate-*r* 27.7%

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

   13. add-sqr-sqrt 51.9%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2, y, x\right)}{\mathsf{hypot}\left(x, 2 \cdot y\right)} \cdot \frac{x + y \cdot -2}{\mathsf{hypot}\left(x, 2 \cdot y\right)}} \]
7. Add Preprocessing

Alternative 2: 65.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x + y \cdot -2}{\mathsf{hypot}\left(x, 2 \cdot y\right)} \cdot \left(1 + 2 \cdot \frac{y}{x}\right)\\ t_2 := t\_0 + x \cdot x\\ \mathbf{if}\;t\_0 \leq 10^{-237}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\left(x + 2 \cdot y\right) \cdot \left(x - 2 \cdot y\right)}{t\_2}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+228}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1
         (* (/ (+ x (* y -2.0)) (hypot x (* 2.0 y))) (+ 1.0 (* 2.0 (/ y x)))))
        (t_2 (+ t_0 (* x x))))
   (if (<= t_0 1e-237)
     t_1
     (if (<= t_0 5e-134)
       (/ (* (+ x (* 2.0 y)) (- x (* 2.0 y))) t_2)
       (if (<= t_0 2e-31)
         t_1
         (if (<= t_0 1e+228)
           (/ (- (* x x) t_0) t_2)
           (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x + (y * -2.0)) / hypot(x, (2.0 * y))) * (1.0 + (2.0 * (y / x)));
	double t_2 = t_0 + (x * x);
	double tmp;
	if (t_0 <= 1e-237) {
		tmp = t_1;
	} else if (t_0 <= 5e-134) {
		tmp = ((x + (2.0 * y)) * (x - (2.0 * y))) / t_2;
	} else if (t_0 <= 2e-31) {
		tmp = t_1;
	} else if (t_0 <= 1e+228) {
		tmp = ((x * x) - t_0) / t_2;
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x + (y * -2.0)) / Math.hypot(x, (2.0 * y))) * (1.0 + (2.0 * (y / x)));
	double t_2 = t_0 + (x * x);
	double tmp;
	if (t_0 <= 1e-237) {
		tmp = t_1;
	} else if (t_0 <= 5e-134) {
		tmp = ((x + (2.0 * y)) * (x - (2.0 * y))) / t_2;
	} else if (t_0 <= 2e-31) {
		tmp = t_1;
	} else if (t_0 <= 1e+228) {
		tmp = ((x * x) - t_0) / t_2;
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x + (y * -2.0)) / math.hypot(x, (2.0 * y))) * (1.0 + (2.0 * (y / x)))
	t_2 = t_0 + (x * x)
	tmp = 0
	if t_0 <= 1e-237:
		tmp = t_1
	elif t_0 <= 5e-134:
		tmp = ((x + (2.0 * y)) * (x - (2.0 * y))) / t_2
	elif t_0 <= 2e-31:
		tmp = t_1
	elif t_0 <= 1e+228:
		tmp = ((x * x) - t_0) / t_2
	else:
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x + Float64(y * -2.0)) / hypot(x, Float64(2.0 * y))) * Float64(1.0 + Float64(2.0 * Float64(y / x))))
	t_2 = Float64(t_0 + Float64(x * x))
	tmp = 0.0
	if (t_0 <= 1e-237)
		tmp = t_1;
	elseif (t_0 <= 5e-134)
		tmp = Float64(Float64(Float64(x + Float64(2.0 * y)) * Float64(x - Float64(2.0 * y))) / t_2);
	elseif (t_0 <= 2e-31)
		tmp = t_1;
	elseif (t_0 <= 1e+228)
		tmp = Float64(Float64(Float64(x * x) - t_0) / t_2);
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = ((x + (y * -2.0)) / hypot(x, (2.0 * y))) * (1.0 + (2.0 * (y / x)));
	t_2 = t_0 + (x * x);
	tmp = 0.0;
	if (t_0 <= 1e-237)
		tmp = t_1;
	elseif (t_0 <= 5e-134)
		tmp = ((x + (2.0 * y)) * (x - (2.0 * y))) / t_2;
	elseif (t_0 <= 2e-31)
		tmp = t_1;
	elseif (t_0 <= 1e+228)
		tmp = ((x * x) - t_0) / t_2;
	else
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x + N[(y * -2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + N[(2.0 * y), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-237], t$95$1, If[LessEqual[t$95$0, 5e-134], N[(N[(N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision] * N[(x - N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 2e-31], t$95$1, If[LessEqual[t$95$0, 1e+228], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 9.9999999999999999e-238 or 5.0000000000000003e-134 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e-31

   1. Initial program 56.0%

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

      1. add-sqr-sqrt 56.0%

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

      2. difference-of-squares 56.0%

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

      3. *-commutative 56.0%

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

      4. associate-*r* 56.0%

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

      5. sqrt-prod 56.0%

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

      6. sqrt-unprod 35.1%

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

      7. add-sqr-sqrt 50.4%

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

      8. metadata-eval 50.4%

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

      9. *-commutative 50.4%

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

      10. associate-*r* 50.4%

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

      11. sqrt-prod 50.4%

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

      12. sqrt-unprod 35.1%

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

      13. add-sqr-sqrt 56.0%

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

      14. metadata-eval 56.0%

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

   4. Applied egg-rr 56.0%

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

      1. add-sqr-sqrt 56.0%

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

      2. times-frac 57.3%

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

      3. +-commutative 57.3%

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

      4. *-commutative 57.3%

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

      5. fma-define 57.3%

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

      6. add-sqr-sqrt 57.3%

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

      7. hypot-define 57.3%

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

      8. sqrt-prod 35.9%

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

      9. *-commutative 35.9%

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

      10. sqrt-prod 35.9%

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

      11. metadata-eval 35.9%

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

      12. associate-*r* 35.9%

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

      13. add-sqr-sqrt 57.3%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 44.4%

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

   if 9.9999999999999999e-238 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 5.0000000000000003e-134

   1. Initial program 96.7%

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

      1. add-sqr-sqrt 96.7%

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

      2. difference-of-squares 96.7%

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

      3. *-commutative 96.7%

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

      4. associate-*r* 96.7%

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

      5. sqrt-prod 96.7%

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

      6. sqrt-unprod 57.9%

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

      7. add-sqr-sqrt 77.7%

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

      8. metadata-eval 77.7%

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

      9. *-commutative 77.7%

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

      10. associate-*r* 77.7%

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

      11. sqrt-prod 77.7%

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

      12. sqrt-unprod 57.9%

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

      13. add-sqr-sqrt 96.7%

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

      14. metadata-eval 96.7%

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

   4. Applied egg-rr 96.7%

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

   if 2e-31 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 9.9999999999999992e227

   1. Initial program 68.5%

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

   if 9.9999999999999992e227 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 13.8%

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

   3. Taylor expanded in x around 0 80.1%

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

      1. pow2 80.1%

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

      2. unpow2 80.1%

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

      3. times-frac 87.0%

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

   5. Applied egg-rr 87.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-237}:\\ \;\;\;\;\frac{x + y \cdot -2}{\mathsf{hypot}\left(x, 2 \cdot y\right)} \cdot \left(1 + 2 \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\left(x + 2 \cdot y\right) \cdot \left(x - 2 \cdot y\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-31}:\\ \;\;\;\;\frac{x + y \cdot -2}{\mathsf{hypot}\left(x, 2 \cdot y\right)} \cdot \left(1 + 2 \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+228}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \left(1 + 2 \cdot \frac{y}{x}\right) \cdot \left(1 + -2 \cdot \frac{y}{x}\right)\\ t_2 := t\_0 + x \cdot x\\ \mathbf{if}\;t\_0 \leq 10^{-237}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\left(x + 2 \cdot y\right) \cdot \left(x - 2 \cdot y\right)}{t\_2}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+228}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (* (+ 1.0 (* 2.0 (/ y x))) (+ 1.0 (* -2.0 (/ y x)))))
        (t_2 (+ t_0 (* x x))))
   (if (<= t_0 1e-237)
     t_1
     (if (<= t_0 5e-134)
       (/ (* (+ x (* 2.0 y)) (- x (* 2.0 y))) t_2)
       (if (<= t_0 2e-31)
         t_1
         (if (<= t_0 1e+228)
           (/ (- (* x x) t_0) t_2)
           (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = (1.0 + (2.0 * (y / x))) * (1.0 + (-2.0 * (y / x)));
	double t_2 = t_0 + (x * x);
	double tmp;
	if (t_0 <= 1e-237) {
		tmp = t_1;
	} else if (t_0 <= 5e-134) {
		tmp = ((x + (2.0 * y)) * (x - (2.0 * y))) / t_2;
	} else if (t_0 <= 2e-31) {
		tmp = t_1;
	} else if (t_0 <= 1e+228) {
		tmp = ((x * x) - t_0) / t_2;
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = (1.0d0 + (2.0d0 * (y / x))) * (1.0d0 + ((-2.0d0) * (y / x)))
    t_2 = t_0 + (x * x)
    if (t_0 <= 1d-237) then
        tmp = t_1
    else if (t_0 <= 5d-134) then
        tmp = ((x + (2.0d0 * y)) * (x - (2.0d0 * y))) / t_2
    else if (t_0 <= 2d-31) then
        tmp = t_1
    else if (t_0 <= 1d+228) then
        tmp = ((x * x) - t_0) / t_2
    else
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-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 t_1 = (1.0 + (2.0 * (y / x))) * (1.0 + (-2.0 * (y / x)));
	double t_2 = t_0 + (x * x);
	double tmp;
	if (t_0 <= 1e-237) {
		tmp = t_1;
	} else if (t_0 <= 5e-134) {
		tmp = ((x + (2.0 * y)) * (x - (2.0 * y))) / t_2;
	} else if (t_0 <= 2e-31) {
		tmp = t_1;
	} else if (t_0 <= 1e+228) {
		tmp = ((x * x) - t_0) / t_2;
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = (1.0 + (2.0 * (y / x))) * (1.0 + (-2.0 * (y / x)))
	t_2 = t_0 + (x * x)
	tmp = 0
	if t_0 <= 1e-237:
		tmp = t_1
	elif t_0 <= 5e-134:
		tmp = ((x + (2.0 * y)) * (x - (2.0 * y))) / t_2
	elif t_0 <= 2e-31:
		tmp = t_1
	elif t_0 <= 1e+228:
		tmp = ((x * x) - t_0) / t_2
	else:
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(1.0 + Float64(2.0 * Float64(y / x))) * Float64(1.0 + Float64(-2.0 * Float64(y / x))))
	t_2 = Float64(t_0 + Float64(x * x))
	tmp = 0.0
	if (t_0 <= 1e-237)
		tmp = t_1;
	elseif (t_0 <= 5e-134)
		tmp = Float64(Float64(Float64(x + Float64(2.0 * y)) * Float64(x - Float64(2.0 * y))) / t_2);
	elseif (t_0 <= 2e-31)
		tmp = t_1;
	elseif (t_0 <= 1e+228)
		tmp = Float64(Float64(Float64(x * x) - t_0) / t_2);
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = (1.0 + (2.0 * (y / x))) * (1.0 + (-2.0 * (y / x)));
	t_2 = t_0 + (x * x);
	tmp = 0.0;
	if (t_0 <= 1e-237)
		tmp = t_1;
	elseif (t_0 <= 5e-134)
		tmp = ((x + (2.0 * y)) * (x - (2.0 * y))) / t_2;
	elseif (t_0 <= 2e-31)
		tmp = t_1;
	elseif (t_0 <= 1e+228)
		tmp = ((x * x) - t_0) / t_2;
	else
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-237], t$95$1, If[LessEqual[t$95$0, 5e-134], N[(N[(N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision] * N[(x - N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 2e-31], t$95$1, If[LessEqual[t$95$0, 1e+228], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 9.9999999999999999e-238 or 5.0000000000000003e-134 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e-31

   1. Initial program 56.0%

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

      1. add-sqr-sqrt 56.0%

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

      2. difference-of-squares 56.0%

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

      3. *-commutative 56.0%

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

      4. associate-*r* 56.0%

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

      5. sqrt-prod 56.0%

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

      6. sqrt-unprod 35.1%

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

      7. add-sqr-sqrt 50.4%

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

      8. metadata-eval 50.4%

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

      9. *-commutative 50.4%

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

      10. associate-*r* 50.4%

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

      11. sqrt-prod 50.4%

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

      12. sqrt-unprod 35.1%

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

      13. add-sqr-sqrt 56.0%

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

      14. metadata-eval 56.0%

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

   4. Applied egg-rr 56.0%

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

      1. add-sqr-sqrt 56.0%

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

      2. times-frac 57.3%

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

      3. +-commutative 57.3%

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

      4. *-commutative 57.3%

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

      5. fma-define 57.3%

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

      6. add-sqr-sqrt 57.3%

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

      7. hypot-define 57.3%

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

      8. sqrt-prod 35.9%

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

      9. *-commutative 35.9%

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

      10. sqrt-prod 35.9%

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

      11. metadata-eval 35.9%

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

      12. associate-*r* 35.9%

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

      13. add-sqr-sqrt 57.3%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 44.4%

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

   8. Taylor expanded in x around inf 85.2%

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

   if 9.9999999999999999e-238 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 5.0000000000000003e-134

   1. Initial program 96.7%

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

      1. add-sqr-sqrt 96.7%

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

      2. difference-of-squares 96.7%

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

      3. *-commutative 96.7%

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

      4. associate-*r* 96.7%

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

      5. sqrt-prod 96.7%

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

      6. sqrt-unprod 57.9%

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

      7. add-sqr-sqrt 77.7%

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

      8. metadata-eval 77.7%

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

      9. *-commutative 77.7%

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

      10. associate-*r* 77.7%

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

      11. sqrt-prod 77.7%

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

      12. sqrt-unprod 57.9%

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

      13. add-sqr-sqrt 96.7%

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

      14. metadata-eval 96.7%

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

   4. Applied egg-rr 96.7%

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

   if 2e-31 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 9.9999999999999992e227

   1. Initial program 68.5%

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

   if 9.9999999999999992e227 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 13.8%

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

   3. Taylor expanded in x around 0 80.1%

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

      1. pow2 80.1%

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

      2. unpow2 80.1%

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

      3. times-frac 87.0%

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

   5. Applied egg-rr 87.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-237}:\\ \;\;\;\;\left(1 + 2 \cdot \frac{y}{x}\right) \cdot \left(1 + -2 \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\left(x + 2 \cdot y\right) \cdot \left(x - 2 \cdot y\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-31}:\\ \;\;\;\;\left(1 + 2 \cdot \frac{y}{x}\right) \cdot \left(1 + -2 \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+228}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t\_0}{t\_0 + x \cdot x}\\ t_2 := \left(1 + 2 \cdot \frac{y}{x}\right) \cdot \left(1 + -2 \cdot \frac{y}{x}\right)\\ \mathbf{if}\;t\_0 \leq 10^{-237}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-31}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+228}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (/ (- (* x x) t_0) (+ t_0 (* x x))))
        (t_2 (* (+ 1.0 (* 2.0 (/ y x))) (+ 1.0 (* -2.0 (/ y x))))))
   (if (<= t_0 1e-237)
     t_2
     (if (<= t_0 5e-134)
       t_1
       (if (<= t_0 2e-31)
         t_2
         (if (<= t_0 1e+228) t_1 (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / (t_0 + (x * x));
	double t_2 = (1.0 + (2.0 * (y / x))) * (1.0 + (-2.0 * (y / x)));
	double tmp;
	if (t_0 <= 1e-237) {
		tmp = t_2;
	} else if (t_0 <= 5e-134) {
		tmp = t_1;
	} else if (t_0 <= 2e-31) {
		tmp = t_2;
	} else if (t_0 <= 1e+228) {
		tmp = t_1;
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x * x) - t_0) / (t_0 + (x * x))
    t_2 = (1.0d0 + (2.0d0 * (y / x))) * (1.0d0 + ((-2.0d0) * (y / x)))
    if (t_0 <= 1d-237) then
        tmp = t_2
    else if (t_0 <= 5d-134) then
        tmp = t_1
    else if (t_0 <= 2d-31) then
        tmp = t_2
    else if (t_0 <= 1d+228) then
        tmp = t_1
    else
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-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 t_1 = ((x * x) - t_0) / (t_0 + (x * x));
	double t_2 = (1.0 + (2.0 * (y / x))) * (1.0 + (-2.0 * (y / x)));
	double tmp;
	if (t_0 <= 1e-237) {
		tmp = t_2;
	} else if (t_0 <= 5e-134) {
		tmp = t_1;
	} else if (t_0 <= 2e-31) {
		tmp = t_2;
	} else if (t_0 <= 1e+228) {
		tmp = t_1;
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x * x) - t_0) / (t_0 + (x * x))
	t_2 = (1.0 + (2.0 * (y / x))) * (1.0 + (-2.0 * (y / x)))
	tmp = 0
	if t_0 <= 1e-237:
		tmp = t_2
	elif t_0 <= 5e-134:
		tmp = t_1
	elif t_0 <= 2e-31:
		tmp = t_2
	elif t_0 <= 1e+228:
		tmp = t_1
	else:
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)))
	t_2 = Float64(Float64(1.0 + Float64(2.0 * Float64(y / x))) * Float64(1.0 + Float64(-2.0 * Float64(y / x))))
	tmp = 0.0
	if (t_0 <= 1e-237)
		tmp = t_2;
	elseif (t_0 <= 5e-134)
		tmp = t_1;
	elseif (t_0 <= 2e-31)
		tmp = t_2;
	elseif (t_0 <= 1e+228)
		tmp = t_1;
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = ((x * x) - t_0) / (t_0 + (x * x));
	t_2 = (1.0 + (2.0 * (y / x))) * (1.0 + (-2.0 * (y / x)));
	tmp = 0.0;
	if (t_0 <= 1e-237)
		tmp = t_2;
	elseif (t_0 <= 5e-134)
		tmp = t_1;
	elseif (t_0 <= 2e-31)
		tmp = t_2;
	elseif (t_0 <= 1e+228)
		tmp = t_1;
	else
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-237], t$95$2, If[LessEqual[t$95$0, 5e-134], t$95$1, If[LessEqual[t$95$0, 2e-31], t$95$2, If[LessEqual[t$95$0, 1e+228], t$95$1, N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 9.9999999999999999e-238 or 5.0000000000000003e-134 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e-31

   1. Initial program 56.0%

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

      1. add-sqr-sqrt 56.0%

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

      2. difference-of-squares 56.0%

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

      3. *-commutative 56.0%

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

      4. associate-*r* 56.0%

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

      5. sqrt-prod 56.0%

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

      6. sqrt-unprod 35.1%

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

      7. add-sqr-sqrt 50.4%

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

      8. metadata-eval 50.4%

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

      9. *-commutative 50.4%

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

      10. associate-*r* 50.4%

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

      11. sqrt-prod 50.4%

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

      12. sqrt-unprod 35.1%

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

      13. add-sqr-sqrt 56.0%

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

      14. metadata-eval 56.0%

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

   4. Applied egg-rr 56.0%

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

      1. add-sqr-sqrt 56.0%

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

      2. times-frac 57.3%

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

      3. +-commutative 57.3%

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

      4. *-commutative 57.3%

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

      5. fma-define 57.3%

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

      6. add-sqr-sqrt 57.3%

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

      7. hypot-define 57.3%

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

      8. sqrt-prod 35.9%

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

      9. *-commutative 35.9%

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

      10. sqrt-prod 35.9%

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

      11. metadata-eval 35.9%

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

      12. associate-*r* 35.9%

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

      13. add-sqr-sqrt 57.3%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 44.4%

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

   8. Taylor expanded in x around inf 85.2%

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

   if 9.9999999999999999e-238 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 5.0000000000000003e-134 or 2e-31 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 9.9999999999999992e227

   1. Initial program 78.8%

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

   if 9.9999999999999992e227 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 13.8%

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

   3. Taylor expanded in x around 0 80.1%

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

      1. pow2 80.1%

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

      2. unpow2 80.1%

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

      3. times-frac 87.0%

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

   5. Applied egg-rr 87.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-237}:\\ \;\;\;\;\left(1 + 2 \cdot \frac{y}{x}\right) \cdot \left(1 + -2 \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-31}:\\ \;\;\;\;\left(1 + 2 \cdot \frac{y}{x}\right) \cdot \left(1 + -2 \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+228}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.95 \cdot 10^{-21} \lor \neg \left(x \leq 1.32 \cdot 10^{+23}\right) \land x \leq 5.9 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 2 \cdot \frac{y}{x}\right) \cdot \left(1 + -2 \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 2.95e-21) (and (not (<= x 1.32e+23)) (<= x 5.9e+52)))
   (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)
   (* (+ 1.0 (* 2.0 (/ y x))) (+ 1.0 (* -2.0 (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 2.95e-21) || (!(x <= 1.32e+23) && (x <= 5.9e+52))) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else {
		tmp = (1.0 + (2.0 * (y / x))) * (1.0 + (-2.0 * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= 2.95d-21) .or. (.not. (x <= 1.32d+23)) .and. (x <= 5.9d+52)) then
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-1.0d0)
    else
        tmp = (1.0d0 + (2.0d0 * (y / x))) * (1.0d0 + ((-2.0d0) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= 2.95e-21) || (!(x <= 1.32e+23) && (x <= 5.9e+52))) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else {
		tmp = (1.0 + (2.0 * (y / x))) * (1.0 + (-2.0 * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 2.95e-21) or (not (x <= 1.32e+23) and (x <= 5.9e+52)):
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	else:
		tmp = (1.0 + (2.0 * (y / x))) * (1.0 + (-2.0 * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 2.95e-21) || (!(x <= 1.32e+23) && (x <= 5.9e+52)))
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	else
		tmp = Float64(Float64(1.0 + Float64(2.0 * Float64(y / x))) * Float64(1.0 + Float64(-2.0 * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 2.95e-21) || (~((x <= 1.32e+23)) && (x <= 5.9e+52)))
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	else
		tmp = (1.0 + (2.0 * (y / x))) * (1.0 + (-2.0 * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 2.95e-21], And[N[Not[LessEqual[x, 1.32e+23]], $MachinePrecision], LessEqual[x, 5.9e+52]]], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9500000000000001e-21 or 1.3199999999999999e23 < x < 5.89999999999999996e52

   1. Initial program 52.5%

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

   3. Taylor expanded in x around 0 58.2%

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

      1. pow2 58.2%

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

      2. unpow2 58.2%

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

      3. times-frac 60.9%

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

   5. Applied egg-rr 60.9%

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

   if 2.9500000000000001e-21 < x < 1.3199999999999999e23 or 5.89999999999999996e52 < x

   1. Initial program 42.9%

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

      1. add-sqr-sqrt 42.9%

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

      2. difference-of-squares 42.9%

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

      3. *-commutative 42.9%

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

      4. associate-*r* 42.9%

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

      5. sqrt-prod 42.9%

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

      6. sqrt-unprod 26.8%

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

      7. add-sqr-sqrt 42.9%

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

      8. metadata-eval 42.9%

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

      9. *-commutative 42.9%

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

      10. associate-*r* 42.9%

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

      11. sqrt-prod 42.9%

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

      12. sqrt-unprod 26.8%

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

      13. add-sqr-sqrt 42.9%

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

      14. metadata-eval 42.9%

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

   4. Applied egg-rr 42.9%

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

      1. add-sqr-sqrt 42.9%

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

      2. times-frac 44.6%

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

      3. +-commutative 44.6%

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

      4. *-commutative 44.6%

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

      5. fma-define 44.6%

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

      6. add-sqr-sqrt 44.6%

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

      7. hypot-define 44.6%

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

      8. sqrt-prod 27.9%

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

      9. *-commutative 27.9%

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

      10. sqrt-prod 27.9%

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

      11. metadata-eval 27.9%

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

      12. associate-*r* 27.9%

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

      13. add-sqr-sqrt 44.6%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 88.3%

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

   8. Taylor expanded in x around inf 88.1%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.95 \cdot 10^{-21} \lor \neg \left(x \leq 1.32 \cdot 10^{+23}\right) \land x \leq 5.9 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 2 \cdot \frac{y}{x}\right) \cdot \left(1 + -2 \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-22} \lor \neg \left(x \leq 300000\right) \land x \leq 5.2 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 8.2e-22) (and (not (<= x 300000.0)) (<= x 5.2e+52)))
   (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)
   1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 8.2e-22) || (!(x <= 300000.0) && (x <= 5.2e+52))) {
		tmp = (0.5 * ((x / y) * (x / y))) + -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 <= 8.2d-22) .or. (.not. (x <= 300000.0d0)) .and. (x <= 5.2d+52)) then
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-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 <= 8.2e-22) || (!(x <= 300000.0) && (x <= 5.2e+52))) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 8.2e-22) or (not (x <= 300000.0) and (x <= 5.2e+52)):
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 8.2e-22) || (!(x <= 300000.0) && (x <= 5.2e+52)))
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 8.2e-22) || (~((x <= 300000.0)) && (x <= 5.2e+52)))
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 8.2e-22], And[N[Not[LessEqual[x, 300000.0]], $MachinePrecision], LessEqual[x, 5.2e+52]]], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.1999999999999999e-22 or 3e5 < x < 5.2e52

   1. Initial program 53.0%

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

   3. Taylor expanded in x around 0 57.7%

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

      1. pow2 57.7%

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

      2. unpow2 57.7%

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

      3. times-frac 60.3%

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

   5. Applied egg-rr 60.3%

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

   if 8.1999999999999999e-22 < x < 3e5 or 5.2e52 < x

   1. Initial program 40.7%

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

   3. Taylor expanded in x around inf 87.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-22} \lor \neg \left(x \leq 300000\right) \land x \leq 5.2 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-22}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+52}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.8e-22) -1.0 (if (<= x 6.5e+22) 1.0 (if (<= x 5.5e+52) -1.0 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.8e-22) {
		tmp = -1.0;
	} else if (x <= 6.5e+22) {
		tmp = 1.0;
	} else if (x <= 5.5e+52) {
		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 <= 5.8d-22) then
        tmp = -1.0d0
    else if (x <= 6.5d+22) then
        tmp = 1.0d0
    else if (x <= 5.5d+52) 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 <= 5.8e-22) {
		tmp = -1.0;
	} else if (x <= 6.5e+22) {
		tmp = 1.0;
	} else if (x <= 5.5e+52) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.8e-22:
		tmp = -1.0
	elif x <= 6.5e+22:
		tmp = 1.0
	elif x <= 5.5e+52:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.8e-22)
		tmp = -1.0;
	elseif (x <= 6.5e+22)
		tmp = 1.0;
	elseif (x <= 5.5e+52)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.8e-22)
		tmp = -1.0;
	elseif (x <= 6.5e+22)
		tmp = 1.0;
	elseif (x <= 5.5e+52)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.8e-22], -1.0, If[LessEqual[x, 6.5e+22], 1.0, If[LessEqual[x, 5.5e+52], -1.0, 1.0]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8000000000000003e-22 or 6.49999999999999979e22 < x < 5.49999999999999996e52

   1. Initial program 52.5%

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

   3. Taylor expanded in x around 0 59.5%

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

   if 5.8000000000000003e-22 < x < 6.49999999999999979e22 or 5.49999999999999996e52 < x

   1. Initial program 42.9%

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

   3. Taylor expanded in x around inf 87.7%

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

Alternative 8: 50.2% 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 50.4%

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

3. Taylor expanded in x around 0 49.5%

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

Developer target: 50.7% 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 2024086 
(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))))