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

Percentage Accurate: 50.1% → 100.0%

Time: 20.8s

Alternatives: 12

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: 50.1% 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: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(y \cdot 2, x\right)\\ t_1 := {\left(\frac{y \cdot 2}{t\_0}\right)}^{2}\\ 0 \cdot t\_1 + \left(\log \left(e^{{\left(\frac{x}{t\_0}\right)}^{2}}\right) - t\_1\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot (* y 2.0) x)) (t_1 (pow (/ (* y 2.0) t_0) 2.0)))
   (+ (* 0.0 t_1) (- (log (exp (pow (/ x t_0) 2.0))) t_1))))

C

double code(double x, double y) {
	double t_0 = hypot((y * 2.0), x);
	double t_1 = pow(((y * 2.0) / t_0), 2.0);
	return (0.0 * t_1) + (log(exp(pow((x / t_0), 2.0))) - t_1);
}

Java

public static double code(double x, double y) {
	double t_0 = Math.hypot((y * 2.0), x);
	double t_1 = Math.pow(((y * 2.0) / t_0), 2.0);
	return (0.0 * t_1) + (Math.log(Math.exp(Math.pow((x / t_0), 2.0))) - t_1);
}

Python

def code(x, y):
	t_0 = math.hypot((y * 2.0), x)
	t_1 = math.pow(((y * 2.0) / t_0), 2.0)
	return (0.0 * t_1) + (math.log(math.exp(math.pow((x / t_0), 2.0))) - t_1)

Julia

function code(x, y)
	t_0 = hypot(Float64(y * 2.0), x)
	t_1 = Float64(Float64(y * 2.0) / t_0) ^ 2.0
	return Float64(Float64(0.0 * t_1) + Float64(log(exp((Float64(x / t_0) ^ 2.0))) - t_1))
end

MATLAB

function tmp = code(x, y)
	t_0 = hypot((y * 2.0), x);
	t_1 = ((y * 2.0) / t_0) ^ 2.0;
	tmp = (0.0 * t_1) + (log(exp(((x / t_0) ^ 2.0))) - t_1);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(y * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(y * 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(0.0 * t$95$1), $MachinePrecision] + N[(N[Log[N[Exp[N[Power[N[(x / t$95$0), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 51.5%

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

   1. div-sub 51.6%

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

   2. associate-/l* 51.9%

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

   3. fma-neg 51.9%

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

   4. +-commutative 51.9%

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

   5. *-commutative 51.9%

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

   6. associate-*l* 51.9%

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

   7. fma-define 51.9%

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

   8. pow2 51.9%

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

   9. pow2 51.9%

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

   10. *-commutative 51.9%

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

   11. associate-*l* 51.9%

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

   12. pow2 51.9%

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

   13. +-commutative 51.9%

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

   14. *-commutative 51.9%

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

   15. associate-*l* 51.9%

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

   16. fma-define 51.9%

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

4. Applied egg-rr 51.9%

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

6. Applied egg-rr 34.5%

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

8. Simplified 52.9%

   \[\leadsto \color{blue}{0 \cdot {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(\left(x \cdot \frac{\sqrt{x}}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right) \cdot \frac{\sqrt{x}}{\mathsf{hypot}\left(y \cdot 2, x\right)} - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}\right)} \]
9. Step-by-step derivation

   1. add-log-exp 53.0%

      \[\leadsto 0 \cdot {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(\color{blue}{\log \left(e^{\left(x \cdot \frac{\sqrt{x}}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right) \cdot \frac{\sqrt{x}}{\mathsf{hypot}\left(y \cdot 2, x\right)}}\right)} - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}\right) \]

   2. associate-*r/ 45.0%

      \[\leadsto 0 \cdot {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(\log \left(e^{\color{blue}{\frac{x \cdot \sqrt{x}}{\mathsf{hypot}\left(y \cdot 2, x\right)}} \cdot \frac{\sqrt{x}}{\mathsf{hypot}\left(y \cdot 2, x\right)}}\right) - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}\right) \]

   3. frac-times 39.0%

      \[\leadsto 0 \cdot {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(\log \left(e^{\color{blue}{\frac{\left(x \cdot \sqrt{x}\right) \cdot \sqrt{x}}{\mathsf{hypot}\left(y \cdot 2, x\right) \cdot \mathsf{hypot}\left(y \cdot 2, x\right)}}}\right) - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}\right) \]

   4. associate-*r* 38.9%

      \[\leadsto 0 \cdot {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(\log \left(e^{\frac{\color{blue}{x \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}}{\mathsf{hypot}\left(y \cdot 2, x\right) \cdot \mathsf{hypot}\left(y \cdot 2, x\right)}}\right) - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}\right) \]

   5. add-sqr-sqrt 74.2%

      \[\leadsto 0 \cdot {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(\log \left(e^{\frac{x \cdot \color{blue}{x}}{\mathsf{hypot}\left(y \cdot 2, x\right) \cdot \mathsf{hypot}\left(y \cdot 2, x\right)}}\right) - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}\right) \]

   6. unpow2 74.2%

      \[\leadsto 0 \cdot {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(\log \left(e^{\frac{\color{blue}{{x}^{2}}}{\mathsf{hypot}\left(y \cdot 2, x\right) \cdot \mathsf{hypot}\left(y \cdot 2, x\right)}}\right) - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}\right) \]

   7. unpow2 74.2%

      \[\leadsto 0 \cdot {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(\log \left(e^{\frac{{x}^{2}}{\color{blue}{{\left(\mathsf{hypot}\left(y \cdot 2, x\right)\right)}^{2}}}}\right) - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}\right) \]

   8. add-sqr-sqrt 74.2%

      \[\leadsto 0 \cdot {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(\log \left(e^{\color{blue}{\sqrt{\frac{{x}^{2}}{{\left(\mathsf{hypot}\left(y \cdot 2, x\right)\right)}^{2}}} \cdot \sqrt{\frac{{x}^{2}}{{\left(\mathsf{hypot}\left(y \cdot 2, x\right)\right)}^{2}}}}}\right) - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}\right) \]

   9. pow2 74.2%

      \[\leadsto 0 \cdot {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(\log \left(e^{\color{blue}{{\left(\sqrt{\frac{{x}^{2}}{{\left(\mathsf{hypot}\left(y \cdot 2, x\right)\right)}^{2}}}\right)}^{2}}}\right) - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto 0 \cdot {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(\color{blue}{\log \left(e^{{\left(\frac{x}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}}\right)} - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2}\right) \]
11. Add Preprocessing

Alternative 2: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(y \cdot 2, x\right)\\ t_1 := \frac{t\_0}{y}\\ {\left(\frac{x}{t\_0}\right)}^{2} + \frac{-1}{0.25 \cdot \left(t\_1 \cdot t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot (* y 2.0) x)) (t_1 (/ t_0 y)))
   (+ (pow (/ x t_0) 2.0) (/ -1.0 (* 0.25 (* t_1 t_1))))))

C

double code(double x, double y) {
	double t_0 = hypot((y * 2.0), x);
	double t_1 = t_0 / y;
	return pow((x / t_0), 2.0) + (-1.0 / (0.25 * (t_1 * t_1)));
}

Java

public static double code(double x, double y) {
	double t_0 = Math.hypot((y * 2.0), x);
	double t_1 = t_0 / y;
	return Math.pow((x / t_0), 2.0) + (-1.0 / (0.25 * (t_1 * t_1)));
}

Python

def code(x, y):
	t_0 = math.hypot((y * 2.0), x)
	t_1 = t_0 / y
	return math.pow((x / t_0), 2.0) + (-1.0 / (0.25 * (t_1 * t_1)))

Julia

function code(x, y)
	t_0 = hypot(Float64(y * 2.0), x)
	t_1 = Float64(t_0 / y)
	return Float64((Float64(x / t_0) ^ 2.0) + Float64(-1.0 / Float64(0.25 * Float64(t_1 * t_1))))
end

MATLAB

function tmp = code(x, y)
	t_0 = hypot((y * 2.0), x);
	t_1 = t_0 / y;
	tmp = ((x / t_0) ^ 2.0) + (-1.0 / (0.25 * (t_1 * t_1)));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(y * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / y), $MachinePrecision]}, N[(N[Power[N[(x / t$95$0), $MachinePrecision], 2.0], $MachinePrecision] + N[(-1.0 / N[(0.25 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 51.5%

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

   1. div-sub 51.6%

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

   2. associate-/l* 51.9%

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

   3. fma-neg 51.9%

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

   4. +-commutative 51.9%

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

   5. *-commutative 51.9%

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

   6. associate-*l* 51.9%

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

   7. fma-define 51.9%

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

   8. pow2 51.9%

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

   9. pow2 51.9%

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

   10. *-commutative 51.9%

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

   11. associate-*l* 51.9%

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

   12. pow2 51.9%

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

   13. +-commutative 51.9%

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

   14. *-commutative 51.9%

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

   15. associate-*l* 51.9%

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

   16. fma-define 51.9%

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

4. Applied egg-rr 51.9%

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

   1. fma-undefine 51.9%

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

   2. unsub-neg 51.9%

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

6. Applied egg-rr 74.2%

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

   1. sub-neg 74.2%

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

   2. add-sqr-sqrt 74.2%

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

   3. pow2 74.2%

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

   4. sqrt-div 74.2%

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

   5. unpow2 74.2%

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

   6. sqrt-prod 39.2%

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

   7. add-sqr-sqrt 75.8%

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

   8. sqrt-pow1 100.0%

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

   9. metadata-eval 100.0%

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

   10. pow1 100.0%

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

   11. add-sqr-sqrt 100.0%

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

   12. add-sqr-sqrt 100.0%

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

   13. *-commutative 100.0%

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

   14. associate-/l* 100.0%

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

8. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. associate-*r/ 100.0%

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

   3. *-commutative 100.0%

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

10. Simplified 100.0%

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

    1. unpow2 100.0%

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

    2. clear-num 100.0%

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

    3. clear-num 100.0%

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

    4. frac-times 100.0%

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

    5. metadata-eval 100.0%

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

    6. *-un-lft-identity 100.0%

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

    7. *-commutative 100.0%

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

    8. times-frac 100.0%

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

    9. metadata-eval 100.0%

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

    10. *-un-lft-identity 100.0%

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

    11. *-commutative 100.0%

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

    12. times-frac 100.0%

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

    13. metadata-eval 100.0%

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

12. Applied egg-rr 100.0%

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

    1. swap-sqr 100.0%

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

    2. metadata-eval 100.0%

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

14. Simplified 100.0%

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

15. Final simplification 100.0%

    \[\leadsto {\left(\frac{x}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \frac{-1}{0.25 \cdot \left(\frac{\mathsf{hypot}\left(y \cdot 2, x\right)}{y} \cdot \frac{\mathsf{hypot}\left(y \cdot 2, x\right)}{y}\right)} \]
16. Add Preprocessing

Alternative 3: 100.0% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y) {
	double t_0 = Math.hypot((y * 2.0), x);
	return Math.pow((x / t_0), 2.0) - Math.pow(((y * 2.0) / t_0), 2.0);
}

Python

def code(x, y):
	t_0 = math.hypot((y * 2.0), x)
	return math.pow((x / t_0), 2.0) - math.pow(((y * 2.0) / t_0), 2.0)

Julia

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

MATLAB

function tmp = code(x, y)
	t_0 = hypot((y * 2.0), x);
	tmp = ((x / t_0) ^ 2.0) - (((y * 2.0) / t_0) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 51.5%

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

   1. div-sub 51.6%

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

   2. associate-/l* 51.9%

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

   3. fma-neg 51.9%

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

   4. +-commutative 51.9%

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

   5. *-commutative 51.9%

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

   6. associate-*l* 51.9%

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

   7. fma-define 51.9%

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

   8. pow2 51.9%

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

   9. pow2 51.9%

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

   10. *-commutative 51.9%

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

   11. associate-*l* 51.9%

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

   12. pow2 51.9%

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

   13. +-commutative 51.9%

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

   14. *-commutative 51.9%

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

   15. associate-*l* 51.9%

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

   16. fma-define 51.9%

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

4. Applied egg-rr 51.9%

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

   1. fma-undefine 51.9%

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

   2. unsub-neg 51.9%

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

6. Applied egg-rr 74.2%

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

   1. sub-neg 74.2%

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

   2. add-sqr-sqrt 74.2%

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

   3. pow2 74.2%

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

   4. sqrt-div 74.2%

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

   5. unpow2 74.2%

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

   6. sqrt-prod 39.2%

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

   7. add-sqr-sqrt 75.8%

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

   8. sqrt-pow1 100.0%

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

   9. metadata-eval 100.0%

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

   10. pow1 100.0%

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

   11. add-sqr-sqrt 100.0%

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

   12. add-sqr-sqrt 100.0%

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

   13. *-commutative 100.0%

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

   14. associate-/l* 100.0%

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

8. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. associate-*r/ 100.0%

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

   3. *-commutative 100.0%

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

10. Simplified 100.0%

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

Alternative 4: 99.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y) {
	double t_0 = Math.hypot((y * 2.0), x);
	return Math.pow((x / t_0), 2.0) - Math.pow((y * (2.0 / t_0)), 2.0);
}

Python

def code(x, y):
	t_0 = math.hypot((y * 2.0), x)
	return math.pow((x / t_0), 2.0) - math.pow((y * (2.0 / t_0)), 2.0)

Julia

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

MATLAB

function tmp = code(x, y)
	t_0 = hypot((y * 2.0), x);
	tmp = ((x / t_0) ^ 2.0) - ((y * (2.0 / t_0)) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 51.5%

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

   1. div-sub 51.6%

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

   2. associate-/l* 51.9%

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

   3. fma-neg 51.9%

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

   4. +-commutative 51.9%

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

   5. *-commutative 51.9%

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

   6. associate-*l* 51.9%

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

   7. fma-define 51.9%

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

   8. pow2 51.9%

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

   9. pow2 51.9%

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

   10. *-commutative 51.9%

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

   11. associate-*l* 51.9%

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

   12. pow2 51.9%

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

   13. +-commutative 51.9%

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

   14. *-commutative 51.9%

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

   15. associate-*l* 51.9%

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

   16. fma-define 51.9%

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

4. Applied egg-rr 51.9%

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

   1. fma-undefine 51.9%

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

   2. unsub-neg 51.9%

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

6. Applied egg-rr 74.2%

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

   1. sub-neg 74.2%

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

   2. add-sqr-sqrt 74.2%

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

   3. pow2 74.2%

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

   4. sqrt-div 74.2%

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

   5. unpow2 74.2%

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

   6. sqrt-prod 39.2%

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

   7. add-sqr-sqrt 75.8%

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

   8. sqrt-pow1 100.0%

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

   9. metadata-eval 100.0%

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

   10. pow1 100.0%

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

   11. add-sqr-sqrt 100.0%

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

   12. add-sqr-sqrt 100.0%

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

   13. *-commutative 100.0%

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

   14. associate-/l* 100.0%

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

8. Applied egg-rr 100.0%

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

   1. sub-neg 100.0%

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

   2. associate-*r/ 100.0%

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

   3. *-commutative 100.0%

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

10. Simplified 100.0%

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

    1. associate-/l* 99.8%

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

    2. *-commutative 99.8%

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

12. Applied egg-rr 99.8%

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

13. Final simplification 99.8%

    \[\leadsto {\left(\frac{x}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} - {\left(y \cdot \frac{2}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} \]
14. Add Preprocessing

Alternative 5: 80.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := {\left(\frac{x}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(-1 - -0.25 \cdot {\left(\frac{x}{y}\right)}^{2}\right)\\ t_2 := \frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-254}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-137}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot x \leq 10^{+223}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1
         (+
          (pow (/ x (hypot (* y 2.0) x)) 2.0)
          (- -1.0 (* -0.25 (pow (/ x y) 2.0)))))
        (t_2 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= (* x x) 2e-254)
     t_1
     (if (<= (* x x) 2e-137)
       t_2
       (if (<= (* x x) 5e-44)
         t_1
         (if (<= (* x x) 1e+223) t_2 (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = pow((x / hypot((y * 2.0), x)), 2.0) + (-1.0 - (-0.25 * pow((x / y), 2.0)));
	double t_2 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if ((x * x) <= 2e-254) {
		tmp = t_1;
	} else if ((x * x) <= 2e-137) {
		tmp = t_2;
	} else if ((x * x) <= 5e-44) {
		tmp = t_1;
	} else if ((x * x) <= 1e+223) {
		tmp = t_2;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = Math.pow((x / Math.hypot((y * 2.0), x)), 2.0) + (-1.0 - (-0.25 * Math.pow((x / y), 2.0)));
	double t_2 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if ((x * x) <= 2e-254) {
		tmp = t_1;
	} else if ((x * x) <= 2e-137) {
		tmp = t_2;
	} else if ((x * x) <= 5e-44) {
		tmp = t_1;
	} else if ((x * x) <= 1e+223) {
		tmp = t_2;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = math.pow((x / math.hypot((y * 2.0), x)), 2.0) + (-1.0 - (-0.25 * math.pow((x / y), 2.0)))
	t_2 = ((x * x) - t_0) / ((x * x) + t_0)
	tmp = 0
	if (x * x) <= 2e-254:
		tmp = t_1
	elif (x * x) <= 2e-137:
		tmp = t_2
	elif (x * x) <= 5e-44:
		tmp = t_1
	elif (x * x) <= 1e+223:
		tmp = t_2
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64((Float64(x / hypot(Float64(y * 2.0), x)) ^ 2.0) + Float64(-1.0 - Float64(-0.25 * (Float64(x / y) ^ 2.0))))
	t_2 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if (Float64(x * x) <= 2e-254)
		tmp = t_1;
	elseif (Float64(x * x) <= 2e-137)
		tmp = t_2;
	elseif (Float64(x * x) <= 5e-44)
		tmp = t_1;
	elseif (Float64(x * x) <= 1e+223)
		tmp = t_2;
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = ((x / hypot((y * 2.0), x)) ^ 2.0) + (-1.0 - (-0.25 * ((x / y) ^ 2.0)));
	t_2 = ((x * x) - t_0) / ((x * x) + t_0);
	tmp = 0.0;
	if ((x * x) <= 2e-254)
		tmp = t_1;
	elseif ((x * x) <= 2e-137)
		tmp = t_2;
	elseif ((x * x) <= 5e-44)
		tmp = t_1;
	elseif ((x * x) <= 1e+223)
		tmp = t_2;
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	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[Power[N[(x / N[Sqrt[N[(y * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(-1.0 - N[(-0.25 * N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e-254], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 2e-137], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 5e-44], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 1e+223], t$95$2, N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.9999999999999998e-254 or 1.99999999999999996e-137 < (*.f64 x x) < 5.00000000000000039e-44

   1. Initial program 50.9%

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

      1. div-sub 51.0%

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

      2. associate-/l* 50.9%

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

      3. fma-neg 50.9%

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

      4. +-commutative 50.9%

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

      5. *-commutative 50.9%

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

      6. associate-*l* 50.9%

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

      7. fma-define 50.9%

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

      8. pow2 50.9%

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

      9. pow2 50.9%

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

      10. *-commutative 50.9%

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

      11. associate-*l* 50.9%

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

      12. pow2 50.9%

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

      13. +-commutative 50.9%

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

      14. *-commutative 50.9%

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

      15. associate-*l* 50.9%

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

      16. fma-define 50.9%

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

   4. Applied egg-rr 50.9%

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

      1. fma-undefine 50.9%

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

      2. unsub-neg 50.9%

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

   6. Applied egg-rr 87.5%

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

      1. sub-neg 87.5%

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

      2. add-sqr-sqrt 87.5%

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

      3. pow2 87.5%

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

      4. sqrt-div 87.5%

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

      5. unpow2 87.5%

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

      6. sqrt-prod 48.2%

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

      7. add-sqr-sqrt 87.7%

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

      8. sqrt-pow1 100.0%

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

      9. metadata-eval 100.0%

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

      10. pow1 100.0%

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

      11. add-sqr-sqrt 100.0%

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

      12. add-sqr-sqrt 100.0%

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

      13. *-commutative 100.0%

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

      14. associate-/l* 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in y around inf 83.0%

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

       1. unpow2 83.0%

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

       2. unpow2 83.0%

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

       3. times-frac 90.8%

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

       4. unpow2 90.8%

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

   13. Simplified 90.8%

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

   if 1.9999999999999998e-254 < (*.f64 x x) < 1.99999999999999996e-137 or 5.00000000000000039e-44 < (*.f64 x x) < 1.00000000000000005e223

   1. Initial program 79.7%

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

   if 1.00000000000000005e223 < (*.f64 x x)

   1. Initial program 17.6%

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

   3. Taylor expanded in y around 0 81.0%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 81.0%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 81.0%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 91.7%

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

   5. Applied egg-rr 91.7%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-254}:\\ \;\;\;\;{\left(\frac{x}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(-1 - -0.25 \cdot {\left(\frac{x}{y}\right)}^{2}\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-137}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-44}:\\ \;\;\;\;{\left(\frac{x}{\mathsf{hypot}\left(y \cdot 2, x\right)}\right)}^{2} + \left(-1 - -0.25 \cdot {\left(\frac{x}{y}\right)}^{2}\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+223}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ t_2 := \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-254}:\\ \;\;\;\;\left({\left(\sqrt{1 + t\_2}\right)}^{2} + -1\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-44}:\\ \;\;\;\;t\_2 + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+223}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0)))
        (t_2 (* (* (/ x y) (/ x y)) 0.5)))
   (if (<= (* x x) 2e-254)
     (+ (+ (pow (sqrt (+ 1.0 t_2)) 2.0) -1.0) -1.0)
     (if (<= (* x x) 2e-137)
       t_1
       (if (<= (* x x) 5e-44)
         (+ t_2 -1.0)
         (if (<= (* x x) 1e+223) t_1 (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double t_2 = ((x / y) * (x / y)) * 0.5;
	double tmp;
	if ((x * x) <= 2e-254) {
		tmp = (pow(sqrt((1.0 + t_2)), 2.0) + -1.0) + -1.0;
	} else if ((x * x) <= 2e-137) {
		tmp = t_1;
	} else if ((x * x) <= 5e-44) {
		tmp = t_2 + -1.0;
	} else if ((x * x) <= 1e+223) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    t_2 = ((x / y) * (x / y)) * 0.5d0
    if ((x * x) <= 2d-254) then
        tmp = ((sqrt((1.0d0 + t_2)) ** 2.0d0) + (-1.0d0)) + (-1.0d0)
    else if ((x * x) <= 2d-137) then
        tmp = t_1
    else if ((x * x) <= 5d-44) then
        tmp = t_2 + (-1.0d0)
    else if ((x * x) <= 1d+223) then
        tmp = t_1
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    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) / ((x * x) + t_0);
	double t_2 = ((x / y) * (x / y)) * 0.5;
	double tmp;
	if ((x * x) <= 2e-254) {
		tmp = (Math.pow(Math.sqrt((1.0 + t_2)), 2.0) + -1.0) + -1.0;
	} else if ((x * x) <= 2e-137) {
		tmp = t_1;
	} else if ((x * x) <= 5e-44) {
		tmp = t_2 + -1.0;
	} else if ((x * x) <= 1e+223) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	t_2 = ((x / y) * (x / y)) * 0.5
	tmp = 0
	if (x * x) <= 2e-254:
		tmp = (math.pow(math.sqrt((1.0 + t_2)), 2.0) + -1.0) + -1.0
	elif (x * x) <= 2e-137:
		tmp = t_1
	elif (x * x) <= 5e-44:
		tmp = t_2 + -1.0
	elif (x * x) <= 1e+223:
		tmp = t_1
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	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(Float64(x * x) + t_0))
	t_2 = Float64(Float64(Float64(x / y) * Float64(x / y)) * 0.5)
	tmp = 0.0
	if (Float64(x * x) <= 2e-254)
		tmp = Float64(Float64((sqrt(Float64(1.0 + t_2)) ^ 2.0) + -1.0) + -1.0);
	elseif (Float64(x * x) <= 2e-137)
		tmp = t_1;
	elseif (Float64(x * x) <= 5e-44)
		tmp = Float64(t_2 + -1.0);
	elseif (Float64(x * x) <= 1e+223)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	t_2 = ((x / y) * (x / y)) * 0.5;
	tmp = 0.0;
	if ((x * x) <= 2e-254)
		tmp = ((sqrt((1.0 + t_2)) ^ 2.0) + -1.0) + -1.0;
	elseif ((x * x) <= 2e-137)
		tmp = t_1;
	elseif ((x * x) <= 5e-44)
		tmp = t_2 + -1.0;
	elseif ((x * x) <= 1e+223)
		tmp = t_1;
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	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[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e-254], N[(N[(N[Power[N[Sqrt[N[(1.0 + t$95$2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e-137], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 5e-44], N[(t$95$2 + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+223], t$95$1, N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 1.9999999999999998e-254

   1. Initial program 51.2%

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

   3. Taylor expanded in x around 0 81.1%

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

      1. expm1-log1p-u 81.1%

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

      2. expm1-undefine 81.1%

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

      3. *-commutative 81.1%

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

      4. add-sqr-sqrt 81.1%

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

      5. pow2 81.1%

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

      6. sqrt-div 81.1%

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

      7. sqrt-pow1 81.3%

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

      8. metadata-eval 81.3%

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

      9. pow1 81.3%

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

      10. sqrt-pow1 90.7%

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

      11. metadata-eval 90.7%

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

      12. pow1 90.7%

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

   5. Applied egg-rr 90.7%

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

      1. add-sqr-sqrt 90.7%

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

      2. pow2 90.7%

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

      3. log1p-undefine 90.7%

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

      4. pow2 90.7%

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

      5. *-commutative 90.7%

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

      6. add-exp-log 90.7%

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

      7. +-commutative 90.7%

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

      8. *-commutative 90.7%

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

      9. pow2 90.7%

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

   7. Applied egg-rr 90.7%

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

      1. pow2 90.7%

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

   9. Applied egg-rr 90.7%

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

   if 1.9999999999999998e-254 < (*.f64 x x) < 1.99999999999999996e-137 or 5.00000000000000039e-44 < (*.f64 x x) < 1.00000000000000005e223

   1. Initial program 79.7%

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

   if 1.99999999999999996e-137 < (*.f64 x x) < 5.00000000000000039e-44

   1. Initial program 49.9%

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

   3. Taylor expanded in x around 0 90.9%

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

      1. unpow2 90.9%

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

      2. unpow2 90.9%

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

      3. times-frac 90.9%

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

   5. Applied egg-rr 90.9%

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

   if 1.00000000000000005e223 < (*.f64 x x)

   1. Initial program 17.6%

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

   3. Taylor expanded in y around 0 81.0%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 81.0%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 81.0%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 91.7%

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

   5. Applied egg-rr 91.7%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-254}:\\ \;\;\;\;\left({\left(\sqrt{1 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5}\right)}^{2} + -1\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-137}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+223}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ t_2 := \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-254}:\\ \;\;\;\;\left(\left(1 + t\_2\right) + -1\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-44}:\\ \;\;\;\;t\_2 + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+223}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0)))
        (t_2 (* (* (/ x y) (/ x y)) 0.5)))
   (if (<= (* x x) 2e-254)
     (+ (+ (+ 1.0 t_2) -1.0) -1.0)
     (if (<= (* x x) 2e-137)
       t_1
       (if (<= (* x x) 5e-44)
         (+ t_2 -1.0)
         (if (<= (* x x) 1e+223) t_1 (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double t_2 = ((x / y) * (x / y)) * 0.5;
	double tmp;
	if ((x * x) <= 2e-254) {
		tmp = ((1.0 + t_2) + -1.0) + -1.0;
	} else if ((x * x) <= 2e-137) {
		tmp = t_1;
	} else if ((x * x) <= 5e-44) {
		tmp = t_2 + -1.0;
	} else if ((x * x) <= 1e+223) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    t_2 = ((x / y) * (x / y)) * 0.5d0
    if ((x * x) <= 2d-254) then
        tmp = ((1.0d0 + t_2) + (-1.0d0)) + (-1.0d0)
    else if ((x * x) <= 2d-137) then
        tmp = t_1
    else if ((x * x) <= 5d-44) then
        tmp = t_2 + (-1.0d0)
    else if ((x * x) <= 1d+223) then
        tmp = t_1
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    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) / ((x * x) + t_0);
	double t_2 = ((x / y) * (x / y)) * 0.5;
	double tmp;
	if ((x * x) <= 2e-254) {
		tmp = ((1.0 + t_2) + -1.0) + -1.0;
	} else if ((x * x) <= 2e-137) {
		tmp = t_1;
	} else if ((x * x) <= 5e-44) {
		tmp = t_2 + -1.0;
	} else if ((x * x) <= 1e+223) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	t_2 = ((x / y) * (x / y)) * 0.5
	tmp = 0
	if (x * x) <= 2e-254:
		tmp = ((1.0 + t_2) + -1.0) + -1.0
	elif (x * x) <= 2e-137:
		tmp = t_1
	elif (x * x) <= 5e-44:
		tmp = t_2 + -1.0
	elif (x * x) <= 1e+223:
		tmp = t_1
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	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(Float64(x * x) + t_0))
	t_2 = Float64(Float64(Float64(x / y) * Float64(x / y)) * 0.5)
	tmp = 0.0
	if (Float64(x * x) <= 2e-254)
		tmp = Float64(Float64(Float64(1.0 + t_2) + -1.0) + -1.0);
	elseif (Float64(x * x) <= 2e-137)
		tmp = t_1;
	elseif (Float64(x * x) <= 5e-44)
		tmp = Float64(t_2 + -1.0);
	elseif (Float64(x * x) <= 1e+223)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	t_2 = ((x / y) * (x / y)) * 0.5;
	tmp = 0.0;
	if ((x * x) <= 2e-254)
		tmp = ((1.0 + t_2) + -1.0) + -1.0;
	elseif ((x * x) <= 2e-137)
		tmp = t_1;
	elseif ((x * x) <= 5e-44)
		tmp = t_2 + -1.0;
	elseif ((x * x) <= 1e+223)
		tmp = t_1;
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	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[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e-254], N[(N[(N[(1.0 + t$95$2), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e-137], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 5e-44], N[(t$95$2 + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+223], t$95$1, N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 1.9999999999999998e-254

   1. Initial program 51.2%

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

   3. Taylor expanded in x around 0 81.1%

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

      1. unpow2 81.1%

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

      2. unpow2 81.1%

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

      3. times-frac 90.7%

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

   5. Applied egg-rr 90.7%

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

      1. *-commutative 90.7%

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

      2. pow2 90.7%

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

      3. expm1-log1p-u 90.7%

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

      4. expm1-define 90.7%

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

      5. log1p-undefine 90.7%

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

      6. pow2 90.7%

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

      7. *-commutative 90.7%

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

      8. add-exp-log 90.7%

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

      9. +-commutative 90.7%

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

      10. *-commutative 90.7%

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

      11. pow2 90.7%

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

   7. Applied egg-rr 90.7%

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

      1. pow2 90.7%

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

   9. Applied egg-rr 90.7%

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

   if 1.9999999999999998e-254 < (*.f64 x x) < 1.99999999999999996e-137 or 5.00000000000000039e-44 < (*.f64 x x) < 1.00000000000000005e223

   1. Initial program 79.7%

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

   if 1.99999999999999996e-137 < (*.f64 x x) < 5.00000000000000039e-44

   1. Initial program 49.9%

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

   3. Taylor expanded in x around 0 90.9%

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

      1. unpow2 90.9%

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

      2. unpow2 90.9%

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

      3. times-frac 90.9%

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

   5. Applied egg-rr 90.9%

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

   if 1.00000000000000005e223 < (*.f64 x x)

   1. Initial program 17.6%

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

   3. Taylor expanded in y around 0 81.0%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 81.0%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 81.0%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 91.7%

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

   5. Applied egg-rr 91.7%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-254}:\\ \;\;\;\;\left(\left(1 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5\right) + -1\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-137}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+223}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-18}:\\ \;\;\;\;\left(\left(1 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5\right) + -1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.05e-18)
   (+ (+ (+ 1.0 (* (* (/ x y) (/ x y)) 0.5)) -1.0) -1.0)
   (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.05e-18) {
		tmp = ((1.0 + (((x / y) * (x / y)) * 0.5)) + -1.0) + -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (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 <= 2.05d-18) then
        tmp = ((1.0d0 + (((x / y) * (x / y)) * 0.5d0)) + (-1.0d0)) + (-1.0d0)
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.05e-18) {
		tmp = ((1.0 + (((x / y) * (x / y)) * 0.5)) + -1.0) + -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.05e-18:
		tmp = ((1.0 + (((x / y) * (x / y)) * 0.5)) + -1.0) + -1.0
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.05e-18)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(x / y) * Float64(x / y)) * 0.5)) + -1.0) + -1.0);
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.05e-18)
		tmp = ((1.0 + (((x / y) * (x / y)) * 0.5)) + -1.0) + -1.0;
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.05e-18], N[(N[(N[(1.0 + N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0499999999999999e-18

   1. Initial program 53.6%

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

   3. Taylor expanded in x around 0 59.3%

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

      1. unpow2 59.3%

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

      2. unpow2 59.3%

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

      3. times-frac 64.9%

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

   5. Applied egg-rr 64.9%

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

      1. *-commutative 64.9%

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

      2. pow2 64.9%

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

      3. expm1-log1p-u 64.9%

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

      4. expm1-define 64.9%

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

      5. log1p-undefine 64.9%

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

      6. pow2 64.9%

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

      7. *-commutative 64.9%

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

      8. add-exp-log 64.9%

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

      9. +-commutative 64.9%

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

      10. *-commutative 64.9%

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

      11. pow2 64.9%

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

   7. Applied egg-rr 64.9%

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

      1. pow2 64.9%

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

   9. Applied egg-rr 64.9%

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

   if 2.0499999999999999e-18 < x

   1. Initial program 45.3%

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

   3. Taylor expanded in y around 0 78.7%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 78.7%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 78.7%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 86.6%

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

   5. Applied egg-rr 86.6%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-18}:\\ \;\;\;\;\left(\left(1 + \left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5\right) + -1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{-19}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.6e-19)
   (+ (* (* (/ x y) (/ x y)) 0.5) -1.0)
   (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.6e-19) {
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (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 <= 7.6d-19) then
        tmp = (((x / y) * (x / y)) * 0.5d0) + (-1.0d0)
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7.6e-19) {
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.6e-19:
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.6e-19)
		tmp = Float64(Float64(Float64(Float64(x / y) * Float64(x / y)) * 0.5) + -1.0);
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7.6e-19)
		tmp = (((x / y) * (x / y)) * 0.5) + -1.0;
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.6e-19], N[(N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.6e-19

   1. Initial program 53.6%

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

   3. Taylor expanded in x around 0 59.3%

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

      1. unpow2 59.3%

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

      2. unpow2 59.3%

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

      3. times-frac 64.9%

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

   5. Applied egg-rr 64.9%

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

   if 7.6e-19 < x

   1. Initial program 45.3%

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

   3. Taylor expanded in y around 0 78.7%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 78.7%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 78.7%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 86.6%

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

   5. Applied egg-rr 86.6%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{-19}:\\ \;\;\;\;\left(\frac{x}{y} \cdot \frac{x}{y}\right) \cdot 0.5 + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-19}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4e-19) -1.0 (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4e-19) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (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 <= 4d-19) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4e-19) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4e-19:
		tmp = -1.0
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4e-19)
		tmp = -1.0;
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4e-19)
		tmp = -1.0;
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4e-19], -1.0, N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999999e-19

   1. Initial program 53.6%

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

   3. Taylor expanded in x around 0 63.6%

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

   if 3.9999999999999999e-19 < x

   1. Initial program 45.3%

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

   3. Taylor expanded in y around 0 78.7%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 78.7%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 78.7%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 86.6%

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

   5. Applied egg-rr 86.6%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 63.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-20}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1e-20) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e-20) {
		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 <= 1d-20) 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 <= 1e-20) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1e-20:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e-20)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1e-20)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e-20], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999945e-21

   1. Initial program 53.6%

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

   3. Taylor expanded in x around 0 63.6%

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

   if 9.99999999999999945e-21 < x

   1. Initial program 45.3%

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

   3. Taylor expanded in x around inf 86.2%

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

Alternative 12: 50.6% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in x around 0 51.5%

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

Developer target: 50.6% 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 2024088 
(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))))