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

Percentage Accurate: 50.7% → 80.9%

Time: 8.1s

Alternatives: 5

Speedup: 3.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t\_0}{x \cdot x + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 50.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t\_0}{x \cdot x + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 80.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 10^{-315}:\\ \;\;\;\;\frac{x\_m}{\mathsf{hypot}\left(y \cdot 2, x\_m\right)}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{x\_m \cdot x\_m - t\_0}{\mathsf{fma}\left(y \cdot 4, y, x\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 1e-315)
     (/ x_m (hypot (* y 2.0) x_m))
     (if (<= t_0 5e+306)
       (/ (- (* x_m x_m) t_0) (fma (* y 4.0) y (* x_m x_m)))
       -1.0))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-315) {
		tmp = x_m / hypot((y * 2.0), x_m);
	} else if (t_0 <= 5e+306) {
		tmp = ((x_m * x_m) - t_0) / fma((y * 4.0), y, (x_m * x_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 1e-315)
		tmp = Float64(x_m / hypot(Float64(y * 2.0), x_m));
	elseif (t_0 <= 5e+306)
		tmp = Float64(Float64(Float64(x_m * x_m) - t_0) / fma(Float64(y * 4.0), y, Float64(x_m * x_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-315], N[(x$95$m / N[Sqrt[N[(y * 2.0), $MachinePrecision] ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+306], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(y * 4.0), $MachinePrecision] * y + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 9.999999985e-316

   1. Initial program 56.2%

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

      1. clear-num 56.2%

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

      2. inv-pow 56.2%

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

      3. +-commutative 56.2%

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

      4. *-commutative 56.2%

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

      5. associate-*l* 56.2%

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

      6. fma-define 56.2%

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

      7. pow2 56.2%

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

      8. pow2 56.2%

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

      9. sub-neg 56.2%

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

      10. +-commutative 56.2%

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

      11. *-commutative 56.2%

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

      12. distribute-rgt-neg-in 56.2%

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

      13. fma-define 56.2%

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

      14. distribute-rgt-neg-in 56.2%

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

      15. metadata-eval 56.2%

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

      16. pow2 56.2%

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

   4. Applied egg-rr 56.2%

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

   6. Applied egg-rr 89.6%

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

   7. Taylor expanded in y around 0 46.5%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(\sqrt{4 \cdot {y}^{2}}, x\right)} \cdot \color{blue}{x} \]
   8. Step-by-step derivation

      1. *-commutative 46.5%

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

      2. clear-num 46.5%

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

      3. un-div-inv 46.7%

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

      4. /-rgt-identity 46.7%

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

      5. *-commutative 46.7%

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

      6. sqrt-prod 46.7%

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

      7. sqrt-pow1 44.2%

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

      8. metadata-eval 44.2%

         \[\leadsto \frac{x}{\mathsf{hypot}\left({y}^{\color{blue}{1}} \cdot \sqrt{4}, x\right)} \]

      9. pow1 44.2%

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

      10. metadata-eval 44.2%

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

   9. Applied egg-rr 44.2%

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

   if 9.999999985e-316 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999993e306

   1. Initial program 78.5%

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

      1. +-commutative 78.5%

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

      2. fma-define 78.6%

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

      3. pow2 78.6%

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

   4. Applied egg-rr 78.6%

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

      1. pow2 78.6%

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

   6. Applied egg-rr 78.6%

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

   if 4.99999999999999993e306 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 92.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-315}:\\ \;\;\;\;\frac{x}{\mathsf{hypot}\left(y \cdot 2, x\right)}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(y \cdot 4, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 10^{-315}:\\ \;\;\;\;\frac{x\_m}{\mathsf{hypot}\left(y \cdot 2, x\_m\right)}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{x\_m \cdot x\_m - t\_0}{t\_0 + x\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 1e-315)
     (/ x_m (hypot (* y 2.0) x_m))
     (if (<= t_0 5e+306) (/ (- (* x_m x_m) t_0) (+ t_0 (* x_m x_m))) -1.0))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-315) {
		tmp = x_m / hypot((y * 2.0), x_m);
	} else if (t_0 <= 5e+306) {
		tmp = ((x_m * x_m) - t_0) / (t_0 + (x_m * x_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-315) {
		tmp = x_m / Math.hypot((y * 2.0), x_m);
	} else if (t_0 <= 5e+306) {
		tmp = ((x_m * x_m) - t_0) / (t_0 + (x_m * x_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 1e-315:
		tmp = x_m / math.hypot((y * 2.0), x_m)
	elif t_0 <= 5e+306:
		tmp = ((x_m * x_m) - t_0) / (t_0 + (x_m * x_m))
	else:
		tmp = -1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 1e-315)
		tmp = Float64(x_m / hypot(Float64(y * 2.0), x_m));
	elseif (t_0 <= 5e+306)
		tmp = Float64(Float64(Float64(x_m * x_m) - t_0) / Float64(t_0 + Float64(x_m * x_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 1e-315)
		tmp = x_m / hypot((y * 2.0), x_m);
	elseif (t_0 <= 5e+306)
		tmp = ((x_m * x_m) - t_0) / (t_0 + (x_m * x_m));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-315], N[(x$95$m / N[Sqrt[N[(y * 2.0), $MachinePrecision] ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+306], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 9.999999985e-316

   1. Initial program 56.2%

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

      1. clear-num 56.2%

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

      2. inv-pow 56.2%

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

      3. +-commutative 56.2%

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

      4. *-commutative 56.2%

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

      5. associate-*l* 56.2%

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

      6. fma-define 56.2%

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

      7. pow2 56.2%

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

      8. pow2 56.2%

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

      9. sub-neg 56.2%

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

      10. +-commutative 56.2%

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

      11. *-commutative 56.2%

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

      12. distribute-rgt-neg-in 56.2%

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

      13. fma-define 56.2%

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

      14. distribute-rgt-neg-in 56.2%

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

      15. metadata-eval 56.2%

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

      16. pow2 56.2%

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

   4. Applied egg-rr 56.2%

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

   6. Applied egg-rr 89.6%

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

   7. Taylor expanded in y around 0 46.5%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(\sqrt{4 \cdot {y}^{2}}, x\right)} \cdot \color{blue}{x} \]
   8. Step-by-step derivation

      1. *-commutative 46.5%

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

      2. clear-num 46.5%

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

      3. un-div-inv 46.7%

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

      4. /-rgt-identity 46.7%

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

      5. *-commutative 46.7%

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

      6. sqrt-prod 46.7%

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

      7. sqrt-pow1 44.2%

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

      8. metadata-eval 44.2%

         \[\leadsto \frac{x}{\mathsf{hypot}\left({y}^{\color{blue}{1}} \cdot \sqrt{4}, x\right)} \]

      9. pow1 44.2%

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

      10. metadata-eval 44.2%

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

   9. Applied egg-rr 44.2%

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

   if 9.999999985e-316 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999993e306

   1. Initial program 78.5%

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

   if 4.99999999999999993e306 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 92.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-315}:\\ \;\;\;\;\frac{x}{\mathsf{hypot}\left(y \cdot 2, x\right)}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 1.6 \cdot 10^{-307}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 4.6 \cdot 10^{+306}:\\ \;\;\;\;\frac{x\_m \cdot x\_m - t\_0}{t\_0 + x\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 1.6e-307)
     1.0
     (if (<= t_0 4.6e+306) (/ (- (* x_m x_m) t_0) (+ t_0 (* x_m x_m))) -1.0))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1.6e-307) {
		tmp = 1.0;
	} else if (t_0 <= 4.6e+306) {
		tmp = ((x_m * x_m) - t_0) / (t_0 + (x_m * x_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 1.6d-307) then
        tmp = 1.0d0
    else if (t_0 <= 4.6d+306) then
        tmp = ((x_m * x_m) - t_0) / (t_0 + (x_m * x_m))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1.6e-307) {
		tmp = 1.0;
	} else if (t_0 <= 4.6e+306) {
		tmp = ((x_m * x_m) - t_0) / (t_0 + (x_m * x_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 1.6e-307:
		tmp = 1.0
	elif t_0 <= 4.6e+306:
		tmp = ((x_m * x_m) - t_0) / (t_0 + (x_m * x_m))
	else:
		tmp = -1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 1.6e-307)
		tmp = 1.0;
	elseif (t_0 <= 4.6e+306)
		tmp = Float64(Float64(Float64(x_m * x_m) - t_0) / Float64(t_0 + Float64(x_m * x_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 1.6e-307)
		tmp = 1.0;
	elseif (t_0 <= 4.6e+306)
		tmp = ((x_m * x_m) - t_0) / (t_0 + (x_m * x_m));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1.6e-307], 1.0, If[LessEqual[t$95$0, 4.6e+306], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.60000000000000005e-307

   1. Initial program 56.2%

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

   3. Taylor expanded in x around inf 89.9%

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

   if 1.60000000000000005e-307 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.6000000000000001e306

   1. Initial program 78.5%

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

   if 4.6000000000000001e306 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 92.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 1.6 \cdot 10^{-307}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 4.6 \cdot 10^{+306}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 (if (<= y 3e-62) 1.0 -1.0))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (y <= 3e-62) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3d-62) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if (y <= 3e-62) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if y <= 3e-62:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (y <= 3e-62)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if (y <= 3e-62)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[y, 3e-62], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 3.0000000000000001e-62

   1. Initial program 58.8%

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

   3. Taylor expanded in x around inf 60.3%

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

   if 3.0000000000000001e-62 < y

   1. Initial program 47.3%

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

   3. Taylor expanded in x around 0 68.6%

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

Alternative 5: 50.3% accurate, 19.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ -1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 -1.0)

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	return -1.0;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	return -1.0

Julia

x_m = abs(x)
function code(x_m, y)
	return -1.0
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := -1.0

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in x around 0 49.7%

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

Developer Target 1: 51.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 4\\ t_1 := x \cdot x + t\_0\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x \cdot x - t\_3}{x \cdot x + t\_3} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{t\_1} - t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{t\_1}}\right)}^{2} - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 4.0))
        (t_1 (+ (* x x) t_0))
        (t_2 (/ t_0 t_1))
        (t_3 (* (* y 4.0) y)))
   (if (< (/ (- (* x x) t_3) (+ (* x x) t_3)) 0.9743233849626781)
     (- (/ (* x x) t_1) t_2)
     (- (pow (/ x (sqrt t_1)) 2.0) t_2))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = pow((x / sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (y * y) * 4.0d0
    t_1 = (x * x) + t_0
    t_2 = t_0 / t_1
    t_3 = (y * 4.0d0) * y
    if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781d0) then
        tmp = ((x * x) / t_1) - t_2
    else
        tmp = ((x / sqrt(t_1)) ** 2.0d0) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = Math.pow((x / Math.sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 4.0
	t_1 = (x * x) + t_0
	t_2 = t_0 / t_1
	t_3 = (y * 4.0) * y
	tmp = 0
	if (((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781:
		tmp = ((x * x) / t_1) - t_2
	else:
		tmp = math.pow((x / math.sqrt(t_1)), 2.0) - t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 4.0)
	t_1 = Float64(Float64(x * x) + t_0)
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_3) / Float64(Float64(x * x) + t_3)) < 0.9743233849626781)
		tmp = Float64(Float64(Float64(x * x) / t_1) - t_2);
	else
		tmp = Float64((Float64(x / sqrt(t_1)) ^ 2.0) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 4.0;
	t_1 = (x * x) + t_0;
	t_2 = t_0 / t_1;
	t_3 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781)
		tmp = ((x * x) / t_1) - t_2;
	else
		tmp = ((x / sqrt(t_1)) ^ 2.0) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[Less[N[(N[(N[(x * x), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 0.9743233849626781], N[(N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Power[N[(x / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2024150 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))) 9743233849626781/10000000000000000) (- (/ (* x x) (+ (* x x) (* (* y y) 4))) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4)))) 2) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4))))))

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