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

Percentage Accurate: 51.2% → 81.8%

Time: 4.7s

Alternatives: 4

Speedup: 0.9×

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: 51.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.65 \cdot 10^{-97}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x\_m \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, y \cdot y, x\_m \cdot x\_m\right)}{\mathsf{fma}\left(4 \cdot y, y, x\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-8}{x\_m} \cdot y}{x\_m}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 3.65e-97)
   -1.0
   (if (<= x_m 2e+117)
     (/ (fma -4.0 (* y y) (* x_m x_m)) (fma (* 4.0 y) y (* x_m x_m)))
     (fma (/ (* (/ -8.0 x_m) y) x_m) y 1.0))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 3.65e-97) {
		tmp = -1.0;
	} else if (x_m <= 2e+117) {
		tmp = fma(-4.0, (y * y), (x_m * x_m)) / fma((4.0 * y), y, (x_m * x_m));
	} else {
		tmp = fma((((-8.0 / x_m) * y) / x_m), y, 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 3.65e-97)
		tmp = -1.0;
	elseif (x_m <= 2e+117)
		tmp = Float64(fma(-4.0, Float64(y * y), Float64(x_m * x_m)) / fma(Float64(4.0 * y), y, Float64(x_m * x_m)));
	else
		tmp = fma(Float64(Float64(Float64(-8.0 / x_m) * y) / x_m), y, 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 3.65e-97], -1.0, If[LessEqual[x$95$m, 2e+117], N[(N[(-4.0 * N[(y * y), $MachinePrecision] + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * y), $MachinePrecision] * y + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-8.0 / x$95$m), $MachinePrecision] * y), $MachinePrecision] / x$95$m), $MachinePrecision] * y + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.64999999999999987e-97

   1. Initial program 53.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 57.2%

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

   if 3.64999999999999987e-97 < x < 2.0000000000000001e117

   1. Initial program 84.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 84.4

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 84.4

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 84.4

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

   4. Applied rewrites 84.4%

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

   if 2.0000000000000001e117 < x

   1. Initial program 10.8%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-out-- N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(8\right)}}{{x}^{2}} \cdot {y}^{2} + 1 \]

      9. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{8}{{x}^{2}}\right)\right)} \cdot {y}^{2} + 1 \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{8 \cdot 1}}{{x}^{2}}\right)\right) \cdot {y}^{2} + 1 \]

      11. associate-*r/ N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{8 \cdot \frac{1}{{x}^{2}}}\right)\right) \cdot {y}^{2} + 1 \]

      12. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(8 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \color{blue}{\left(y \cdot y\right)} + 1 \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(8 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y\right) \cdot y} + 1 \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(8 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y, y, 1\right)} \]

   5. Applied rewrites 85.9%

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

   7. Applied rewrites 87.1%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-8}{x} \cdot y}{x}, y, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 76.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x\_m \cdot x\_m - t\_0}{x\_m \cdot x\_m + t\_0} \leq -1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-8}{x\_m} \cdot y}{x\_m}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y)))
   (if (<= (/ (- (* x_m x_m) t_0) (+ (* x_m x_m) t_0)) -1.0)
     -1.0
     (fma (/ (* (/ -8.0 x_m) y) x_m) y 1.0))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (y * 4.0) * y;
	double tmp;
	if ((((x_m * x_m) - t_0) / ((x_m * x_m) + t_0)) <= -1.0) {
		tmp = -1.0;
	} else {
		tmp = fma((((-8.0 / x_m) * y) / x_m), y, 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x_m * x_m) - t_0) / Float64(Float64(x_m * x_m) + t_0)) <= -1.0)
		tmp = -1.0;
	else
		tmp = fma(Float64(Float64(Float64(-8.0 / x_m) * y) / x_m), y, 1.0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], -1.0, N[(N[(N[(N[(-8.0 / x$95$m), $MachinePrecision] * y), $MachinePrecision] / x$95$m), $MachinePrecision] * y + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

   if -1 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)))

   1. Initial program 34.6%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-out-- N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(8\right)}}{{x}^{2}} \cdot {y}^{2} + 1 \]

      9. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{8}{{x}^{2}}\right)\right)} \cdot {y}^{2} + 1 \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{8 \cdot 1}}{{x}^{2}}\right)\right) \cdot {y}^{2} + 1 \]

      11. associate-*r/ N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{8 \cdot \frac{1}{{x}^{2}}}\right)\right) \cdot {y}^{2} + 1 \]

      12. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(8 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \color{blue}{\left(y \cdot y\right)} + 1 \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(8 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y\right) \cdot y} + 1 \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(8 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot y, y, 1\right)} \]

   5. Applied rewrites 66.0%

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

   7. Applied rewrites 72.1%

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

Alternative 3: 75.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (y * 4.0) * y;
	double tmp;
	if ((((x_m * x_m) - t_0) / ((x_m * x_m) + t_0)) <= -1.0) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (y * 4.0d0) * y
    if ((((x_m * x_m) - t_0) / ((x_m * x_m) + t_0)) <= (-1.0d0)) 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 t_0 = (y * 4.0) * y;
	double tmp;
	if ((((x_m * x_m) - t_0) / ((x_m * x_m) + t_0)) <= -1.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	t_0 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x_m * x_m) - t_0) / ((x_m * x_m) + t_0)) <= -1.0)
		tmp = -1.0;
	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[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

   if -1 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)))

   1. Initial program 34.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 70.7%

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

Alternative 4: 50.3% accurate, 48.0× 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 52.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Applied rewrites 49.0%

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

Developer Target 1: 51.6% accurate, 0.2× 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 2024312 
(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))))