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

Percentage Accurate: 50.6% → 80.5%

Time: 4.9s

Alternatives: 4

Speedup: 4.0×

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: 50.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-134}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot 0.5}{y}, \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+259}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-134)
     (fma (/ (* x 0.5) y) (/ x y) -1.0)
     (if (<= (* x x) 4e+259) (/ (- (* x x) t_0) (+ (* x x) t_0)) 1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-134) {
		tmp = fma(((x * 0.5) / y), (x / y), -1.0);
	} else if ((x * x) <= 4e+259) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 1e-134)
		tmp = fma(Float64(Float64(x * 0.5) / y), Float64(x / y), -1.0);
	elseif (Float64(x * x) <= 4e+259)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e-134], N[(N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 4e+259], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.00000000000000004e-134

   1. Initial program 53.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. metadata-eval 75.6

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

   5. Simplified 75.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 83.9

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

   7. Applied egg-rr 83.9%

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

   if 1.00000000000000004e-134 < (*.f64 x x) < 4e259

   1. Initial program 79.2%

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

   if 4e259 < (*.f64 x x)

   1. Initial program 7.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 84.3%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-134}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot 0.5}{y}, \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+259}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-134}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot 0.5}{y}, \frac{x}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e-134) (fma (/ (* x 0.5) y) (/ x y) -1.0) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e-134) {
		tmp = fma(((x * 0.5) / y), (x / y), -1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e-134)
		tmp = fma(Float64(Float64(x * 0.5) / y), Float64(x / y), -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-134], N[(N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000004e-134

   1. Initial program 53.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. metadata-eval 75.6

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

   5. Simplified 75.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 83.9

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

   7. Applied egg-rr 83.9%

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

   if 1.00000000000000004e-134 < (*.f64 x x)

   1. Initial program 40.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.9%

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

Alternative 3: 74.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-134}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* x x) 1e-134) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e-134) {
		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 * x) <= 1d-134) 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 * x) <= 1e-134) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 1e-134:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e-134)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 1e-134)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-134], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000004e-134

   1. Initial program 53.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 82.7%

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

   if 1.00000000000000004e-134 < (*.f64 x x)

   1. Initial program 40.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.9%

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

Alternative 4: 49.8% accurate, 48.0× 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 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 0

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

5. Simplified 48.5%

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

Developer Target 1: 51.0% 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 2024195 
(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))))