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

Percentage Accurate: 50.3% → 76.3%

Time: 7.5s

Alternatives: 5

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.3% 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: 76.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{if}\;t_1 \leq 2:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{log1p}\left(0.3333333333333333 \cdot {\left(\frac{x}{y}\right)}^{2}\right) \cdot 3\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= t_1 2.0)
     t_1
     (+
      (* 0.5 (* (log1p (* 0.3333333333333333 (pow (/ x y) 2.0))) 3.0))
      -1.0))))

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 tmp;
	if (t_1 <= 2.0) {
		tmp = t_1;
	} else {
		tmp = (0.5 * (log1p((0.3333333333333333 * pow((x / y), 2.0))) * 3.0)) + -1.0;
	}
	return tmp;
}

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 tmp;
	if (t_1 <= 2.0) {
		tmp = t_1;
	} else {
		tmp = (0.5 * (Math.log1p((0.3333333333333333 * Math.pow((x / y), 2.0))) * 3.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	tmp = 0
	if t_1 <= 2.0:
		tmp = t_1
	else:
		tmp = (0.5 * (math.log1p((0.3333333333333333 * math.pow((x / y), 2.0))) * 3.0)) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if (t_1 <= 2.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(0.5 * Float64(log1p(Float64(0.3333333333333333 * (Float64(x / y) ^ 2.0))) * 3.0)) + -1.0);
	end
	return 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]}, If[LessEqual[t$95$1, 2.0], t$95$1, N[(N[(0.5 * N[(N[Log[1 + N[(0.3333333333333333 * N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y 4) y))) < 2

   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

   if 2 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y 4) 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 48.5%

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

      1. add-log-exp 48.5%

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

      2. add-cube-cbrt 48.5%

         \[\leadsto 0.5 \cdot \log \color{blue}{\left(\left(\sqrt[3]{e^{\frac{{x}^{2}}{{y}^{2}}}} \cdot \sqrt[3]{e^{\frac{{x}^{2}}{{y}^{2}}}}\right) \cdot \sqrt[3]{e^{\frac{{x}^{2}}{{y}^{2}}}}\right)} - 1 \]

      3. log-prod 48.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\log \left(\sqrt[3]{e^{\frac{{x}^{2}}{{y}^{2}}}} \cdot \sqrt[3]{e^{\frac{{x}^{2}}{{y}^{2}}}}\right) + \log \left(\sqrt[3]{e^{\frac{{x}^{2}}{{y}^{2}}}}\right)\right)} - 1 \]

   5. Applied egg-rr 62.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right)\right)} - 1 \]
   6. Step-by-step derivation

      1. log-pow 62.3%

         \[\leadsto 0.5 \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right)} + \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right)\right) - 1 \]

      2. distribute-lft1-in 62.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right)\right)} - 1 \]

      3. metadata-eval 62.3%

         \[\leadsto 0.5 \cdot \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right)\right) - 1 \]

      4. *-commutative 62.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right) \cdot 3\right)} - 1 \]

   7. Simplified 62.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\log \left(\sqrt[3]{e^{{\left(\frac{x}{y}\right)}^{2}}}\right) \cdot 3\right)} - 1 \]

   8. Taylor expanded in x around 0 48.5%

      \[\leadsto 0.5 \cdot \left(\log \color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{{x}^{2}}{{y}^{2}}\right)} \cdot 3\right) - 1 \]
   9. Step-by-step derivation

      1. +-commutative 48.5%

         \[\leadsto 0.5 \cdot \left(\log \color{blue}{\left(0.3333333333333333 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right)} \cdot 3\right) - 1 \]

      2. fma-def 48.5%

         \[\leadsto 0.5 \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, \frac{{x}^{2}}{{y}^{2}}, 1\right)\right)} \cdot 3\right) - 1 \]

      3. unpow2 48.5%

         \[\leadsto 0.5 \cdot \left(\log \left(\mathsf{fma}\left(0.3333333333333333, \frac{\color{blue}{x \cdot x}}{{y}^{2}}, 1\right)\right) \cdot 3\right) - 1 \]

      4. unpow2 48.5%

         \[\leadsto 0.5 \cdot \left(\log \left(\mathsf{fma}\left(0.3333333333333333, \frac{x \cdot x}{\color{blue}{y \cdot y}}, 1\right)\right) \cdot 3\right) - 1 \]

      5. times-frac 63.9%

         \[\leadsto 0.5 \cdot \left(\log \left(\mathsf{fma}\left(0.3333333333333333, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, 1\right)\right) \cdot 3\right) - 1 \]

      6. unpow2 63.9%

         \[\leadsto 0.5 \cdot \left(\log \left(\mathsf{fma}\left(0.3333333333333333, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}, 1\right)\right) \cdot 3\right) - 1 \]

   10. Simplified 63.9%

       \[\leadsto 0.5 \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, {\left(\frac{x}{y}\right)}^{2}, 1\right)\right)} \cdot 3\right) - 1 \]
   11. Step-by-step derivation

       1. *-un-lft-identity 63.9%

          \[\leadsto 0.5 \cdot \left(\log \color{blue}{\left(1 \cdot \mathsf{fma}\left(0.3333333333333333, {\left(\frac{x}{y}\right)}^{2}, 1\right)\right)} \cdot 3\right) - 1 \]

       2. *-commutative 63.9%

          \[\leadsto 0.5 \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, {\left(\frac{x}{y}\right)}^{2}, 1\right) \cdot 1\right)} \cdot 3\right) - 1 \]

       3. log-prod 63.9%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\log \left(\mathsf{fma}\left(0.3333333333333333, {\left(\frac{x}{y}\right)}^{2}, 1\right)\right) + \log 1\right)} \cdot 3\right) - 1 \]

       4. fma-udef 63.9%

          \[\leadsto 0.5 \cdot \left(\left(\log \color{blue}{\left(0.3333333333333333 \cdot {\left(\frac{x}{y}\right)}^{2} + 1\right)} + \log 1\right) \cdot 3\right) - 1 \]

       5. +-commutative 63.9%

          \[\leadsto 0.5 \cdot \left(\left(\log \color{blue}{\left(1 + 0.3333333333333333 \cdot {\left(\frac{x}{y}\right)}^{2}\right)} + \log 1\right) \cdot 3\right) - 1 \]

       6. log1p-udef 63.9%

          \[\leadsto 0.5 \cdot \left(\left(\color{blue}{\mathsf{log1p}\left(0.3333333333333333 \cdot {\left(\frac{x}{y}\right)}^{2}\right)} + \log 1\right) \cdot 3\right) - 1 \]

       7. metadata-eval 63.9%

          \[\leadsto 0.5 \cdot \left(\left(\mathsf{log1p}\left(0.3333333333333333 \cdot {\left(\frac{x}{y}\right)}^{2}\right) + \color{blue}{0}\right) \cdot 3\right) - 1 \]

   12. Applied egg-rr 63.9%

       \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\mathsf{log1p}\left(0.3333333333333333 \cdot {\left(\frac{x}{y}\right)}^{2}\right) + 0\right)} \cdot 3\right) - 1 \]
   13. Step-by-step derivation

       1. +-rgt-identity 63.9%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\mathsf{log1p}\left(0.3333333333333333 \cdot {\left(\frac{x}{y}\right)}^{2}\right)} \cdot 3\right) - 1 \]

   14. Simplified 63.9%

       \[\leadsto 0.5 \cdot \left(\color{blue}{\mathsf{log1p}\left(0.3333333333333333 \cdot {\left(\frac{x}{y}\right)}^{2}\right)} \cdot 3\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)} \leq 2:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{log1p}\left(0.3333333333333333 \cdot {\left(\frac{x}{y}\right)}^{2}\right) \cdot 3\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{if}\;t_1 \leq 2:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= t_1 2.0) t_1 (+ (* 0.5 (* (/ x y) (/ x y))) -1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (t_1 <= 2.0) {
		tmp = t_1;
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    if (t_1 <= 2.0d0) then
        tmp = t_1
    else
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (t_1 <= 2.0) {
		tmp = t_1;
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	tmp = 0
	if t_1 <= 2.0:
		tmp = t_1
	else:
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if (t_1 <= 2.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	tmp = 0.0;
	if (t_1 <= 2.0)
		tmp = t_1;
	else
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y 4) y))) < 2

   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

   if 2 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y 4) 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 48.5%

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

      1. unpow2 48.5%

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

      2. unpow2 48.5%

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

      3. times-frac 62.6%

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

   5. Applied egg-rr 62.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

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

Alternative 3: 62.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-58}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-20} \lor \neg \left(y \leq 3.8 \cdot 10^{-9}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.45e-58)
   1.0
   (if (or (<= y 3e-20) (not (<= y 3.8e-9)))
     (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)
     1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-58) {
		tmp = 1.0;
	} else if ((y <= 3e-20) || !(y <= 3.8e-9)) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.45d-58) then
        tmp = 1.0d0
    else if ((y <= 3d-20) .or. (.not. (y <= 3.8d-9))) then
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-58) {
		tmp = 1.0;
	} else if ((y <= 3e-20) || !(y <= 3.8e-9)) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.45e-58:
		tmp = 1.0
	elif (y <= 3e-20) or not (y <= 3.8e-9):
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.45e-58)
		tmp = 1.0;
	elseif ((y <= 3e-20) || !(y <= 3.8e-9))
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.45e-58)
		tmp = 1.0;
	elseif ((y <= 3e-20) || ~((y <= 3.8e-9)))
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.45e-58], 1.0, If[Or[LessEqual[y, 3e-20], N[Not[LessEqual[y, 3.8e-9]], $MachinePrecision]], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < 1.44999999999999995e-58 or 3.00000000000000029e-20 < y < 3.80000000000000011e-9

   1. Initial program 54.8%

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

   3. Taylor expanded in x around inf 52.5%

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

   if 1.44999999999999995e-58 < y < 3.00000000000000029e-20 or 3.80000000000000011e-9 < y

   1. Initial program 45.7%

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

   3. Taylor expanded in x around 0 83.3%

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

      1. unpow2 83.3%

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

      2. unpow2 83.3%

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

      3. times-frac 87.1%

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

   5. Applied egg-rr 87.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-58}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-20} \lor \neg \left(y \leq 3.8 \cdot 10^{-9}\right):\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-57}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-20}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.9e-57) 1.0 (if (<= y 1e-20) -1.0 (if (<= y 1e-8) 1.0 -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-57) {
		tmp = 1.0;
	} else if (y <= 1e-20) {
		tmp = -1.0;
	} else if (y <= 1e-8) {
		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 (y <= 3.9d-57) then
        tmp = 1.0d0
    else if (y <= 1d-20) then
        tmp = -1.0d0
    else if (y <= 1d-8) 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 (y <= 3.9e-57) {
		tmp = 1.0;
	} else if (y <= 1e-20) {
		tmp = -1.0;
	} else if (y <= 1e-8) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.9e-57:
		tmp = 1.0
	elif y <= 1e-20:
		tmp = -1.0
	elif y <= 1e-8:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.9e-57)
		tmp = 1.0;
	elseif (y <= 1e-20)
		tmp = -1.0;
	elseif (y <= 1e-8)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.9e-57)
		tmp = 1.0;
	elseif (y <= 1e-20)
		tmp = -1.0;
	elseif (y <= 1e-8)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.9e-57], 1.0, If[LessEqual[y, 1e-20], -1.0, If[LessEqual[y, 1e-8], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 3.90000000000000006e-57 or 9.99999999999999945e-21 < y < 1e-8

   1. Initial program 54.8%

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

   3. Taylor expanded in x around inf 52.5%

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

   if 3.90000000000000006e-57 < y < 9.99999999999999945e-21 or 1e-8 < y

   1. Initial program 45.7%

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

   3. Taylor expanded in x around 0 86.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-57}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-20}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.8% 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 52.3%

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

3. Taylor expanded in x around 0 58.2%

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

4. Final simplification 58.2%

   \[\leadsto -1 \]
5. Add Preprocessing

Developer target: 50.8% 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 2024021 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (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))))