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

Percentage Accurate: 50.4% → 79.8%

Time: 9.8s

Alternatives: 6

Speedup: 19.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.4% 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: 79.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \log \left(\mathsf{fma}\left(0.5, e^{-1} \cdot {\left(\frac{x}{y}\right)}^{2}, e^{-1}\right)\right)\\ t_2 := x \cdot x + t\_0\\ t_3 := \frac{x \cdot x - t\_0}{t\_2}\\ \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-289}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-242}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot x \leq 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{t\_2}\\ \mathbf{elif}\;x \cdot x \leq 50000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot x \leq 1.08 \cdot 10^{+284}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (log (fma 0.5 (* (exp -1.0) (pow (/ x y) 2.0)) (exp -1.0))))
        (t_2 (+ (* x x) t_0))
        (t_3 (/ (- (* x x) t_0) t_2)))
   (if (<= (* x x) 4e-289)
     t_1
     (if (<= (* x x) 5e-242)
       t_3
       (if (<= (* x x) 5e-151)
         t_1
         (if (<= (* x x) 1e-121)
           (/ (fma (* y -4.0) y (pow x 2.0)) t_2)
           (if (<= (* x x) 50000.0)
             t_1
             (if (<= (* x x) 1.08e+284) t_3 1.0))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = log(fma(0.5, (exp(-1.0) * pow((x / y), 2.0)), exp(-1.0)));
	double t_2 = (x * x) + t_0;
	double t_3 = ((x * x) - t_0) / t_2;
	double tmp;
	if ((x * x) <= 4e-289) {
		tmp = t_1;
	} else if ((x * x) <= 5e-242) {
		tmp = t_3;
	} else if ((x * x) <= 5e-151) {
		tmp = t_1;
	} else if ((x * x) <= 1e-121) {
		tmp = fma((y * -4.0), y, pow(x, 2.0)) / t_2;
	} else if ((x * x) <= 50000.0) {
		tmp = t_1;
	} else if ((x * x) <= 1.08e+284) {
		tmp = t_3;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = log(fma(0.5, Float64(exp(-1.0) * (Float64(x / y) ^ 2.0)), exp(-1.0)))
	t_2 = Float64(Float64(x * x) + t_0)
	t_3 = Float64(Float64(Float64(x * x) - t_0) / t_2)
	tmp = 0.0
	if (Float64(x * x) <= 4e-289)
		tmp = t_1;
	elseif (Float64(x * x) <= 5e-242)
		tmp = t_3;
	elseif (Float64(x * x) <= 5e-151)
		tmp = t_1;
	elseif (Float64(x * x) <= 1e-121)
		tmp = Float64(fma(Float64(y * -4.0), y, (x ^ 2.0)) / t_2);
	elseif (Float64(x * x) <= 50000.0)
		tmp = t_1;
	elseif (Float64(x * x) <= 1.08e+284)
		tmp = t_3;
	else
		tmp = 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[Log[N[(0.5 * N[(N[Exp[-1.0], $MachinePrecision] * N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Exp[-1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 4e-289], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 5e-242], t$95$3, If[LessEqual[N[(x * x), $MachinePrecision], 5e-151], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 1e-121], N[(N[(N[(y * -4.0), $MachinePrecision] * y + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 50000.0], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 1.08e+284], t$95$3, 1.0]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 4e-289 or 4.9999999999999998e-242 < (*.f64 x x) < 5.00000000000000003e-151 or 9.9999999999999998e-122 < (*.f64 x x) < 5e4

   1. Initial program 59.0%

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

      1. sub-neg 59.0%

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

      2. +-commutative 59.0%

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

      3. distribute-lft-neg-in 59.0%

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

      4. fma-define 59.0%

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

      5. distribute-rgt-neg-in 59.0%

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

      6. metadata-eval 59.0%

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

      7. pow2 59.0%

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

   4. Applied egg-rr 59.0%

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

   5. Taylor expanded in x around 0 72.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-0.125 \cdot \frac{{x}^{2}}{{y}^{4}} + 0.5 \cdot \frac{1}{{y}^{2}}\right) - 1} \]
   6. Step-by-step derivation

      1. distribute-lft-in 72.0%

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(-0.125 \cdot \frac{{x}^{2}}{{y}^{4}}\right) + {x}^{2} \cdot \left(0.5 \cdot \frac{1}{{y}^{2}}\right)\right)} - 1 \]

      2. associate--l+ 72.0%

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(-0.125 \cdot \frac{{x}^{2}}{{y}^{4}}\right) + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{y}^{2}}\right) - 1\right)} \]

      3. *-commutative 72.0%

         \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{x}^{2}}{{y}^{4}}\right) \cdot {x}^{2}} + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{y}^{2}}\right) - 1\right) \]

      4. associate-*l* 72.0%

         \[\leadsto \color{blue}{-0.125 \cdot \left(\frac{{x}^{2}}{{y}^{4}} \cdot {x}^{2}\right)} + \left({x}^{2} \cdot \left(0.5 \cdot \frac{1}{{y}^{2}}\right) - 1\right) \]

      5. fma-define 72.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, \frac{{x}^{2}}{{y}^{4}} \cdot {x}^{2}, {x}^{2} \cdot \left(0.5 \cdot \frac{1}{{y}^{2}}\right) - 1\right)} \]

   7. Simplified 89.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{4}, \mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, 0.5, -1\right)\right)} \]
   8. Step-by-step derivation

      1. add-log-exp 89.1%

         \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{4}, \mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, 0.5, -1\right)\right)}\right)} \]

   9. Applied egg-rr 89.1%

      \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{4}, \mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, 0.5, -1\right)\right)}\right)} \]

   10. Taylor expanded in x around 0 83.4%

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

       1. +-commutative 83.4%

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

       2. fma-define 83.4%

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

       3. *-commutative 83.4%

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

       4. associate-/l* 83.4%

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

       5. unpow2 83.4%

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

       6. unpow2 83.4%

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

       7. times-frac 90.2%

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

       8. unpow2 90.2%

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

   12. Simplified 90.2%

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

   if 4e-289 < (*.f64 x x) < 4.9999999999999998e-242 or 5e4 < (*.f64 x x) < 1.07999999999999993e284

   1. Initial program 78.3%

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

   if 5.00000000000000003e-151 < (*.f64 x x) < 9.9999999999999998e-122

   1. Initial program 91.5%

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

      1. sub-neg 91.5%

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

      2. +-commutative 91.5%

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

      3. distribute-lft-neg-in 91.5%

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

      4. fma-define 91.7%

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

      5. distribute-rgt-neg-in 91.7%

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

      6. metadata-eval 91.7%

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

      7. pow2 91.7%

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

   4. Applied egg-rr 91.7%

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

   if 1.07999999999999993e284 < (*.f64 x x)

   1. Initial program 2.4%

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

   3. Taylor expanded in x around inf 86.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-289}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(0.5, e^{-1} \cdot {\left(\frac{x}{y}\right)}^{2}, e^{-1}\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-151}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(0.5, e^{-1} \cdot {\left(\frac{x}{y}\right)}^{2}, e^{-1}\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 50000:\\ \;\;\;\;\log \left(\mathsf{fma}\left(0.5, e^{-1} \cdot {\left(\frac{x}{y}\right)}^{2}, e^{-1}\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 1.08 \cdot 10^{+284}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := x \cdot x + t\_0\\ t_2 := \frac{x \cdot x - t\_0}{t\_1}\\ t_3 := -1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-289}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-242}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-151}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \cdot x \leq 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{t\_1}\\ \mathbf{elif}\;x \cdot x \leq 50000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \cdot x \leq 1.08 \cdot 10^{+284}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (+ (* x x) t_0))
        (t_2 (/ (- (* x x) t_0) t_1))
        (t_3 (+ -1.0 (* 0.5 (* (/ x y) (/ x y))))))
   (if (<= (* x x) 4e-289)
     t_3
     (if (<= (* x x) 5e-242)
       t_2
       (if (<= (* x x) 5e-151)
         t_3
         (if (<= (* x x) 1e-121)
           (/ (fma (* y -4.0) y (pow x 2.0)) t_1)
           (if (<= (* x x) 50000.0)
             t_3
             (if (<= (* x x) 1.08e+284) t_2 1.0))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = (x * x) + t_0;
	double t_2 = ((x * x) - t_0) / t_1;
	double t_3 = -1.0 + (0.5 * ((x / y) * (x / y)));
	double tmp;
	if ((x * x) <= 4e-289) {
		tmp = t_3;
	} else if ((x * x) <= 5e-242) {
		tmp = t_2;
	} else if ((x * x) <= 5e-151) {
		tmp = t_3;
	} else if ((x * x) <= 1e-121) {
		tmp = fma((y * -4.0), y, pow(x, 2.0)) / t_1;
	} else if ((x * x) <= 50000.0) {
		tmp = t_3;
	} else if ((x * x) <= 1.08e+284) {
		tmp = t_2;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(x * x) + t_0)
	t_2 = Float64(Float64(Float64(x * x) - t_0) / t_1)
	t_3 = Float64(-1.0 + Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))))
	tmp = 0.0
	if (Float64(x * x) <= 4e-289)
		tmp = t_3;
	elseif (Float64(x * x) <= 5e-242)
		tmp = t_2;
	elseif (Float64(x * x) <= 5e-151)
		tmp = t_3;
	elseif (Float64(x * x) <= 1e-121)
		tmp = Float64(fma(Float64(y * -4.0), y, (x ^ 2.0)) / t_1);
	elseif (Float64(x * x) <= 50000.0)
		tmp = t_3;
	elseif (Float64(x * x) <= 1.08e+284)
		tmp = t_2;
	else
		tmp = 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[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 + N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 4e-289], t$95$3, If[LessEqual[N[(x * x), $MachinePrecision], 5e-242], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 5e-151], t$95$3, If[LessEqual[N[(x * x), $MachinePrecision], 1e-121], N[(N[(N[(y * -4.0), $MachinePrecision] * y + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 50000.0], t$95$3, If[LessEqual[N[(x * x), $MachinePrecision], 1.08e+284], t$95$2, 1.0]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 4e-289 or 4.9999999999999998e-242 < (*.f64 x x) < 5.00000000000000003e-151 or 9.9999999999999998e-122 < (*.f64 x x) < 5e4

   1. Initial program 59.0%

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

   3. Taylor expanded in x around 0 83.3%

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

      1. pow2 83.3%

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

      2. unpow2 83.3%

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

      3. times-frac 89.6%

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

   5. Applied egg-rr 89.6%

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

   if 4e-289 < (*.f64 x x) < 4.9999999999999998e-242 or 5e4 < (*.f64 x x) < 1.07999999999999993e284

   1. Initial program 78.3%

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

   if 5.00000000000000003e-151 < (*.f64 x x) < 9.9999999999999998e-122

   1. Initial program 91.5%

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

      1. sub-neg 91.5%

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

      2. +-commutative 91.5%

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

      3. distribute-lft-neg-in 91.5%

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

      4. fma-define 91.7%

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

      5. distribute-rgt-neg-in 91.7%

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

      6. metadata-eval 91.7%

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

      7. pow2 91.7%

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

   4. Applied egg-rr 91.7%

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

   if 1.07999999999999993e284 < (*.f64 x x)

   1. Initial program 2.4%

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

   3. Taylor expanded in x around inf 86.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-289}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-151}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 50000:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 1.08 \cdot 10^{+284}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ t_2 := -1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-289}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-242}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-151}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot x \leq 10^{-121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot x \leq 50000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot x \leq 1.08 \cdot 10^{+284}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0)))
        (t_2 (+ -1.0 (* 0.5 (* (/ x y) (/ x y))))))
   (if (<= (* x x) 4e-289)
     t_2
     (if (<= (* x x) 5e-242)
       t_1
       (if (<= (* x x) 5e-151)
         t_2
         (if (<= (* x x) 1e-121)
           t_1
           (if (<= (* x x) 50000.0)
             t_2
             (if (<= (* x x) 1.08e+284) t_1 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 t_2 = -1.0 + (0.5 * ((x / y) * (x / y)));
	double tmp;
	if ((x * x) <= 4e-289) {
		tmp = t_2;
	} else if ((x * x) <= 5e-242) {
		tmp = t_1;
	} else if ((x * x) <= 5e-151) {
		tmp = t_2;
	} else if ((x * x) <= 1e-121) {
		tmp = t_1;
	} else if ((x * x) <= 50000.0) {
		tmp = t_2;
	} else if ((x * x) <= 1.08e+284) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    t_2 = (-1.0d0) + (0.5d0 * ((x / y) * (x / y)))
    if ((x * x) <= 4d-289) then
        tmp = t_2
    else if ((x * x) <= 5d-242) then
        tmp = t_1
    else if ((x * x) <= 5d-151) then
        tmp = t_2
    else if ((x * x) <= 1d-121) then
        tmp = t_1
    else if ((x * x) <= 50000.0d0) then
        tmp = t_2
    else if ((x * x) <= 1.08d+284) then
        tmp = t_1
    else
        tmp = 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 t_2 = -1.0 + (0.5 * ((x / y) * (x / y)));
	double tmp;
	if ((x * x) <= 4e-289) {
		tmp = t_2;
	} else if ((x * x) <= 5e-242) {
		tmp = t_1;
	} else if ((x * x) <= 5e-151) {
		tmp = t_2;
	} else if ((x * x) <= 1e-121) {
		tmp = t_1;
	} else if ((x * x) <= 50000.0) {
		tmp = t_2;
	} else if ((x * x) <= 1.08e+284) {
		tmp = t_1;
	} else {
		tmp = 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)
	t_2 = -1.0 + (0.5 * ((x / y) * (x / y)))
	tmp = 0
	if (x * x) <= 4e-289:
		tmp = t_2
	elif (x * x) <= 5e-242:
		tmp = t_1
	elif (x * x) <= 5e-151:
		tmp = t_2
	elif (x * x) <= 1e-121:
		tmp = t_1
	elif (x * x) <= 50000.0:
		tmp = t_2
	elif (x * x) <= 1.08e+284:
		tmp = t_1
	else:
		tmp = 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))
	t_2 = Float64(-1.0 + Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))))
	tmp = 0.0
	if (Float64(x * x) <= 4e-289)
		tmp = t_2;
	elseif (Float64(x * x) <= 5e-242)
		tmp = t_1;
	elseif (Float64(x * x) <= 5e-151)
		tmp = t_2;
	elseif (Float64(x * x) <= 1e-121)
		tmp = t_1;
	elseif (Float64(x * x) <= 50000.0)
		tmp = t_2;
	elseif (Float64(x * x) <= 1.08e+284)
		tmp = t_1;
	else
		tmp = 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);
	t_2 = -1.0 + (0.5 * ((x / y) * (x / y)));
	tmp = 0.0;
	if ((x * x) <= 4e-289)
		tmp = t_2;
	elseif ((x * x) <= 5e-242)
		tmp = t_1;
	elseif ((x * x) <= 5e-151)
		tmp = t_2;
	elseif ((x * x) <= 1e-121)
		tmp = t_1;
	elseif ((x * x) <= 50000.0)
		tmp = t_2;
	elseif ((x * x) <= 1.08e+284)
		tmp = t_1;
	else
		tmp = 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]}, Block[{t$95$2 = N[(-1.0 + N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 4e-289], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 5e-242], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 5e-151], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 1e-121], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 50000.0], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 1.08e+284], t$95$1, 1.0]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4e-289 or 4.9999999999999998e-242 < (*.f64 x x) < 5.00000000000000003e-151 or 9.9999999999999998e-122 < (*.f64 x x) < 5e4

   1. Initial program 59.0%

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

   3. Taylor expanded in x around 0 83.3%

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

      1. pow2 83.3%

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

      2. unpow2 83.3%

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

      3. times-frac 89.6%

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

   5. Applied egg-rr 89.6%

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

   if 4e-289 < (*.f64 x x) < 4.9999999999999998e-242 or 5.00000000000000003e-151 < (*.f64 x x) < 9.9999999999999998e-122 or 5e4 < (*.f64 x x) < 1.07999999999999993e284

   1. Initial program 80.5%

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

   if 1.07999999999999993e284 < (*.f64 x x)

   1. Initial program 2.4%

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

   3. Taylor expanded in x around inf 86.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-289}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-151}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{-121}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 50000:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 1.08 \cdot 10^{+284}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7200:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7200.0) (+ -1.0 (* 0.5 (* (/ x y) (/ x y)))) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7200.0) {
		tmp = -1.0 + (0.5 * ((x / y) * (x / y)));
	} 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 <= 7200.0d0) then
        tmp = (-1.0d0) + (0.5d0 * ((x / y) * (x / y)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7200.0) {
		tmp = -1.0 + (0.5 * ((x / y) * (x / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7200.0:
		tmp = -1.0 + (0.5 * ((x / y) * (x / y)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7200.0)
		tmp = Float64(-1.0 + Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7200.0)
		tmp = -1.0 + (0.5 * ((x / y) * (x / y)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7200.0], N[(-1.0 + N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 7200

   1. Initial program 52.8%

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

   3. Taylor expanded in x around 0 54.3%

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

      1. pow2 54.3%

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

      2. unpow2 54.3%

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

      3. times-frac 59.7%

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

   5. Applied egg-rr 59.7%

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

   if 7200 < x

   1. Initial program 26.2%

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

   3. Taylor expanded in x around inf 75.4%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7200:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5000:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 5000.0) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5000.0) {
		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 <= 5000.0d0) 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 <= 5000.0) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5000.0:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5000.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5000.0)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5000.0], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 52.8%

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

   3. Taylor expanded in x around 0 58.4%

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

   if 5e3 < x

   1. Initial program 26.2%

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

   3. Taylor expanded in x around inf 75.4%

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

Alternative 6: 50.4% 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 46.5%

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

3. Taylor expanded in x around 0 49.9%

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

Developer target: 50.9% 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 2024094 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

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