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

Percentage Accurate: 50.5% → 79.5%

Time: 5.9s

Alternatives: 5

Speedup: 3.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.5% 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: 79.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-230}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 10^{+47}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + 4 \cdot \left(y \cdot y\right)}{x \cdot x + \left(y \cdot y\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 5e-230)
     1.0
     (if (<= t_0 1e+47)
       (/ 1.0 (/ (+ (* x x) (* 4.0 (* y y))) (+ (* x x) (* (* y y) -4.0))))
       (+ -1.0 (* 0.5 (/ (* x (/ x y)) y)))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 5e-230) {
		tmp = 1.0;
	} else if (t_0 <= 1e+47) {
		tmp = 1.0 / (((x * x) + (4.0 * (y * y))) / ((x * x) + ((y * y) * -4.0)));
	} else {
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y));
	}
	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) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 5d-230) then
        tmp = 1.0d0
    else if (t_0 <= 1d+47) then
        tmp = 1.0d0 / (((x * x) + (4.0d0 * (y * y))) / ((x * x) + ((y * y) * (-4.0d0))))
    else
        tmp = (-1.0d0) + (0.5d0 * ((x * (x / y)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 5e-230) {
		tmp = 1.0;
	} else if (t_0 <= 1e+47) {
		tmp = 1.0 / (((x * x) + (4.0 * (y * y))) / ((x * x) + ((y * y) * -4.0)));
	} else {
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 5e-230:
		tmp = 1.0
	elif t_0 <= 1e+47:
		tmp = 1.0 / (((x * x) + (4.0 * (y * y))) / ((x * x) + ((y * y) * -4.0)))
	else:
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 5e-230)
		tmp = 1.0;
	elseif (t_0 <= 1e+47)
		tmp = Float64(1.0 / Float64(Float64(Float64(x * x) + Float64(4.0 * Float64(y * y))) / Float64(Float64(x * x) + Float64(Float64(y * y) * -4.0))));
	else
		tmp = Float64(-1.0 + Float64(0.5 * Float64(Float64(x * Float64(x / y)) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 5e-230)
		tmp = 1.0;
	elseif (t_0 <= 1e+47)
		tmp = 1.0 / (((x * x) + (4.0 * (y * y))) / ((x * x) + ((y * y) * -4.0)));
	else
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-230], 1.0, If[LessEqual[t$95$0, 1e+47], N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] + N[(4.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(0.5 * N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 5.00000000000000035e-230

   1. Initial program 50.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 85.8%

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

   if 5.00000000000000035e-230 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1e47

   1. Initial program 80.6%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x - \left(y \cdot 4\right) \cdot y}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot x + \left(y \cdot 4\right) \cdot y\right), \color{blue}{\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), \left(\left(y \cdot 4\right) \cdot y\right)\right), \left(\color{blue}{x \cdot x} - \left(y \cdot 4\right) \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot 4\right) \cdot y\right)\right), \left(\color{blue}{x} \cdot x - \left(y \cdot 4\right) \cdot y\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(4 \cdot y\right) \cdot y\right)\right), \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(4 \cdot \left(y \cdot y\right)\right)\right), \left(x \cdot \color{blue}{x} - \left(y \cdot 4\right) \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(4, \left(y \cdot y\right)\right)\right), \left(x \cdot \color{blue}{x} - \left(y \cdot 4\right) \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, y\right)\right)\right), \left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, y\right)\right)\right), \left(x \cdot x + \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 4\right) \cdot y\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 4\right) \cdot y\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot 4\right) \cdot y}\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\mathsf{neg}\left(y \cdot \left(y \cdot 4\right)\right)\right)\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\mathsf{neg}\left(\left(y \cdot y\right) \cdot 4\right)\right)\right)\right)\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot y\right) \cdot \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(y \cdot y\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\mathsf{neg}\left(\color{blue}{4}\right)\right)\right)\right)\right)\right) \]

      18. metadata-eval 80.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), -4\right)\right)\right)\right) \]

   4. Applied egg-rr 80.6%

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

   if 1e47 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{{y}^{2}} \cdot {x}^{2}\right)}\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{1 \cdot {x}^{2}}{\color{blue}{{y}^{2}}}\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{{x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{{x}^{2}}{{y}^{2}}\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{x}^{2}}{y \cdot \color{blue}{y}}\right)\right)\right) \]

      16. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\frac{{x}^{2}}{y}}{\color{blue}{y}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{{x}^{2}}{y}\right), \color{blue}{y}\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2}\right), y\right), y\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x\right), y\right), y\right)\right)\right) \]

      20. *-lowering-*.f64 78.8%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), y\right), y\right)\right)\right) \]

   5. Simplified 78.8%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(x \cdot \frac{x}{y}\right), y\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{x}{y} \cdot x\right), y\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{x}{y}\right), x\right), y\right)\right)\right) \]

      4. /-lowering-/.f64 87.3%

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), x\right), y\right)\right)\right) \]

   7. Applied egg-rr 87.3%

      \[\leadsto -1 + 0.5 \cdot \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{-230}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+47}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + 4 \cdot \left(y \cdot y\right)}{x \cdot x + \left(y \cdot y\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-230}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 10^{+47}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{t\_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 5e-230)
     1.0
     (if (<= t_0 1e+47)
       (/ (- (* x x) t_0) (+ t_0 (* x x)))
       (+ -1.0 (* 0.5 (/ (* x (/ x y)) y)))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 5e-230) {
		tmp = 1.0;
	} else if (t_0 <= 1e+47) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y));
	}
	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) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 5d-230) then
        tmp = 1.0d0
    else if (t_0 <= 1d+47) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    else
        tmp = (-1.0d0) + (0.5d0 * ((x * (x / y)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 5e-230) {
		tmp = 1.0;
	} else if (t_0 <= 1e+47) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 5e-230:
		tmp = 1.0
	elif t_0 <= 1e+47:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 5e-230)
		tmp = 1.0;
	elseif (t_0 <= 1e+47)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(-1.0 + Float64(0.5 * Float64(Float64(x * Float64(x / y)) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 5e-230)
		tmp = 1.0;
	elseif (t_0 <= 1e+47)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	else
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-230], 1.0, If[LessEqual[t$95$0, 1e+47], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(0.5 * N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 5.00000000000000035e-230

   1. Initial program 50.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 85.8%

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

   if 5.00000000000000035e-230 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1e47

   1. Initial program 80.6%

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

   if 1e47 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{{y}^{2}} \cdot {x}^{2}\right)}\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{1 \cdot {x}^{2}}{\color{blue}{{y}^{2}}}\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{{x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{{x}^{2}}{{y}^{2}}\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{x}^{2}}{y \cdot \color{blue}{y}}\right)\right)\right) \]

      16. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\frac{{x}^{2}}{y}}{\color{blue}{y}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{{x}^{2}}{y}\right), \color{blue}{y}\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2}\right), y\right), y\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x\right), y\right), y\right)\right)\right) \]

      20. *-lowering-*.f64 78.8%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), y\right), y\right)\right)\right) \]

   5. Simplified 78.8%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(x \cdot \frac{x}{y}\right), y\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{x}{y} \cdot x\right), y\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{x}{y}\right), x\right), y\right)\right)\right) \]

      4. /-lowering-/.f64 87.3%

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), x\right), y\right)\right)\right) \]

   7. Applied egg-rr 87.3%

      \[\leadsto -1 + 0.5 \cdot \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{-230}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+47}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9e+24) 1.0 (+ -1.0 (* 0.5 (/ (* x (/ x y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9e+24) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9d+24) then
        tmp = 1.0d0
    else
        tmp = (-1.0d0) + (0.5d0 * ((x * (x / y)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9e+24) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9e+24:
		tmp = 1.0
	else:
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9e+24)
		tmp = 1.0;
	else
		tmp = Float64(-1.0 + Float64(0.5 * Float64(Float64(x * Float64(x / y)) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e+24)
		tmp = 1.0;
	else
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9e+24], 1.0, N[(-1.0 + N[(0.5 * N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.00000000000000039e24

   1. Initial program 57.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 57.6%

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

   if 9.00000000000000039e24 < y

   1. Initial program 32.7%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(\color{blue}{{x}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\left(\frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{{y}^{2}} \cdot {x}^{2}\right)}\right)\right) \]

      12. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{1 \cdot {x}^{2}}{\color{blue}{{y}^{2}}}\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{{x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{{x}^{2}}{{y}^{2}}\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{{x}^{2}}{y \cdot \color{blue}{y}}\right)\right)\right) \]

      16. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\frac{{x}^{2}}{y}}{\color{blue}{y}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{{x}^{2}}{y}\right), \color{blue}{y}\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2}\right), y\right), y\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x\right), y\right), y\right)\right)\right) \]

      20. *-lowering-*.f64 74.0%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), y\right), y\right)\right)\right) \]

   5. Simplified 74.0%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(x \cdot \frac{x}{y}\right), y\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{x}{y} \cdot x\right), y\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{x}{y}\right), x\right), y\right)\right)\right) \]

      4. /-lowering-/.f64 84.3%

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), x\right), y\right)\right)\right) \]

   7. Applied egg-rr 84.3%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+31}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1e+31) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+31) {
		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 <= 1d+31) 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 <= 1e+31) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+31:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+31)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e+31)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+31], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999996e30

   1. Initial program 57.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 57.6%

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

   if 9.9999999999999996e30 < y

   1. Initial program 32.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 83.4%

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

Alternative 5: 50.5% 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

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

5. Simplified 50.8%

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

Developer Target 1: 51.0% 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 2024158 
(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))))