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

Percentage Accurate: 50.7% → 81.1%

Time: 6.4s

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: 50.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 10^{-315}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{1}{\frac{t\_0 + x \cdot x}{x \cdot x + y \cdot \left(y \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot 0.5}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 1e-315)
     1.0
     (if (<= t_0 5e+306)
       (/ 1.0 (/ (+ t_0 (* x x)) (+ (* x x) (* y (* y -4.0)))))
       (+ -1.0 (* (/ (* x 0.5) y) (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-315) {
		tmp = 1.0;
	} else if (t_0 <= 5e+306) {
		tmp = 1.0 / ((t_0 + (x * x)) / ((x * x) + (y * (y * -4.0))));
	} else {
		tmp = -1.0 + (((x * 0.5) / y) * (x / 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 <= 1d-315) then
        tmp = 1.0d0
    else if (t_0 <= 5d+306) then
        tmp = 1.0d0 / ((t_0 + (x * x)) / ((x * x) + (y * (y * (-4.0d0)))))
    else
        tmp = (-1.0d0) + (((x * 0.5d0) / y) * (x / 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 <= 1e-315) {
		tmp = 1.0;
	} else if (t_0 <= 5e+306) {
		tmp = 1.0 / ((t_0 + (x * x)) / ((x * x) + (y * (y * -4.0))));
	} else {
		tmp = -1.0 + (((x * 0.5) / y) * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 1e-315:
		tmp = 1.0
	elif t_0 <= 5e+306:
		tmp = 1.0 / ((t_0 + (x * x)) / ((x * x) + (y * (y * -4.0))))
	else:
		tmp = -1.0 + (((x * 0.5) / y) * (x / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 1e-315)
		tmp = 1.0;
	elseif (t_0 <= 5e+306)
		tmp = Float64(1.0 / Float64(Float64(t_0 + Float64(x * x)) / Float64(Float64(x * x) + Float64(y * Float64(y * -4.0)))));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(x * 0.5) / y) * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 1e-315)
		tmp = 1.0;
	elseif (t_0 <= 5e+306)
		tmp = 1.0 / ((t_0 + (x * x)) / ((x * x) + (y * (y * -4.0))));
	else
		tmp = -1.0 + (((x * 0.5) / y) * (x / 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, 1e-315], 1.0, If[LessEqual[t$95$0, 5e+306], N[(1.0 / N[(N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 9.999999985e-316

   1. Initial program 56.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 89.9%

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

   if 9.999999985e-316 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999993e306

   1. Initial program 78.5%

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

      1. 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

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

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

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

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

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

      9. sub-neg N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

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

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

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

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

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

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

      17. metadata-eval 78.6%

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

   4. Applied egg-rr 78.6%

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

   if 4.99999999999999993e306 < (*.f64 (*.f64 y #s(literal 4 binary64)) 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

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   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. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)}{{y}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\frac{1}{4} \cdot {x}^{2} - \frac{-1}{4} \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot {x}^{2} - \frac{-1}{4} \cdot {x}^{2}\right), \color{blue}{\left({y}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)\right), \left({\color{blue}{y}}^{2}\right)\right)\right) \]

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

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

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

      20. unpow2 N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 78.9%

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

   5. Simplified 78.9%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. /-lowering-/.f64 92.6%

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

   7. Applied egg-rr 92.6%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-315}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y \cdot 4\right) + x \cdot x}{x \cdot x + y \cdot \left(y \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot 0.5}{y} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 10^{-315}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{t\_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot 0.5}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 1e-315)
     1.0
     (if (<= t_0 5e+306)
       (/ (- (* x x) t_0) (+ t_0 (* x x)))
       (+ -1.0 (* (/ (* x 0.5) y) (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-315) {
		tmp = 1.0;
	} else if (t_0 <= 5e+306) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + (((x * 0.5) / y) * (x / 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 <= 1d-315) then
        tmp = 1.0d0
    else if (t_0 <= 5d+306) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    else
        tmp = (-1.0d0) + (((x * 0.5d0) / y) * (x / 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 <= 1e-315) {
		tmp = 1.0;
	} else if (t_0 <= 5e+306) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + (((x * 0.5) / y) * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 1e-315:
		tmp = 1.0
	elif t_0 <= 5e+306:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = -1.0 + (((x * 0.5) / y) * (x / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 1e-315)
		tmp = 1.0;
	elseif (t_0 <= 5e+306)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(x * 0.5) / y) * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 1e-315)
		tmp = 1.0;
	elseif (t_0 <= 5e+306)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	else
		tmp = -1.0 + (((x * 0.5) / y) * (x / 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, 1e-315], 1.0, If[LessEqual[t$95$0, 5e+306], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 9.999999985e-316

   1. Initial program 56.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 89.9%

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

   if 9.999999985e-316 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999993e306

   1. Initial program 78.5%

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

   if 4.99999999999999993e306 < (*.f64 (*.f64 y #s(literal 4 binary64)) 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

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   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. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)}{{y}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\frac{1}{4} \cdot {x}^{2} - \frac{-1}{4} \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot {x}^{2} - \frac{-1}{4} \cdot {x}^{2}\right), \color{blue}{\left({y}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)\right), \left({\color{blue}{y}}^{2}\right)\right)\right) \]

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

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

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

      20. unpow2 N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 78.9%

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

   5. Simplified 78.9%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. /-lowering-/.f64 92.6%

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

   7. Applied egg-rr 92.6%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-315}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot 0.5}{y} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-81}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x \cdot 0.5}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2e-81) 1.0 (+ -1.0 (* (/ (* x 0.5) y) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e-81) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + (((x * 0.5) / y) * (x / 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 <= 2d-81) then
        tmp = 1.0d0
    else
        tmp = (-1.0d0) + (((x * 0.5d0) / y) * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2e-81) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + (((x * 0.5) / y) * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e-81:
		tmp = 1.0
	else:
		tmp = -1.0 + (((x * 0.5) / y) * (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e-81)
		tmp = 1.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(x * 0.5) / y) * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e-81)
		tmp = 1.0;
	else
		tmp = -1.0 + (((x * 0.5) / y) * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e-81], 1.0, N[(-1.0 + N[(N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e-81

   1. Initial program 58.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 60.8%

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

   if 1.9999999999999999e-81 < y

   1. Initial program 48.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   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. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)}{{y}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\frac{1}{4} \cdot {x}^{2} - \frac{-1}{4} \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot {x}^{2} - \frac{-1}{4} \cdot {x}^{2}\right), \color{blue}{\left({y}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)\right), \left({\color{blue}{y}}^{2}\right)\right)\right) \]

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

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

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

      20. unpow2 N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 66.0%

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

   5. Simplified 66.0%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. /-lowering-/.f64 69.4%

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

   7. Applied egg-rr 69.4%

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

Alternative 4: 61.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3e-62) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3e-62) {
		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 <= 3d-62) 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 <= 3e-62) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3e-62:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3e-62)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e-62)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3e-62], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 3.0000000000000001e-62

   1. Initial program 58.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 60.3%

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

   if 3.0000000000000001e-62 < y

   1. Initial program 47.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.6%

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

Alternative 5: 50.3% 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 54.7%

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

3. Taylor expanded in x around 0

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

5. Simplified 49.7%

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

Developer Target 1: 51.2% 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 2024150 
(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))))