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

Percentage Accurate: 50.5% → 81.3%

Time: 8.7s

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: 50.5% 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: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-305}:\\ \;\;\;\;1 - \frac{\left(y \cdot 4\right) \cdot \frac{y}{x}}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+208}:\\ \;\;\;\;\frac{\frac{1}{x \cdot x + 4 \cdot \left(y \cdot y\right)}}{\frac{1}{x \cdot x + \left(y \cdot y\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 2e-305)
     (- 1.0 (/ (* (* y 4.0) (/ y x)) x))
     (if (<= t_0 2e+208)
       (/
        (/ 1.0 (+ (* x x) (* 4.0 (* y y))))
        (/ 1.0 (+ (* 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 <= 2e-305) {
		tmp = 1.0 - (((y * 4.0) * (y / x)) / x);
	} else if (t_0 <= 2e+208) {
		tmp = (1.0 / ((x * x) + (4.0 * (y * y)))) / (1.0 / ((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 <= 2d-305) then
        tmp = 1.0d0 - (((y * 4.0d0) * (y / x)) / x)
    else if (t_0 <= 2d+208) then
        tmp = (1.0d0 / ((x * x) + (4.0d0 * (y * y)))) / (1.0d0 / ((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 <= 2e-305) {
		tmp = 1.0 - (((y * 4.0) * (y / x)) / x);
	} else if (t_0 <= 2e+208) {
		tmp = (1.0 / ((x * x) + (4.0 * (y * y)))) / (1.0 / ((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 <= 2e-305:
		tmp = 1.0 - (((y * 4.0) * (y / x)) / x)
	elif t_0 <= 2e+208:
		tmp = (1.0 / ((x * x) + (4.0 * (y * y)))) / (1.0 / ((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 <= 2e-305)
		tmp = Float64(1.0 - Float64(Float64(Float64(y * 4.0) * Float64(y / x)) / x));
	elseif (t_0 <= 2e+208)
		tmp = Float64(Float64(1.0 / Float64(Float64(x * x) + Float64(4.0 * Float64(y * y)))) / Float64(1.0 / 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 <= 2e-305)
		tmp = 1.0 - (((y * 4.0) * (y / x)) / x);
	elseif (t_0 <= 2e+208)
		tmp = (1.0 / ((x * x) + (4.0 * (y * y)))) / (1.0 / ((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, 2e-305], N[(1.0 - N[(N[(N[(y * 4.0), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+208], N[(N[(1.0 / N[(N[(x * x), $MachinePrecision] + N[(4.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / 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) < 1.99999999999999999e-305

   1. Initial program 61.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), y\right)\right), \color{blue}{\left({x}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 60.0%

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

   5. Simplified 60.0%

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

      1. sub-div N/A

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

      2. *-inverses N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 77.4%

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

   7. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{1 - \frac{y \cdot \left(y \cdot 4\right)}{x \cdot x}} \]
   8. Step-by-step derivation

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 88.8%

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

   9. Applied egg-rr 88.8%

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

   if 1.99999999999999999e-305 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e208

   1. Initial program 79.6%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

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

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

   4. Applied egg-rr 79.6%

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

   if 2e208 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 23.1%

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

   3. Taylor expanded in x around 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 74.9%

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

   5. Simplified 74.9%

      \[\leadsto \color{blue}{-1 + 0.5 \cdot \frac{\frac{x \cdot x}{y}}{y}} \]
   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 85.0%

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

   7. Applied egg-rr 85.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-305}:\\ \;\;\;\;1 - \frac{\left(y \cdot 4\right) \cdot \frac{y}{x}}{x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+208}:\\ \;\;\;\;\frac{\frac{1}{x \cdot x + 4 \cdot \left(y \cdot y\right)}}{\frac{1}{x \cdot x + \left(y \cdot y\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-305}:\\ \;\;\;\;1 - \frac{\left(y \cdot 4\right) \cdot \frac{y}{x}}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+208}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{t\_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 2e-305)
     (- 1.0 (/ (* (* y 4.0) (/ y x)) x))
     (if (<= t_0 2e+208)
       (/ (- (* 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 <= 2e-305) {
		tmp = 1.0 - (((y * 4.0) * (y / x)) / x);
	} else if (t_0 <= 2e+208) {
		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 <= 2d-305) then
        tmp = 1.0d0 - (((y * 4.0d0) * (y / x)) / x)
    else if (t_0 <= 2d+208) 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 <= 2e-305) {
		tmp = 1.0 - (((y * 4.0) * (y / x)) / x);
	} else if (t_0 <= 2e+208) {
		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 <= 2e-305:
		tmp = 1.0 - (((y * 4.0) * (y / x)) / x)
	elif t_0 <= 2e+208:
		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 <= 2e-305)
		tmp = Float64(1.0 - Float64(Float64(Float64(y * 4.0) * Float64(y / x)) / x));
	elseif (t_0 <= 2e+208)
		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 <= 2e-305)
		tmp = 1.0 - (((y * 4.0) * (y / x)) / x);
	elseif (t_0 <= 2e+208)
		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, 2e-305], N[(1.0 - N[(N[(N[(y * 4.0), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+208], 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) < 1.99999999999999999e-305

   1. Initial program 61.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 4\right), y\right)\right), \color{blue}{\left({x}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 60.0%

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

   5. Simplified 60.0%

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

      1. sub-div N/A

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

      2. *-inverses N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 77.4%

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

   7. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{1 - \frac{y \cdot \left(y \cdot 4\right)}{x \cdot x}} \]
   8. Step-by-step derivation

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 88.8%

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

   9. Applied egg-rr 88.8%

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

   if 1.99999999999999999e-305 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e208

   1. Initial program 79.6%

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

   if 2e208 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 23.1%

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

   3. Taylor expanded in x around 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 74.9%

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

   5. Simplified 74.9%

      \[\leadsto \color{blue}{-1 + 0.5 \cdot \frac{\frac{x \cdot x}{y}}{y}} \]
   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 85.0%

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

   7. Applied egg-rr 85.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-305}:\\ \;\;\;\;1 - \frac{\left(y \cdot 4\right) \cdot \frac{y}{x}}{x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+208}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{-24}:\\ \;\;\;\;1 + \frac{y}{x} \cdot \frac{y \cdot -8}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* y 4.0)) 5e-24)
   (+ 1.0 (* (/ y x) (/ (* y -8.0) x)))
   (+ -1.0 (* 0.5 (/ (* x (/ x y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (y * 4.0)) <= 5e-24) {
		tmp = 1.0 + ((y / x) * ((y * -8.0) / 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) :: tmp
    if ((y * (y * 4.0d0)) <= 5d-24) then
        tmp = 1.0d0 + ((y / x) * ((y * (-8.0d0)) / 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 tmp;
	if ((y * (y * 4.0)) <= 5e-24) {
		tmp = 1.0 + ((y / x) * ((y * -8.0) / x));
	} else {
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (y * 4.0)) <= 5e-24:
		tmp = 1.0 + ((y / x) * ((y * -8.0) / x))
	else:
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(y * 4.0)) <= 5e-24)
		tmp = Float64(1.0 + Float64(Float64(y / x) * Float64(Float64(y * -8.0) / 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)
	tmp = 0.0;
	if ((y * (y * 4.0)) <= 5e-24)
		tmp = 1.0 + ((y / x) * ((y * -8.0) / x));
	else
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 5e-24], N[(1.0 + N[(N[(y / x), $MachinePrecision] * N[(N[(y * -8.0), $MachinePrecision] / 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 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.9999999999999998e-24

   1. Initial program 66.9%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

         \[\leadsto 1 + \frac{{y}^{2}}{{x}^{2}} \cdot -8 \]

      4. *-commutative N/A

         \[\leadsto 1 + -8 \cdot \color{blue}{\frac{{y}^{2}}{{x}^{2}}} \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(-8 \cdot \frac{{y}^{2}}{{x}^{2}}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{{y}^{2}}{{x}^{2}} \cdot \color{blue}{-8}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{{y}^{2} \cdot -8}{\color{blue}{{x}^{2}}}\right)\right) \]

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left({y}^{2} \cdot -8\right), \left({x}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), -8\right), \left({\color{blue}{x}}^{2}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), -8\right), \left({x}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), -8\right), \left({x}^{2}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), -8\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      17. *-lowering-*.f64 71.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), -8\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{1 + \frac{\left(y \cdot y\right) \cdot -8}{x \cdot x}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{y \cdot \left(y \cdot -8\right)}{\color{blue}{x} \cdot x}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{y}{x} \cdot \color{blue}{\frac{y \cdot -8}{x}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(\frac{y \cdot -8}{x}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\frac{\color{blue}{y \cdot -8}}{x}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{/.f64}\left(\left(y \cdot -8\right), \color{blue}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 78.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, -8\right), x\right)\right)\right) \]

   7. Applied egg-rr 78.3%

      \[\leadsto 1 + \color{blue}{\frac{y}{x} \cdot \frac{y \cdot -8}{x}} \]

   if 4.9999999999999998e-24 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 46.8%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   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 73.4%

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

   5. Simplified 73.4%

      \[\leadsto \color{blue}{-1 + 0.5 \cdot \frac{\frac{x \cdot x}{y}}{y}} \]
   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 79.6%

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

   7. Applied egg-rr 79.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{-24}:\\ \;\;\;\;1 + \frac{y}{x} \cdot \frac{y \cdot -8}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.4e-18) 1.0 (+ -1.0 (* 0.5 (/ (* x (/ x y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.4e-18) {
		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 <= 5.4d-18) 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 <= 5.4e-18) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + (0.5 * ((x * (x / y)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.4e-18:
		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 <= 5.4e-18)
		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 <= 5.4e-18)
		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, 5.4e-18], 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 < 5.39999999999999977e-18

   1. Initial program 62.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 57.5%

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

   if 5.39999999999999977e-18 < y

   1. Initial program 42.6%

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

   3. Taylor expanded in x around 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 70.2%

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

   5. Simplified 70.2%

      \[\leadsto \color{blue}{-1 + 0.5 \cdot \frac{\frac{x \cdot x}{y}}{y}} \]
   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 73.6%

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

   7. Applied egg-rr 73.6%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4e-18) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e-18) {
		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 <= 4d-18) 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 <= 4e-18) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e-18:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e-18)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e-18)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e-18], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 4.0000000000000003e-18

   1. Initial program 62.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 57.5%

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

   if 4.0000000000000003e-18 < y

   1. Initial program 42.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.5%

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

Alternative 6: 50.6% 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 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 0

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

5. Simplified 50.6%

   \[\leadsto \color{blue}{-1} \]
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 2024163 
(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))))