Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A

Percentage Accurate: 68.9% → 99.9%

Time: 10.3s

Alternatives: 11

Speedup: 1.2×

• 41.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{y + \left(z + x\right) \cdot \frac{x - z}{y}}{2} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ y (* (+ z x) (/ (- x z) y))) 2.0))

C

double code(double x, double y, double z) {
	return (y + ((z + x) * ((x - z) / y))) / 2.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y + ((z + x) * ((x - z) / y))) / 2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return (y + ((z + x) * ((x - z) / y))) / 2.0;
}

Python

def code(x, y, z):
	return (y + ((z + x) * ((x - z) / y))) / 2.0

Julia

function code(x, y, z)
	return Float64(Float64(y + Float64(Float64(z + x) * Float64(Float64(x - z) / y))) / 2.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y + ((z + x) * ((x - z) / y))) / 2.0;
end

Wolfram

code[x_, y_, z_] := N[(N[(y + N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 66.3%

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

   1. associate-/r* N/A

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

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

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

   5. associate-+l- N/A

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

   6. div-sub N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. *-inverses N/A

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

   10. *-lft-identity N/A

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

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

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

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

   15. *-lowering-*.f64 80.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

3. Simplified 80.4%

   \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. difference-of-squares N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

   6. --lowering--.f64 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 41.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-220}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot -0.5}{y}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-97}:\\ \;\;\;\;\frac{0.5}{\frac{y}{x \cdot x}}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.3e-220)
   (/ (* (* z z) -0.5) y)
   (if (<= y 1.02e-97)
     (/ 0.5 (/ y (* x x)))
     (if (<= y 3.5e+37)
       (* z (/ (* z -0.5) y))
       (if (<= y 9.5e+106) (/ (* x (/ x y)) 2.0) (/ y 2.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.3e-220) {
		tmp = ((z * z) * -0.5) / y;
	} else if (y <= 1.02e-97) {
		tmp = 0.5 / (y / (x * x));
	} else if (y <= 3.5e+37) {
		tmp = z * ((z * -0.5) / y);
	} else if (y <= 9.5e+106) {
		tmp = (x * (x / y)) / 2.0;
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.3d-220) then
        tmp = ((z * z) * (-0.5d0)) / y
    else if (y <= 1.02d-97) then
        tmp = 0.5d0 / (y / (x * x))
    else if (y <= 3.5d+37) then
        tmp = z * ((z * (-0.5d0)) / y)
    else if (y <= 9.5d+106) then
        tmp = (x * (x / y)) / 2.0d0
    else
        tmp = y / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.3e-220) {
		tmp = ((z * z) * -0.5) / y;
	} else if (y <= 1.02e-97) {
		tmp = 0.5 / (y / (x * x));
	} else if (y <= 3.5e+37) {
		tmp = z * ((z * -0.5) / y);
	} else if (y <= 9.5e+106) {
		tmp = (x * (x / y)) / 2.0;
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.3e-220:
		tmp = ((z * z) * -0.5) / y
	elif y <= 1.02e-97:
		tmp = 0.5 / (y / (x * x))
	elif y <= 3.5e+37:
		tmp = z * ((z * -0.5) / y)
	elif y <= 9.5e+106:
		tmp = (x * (x / y)) / 2.0
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.3e-220)
		tmp = Float64(Float64(Float64(z * z) * -0.5) / y);
	elseif (y <= 1.02e-97)
		tmp = Float64(0.5 / Float64(y / Float64(x * x)));
	elseif (y <= 3.5e+37)
		tmp = Float64(z * Float64(Float64(z * -0.5) / y));
	elseif (y <= 9.5e+106)
		tmp = Float64(Float64(x * Float64(x / y)) / 2.0);
	else
		tmp = Float64(y / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.3e-220)
		tmp = ((z * z) * -0.5) / y;
	elseif (y <= 1.02e-97)
		tmp = 0.5 / (y / (x * x));
	elseif (y <= 3.5e+37)
		tmp = z * ((z * -0.5) / y);
	elseif (y <= 9.5e+106)
		tmp = (x * (x / y)) / 2.0;
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.3e-220], N[(N[(N[(z * z), $MachinePrecision] * -0.5), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.02e-97], N[(0.5 / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e+37], N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e+106], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(y / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 4.29999999999999979e-220

   1. Initial program 72.7%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 84.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 84.0%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot -1\right) \cdot {z}^{2}}{y} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(-1 \cdot {z}^{2}\right)}{y} \]

      4. neg-mul-1 N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 37.0%

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

   7. Simplified 37.0%

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

   if 4.29999999999999979e-220 < y < 1.02000000000000004e-97

   1. Initial program 94.5%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 94.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2}\right), y\right), 2\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x\right), y\right), 2\right) \]

      3. *-lowering-*.f64 68.9%

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

   7. Simplified 68.9%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{1 \cdot \frac{1}{2}}{\frac{y}{x \cdot x}} \]

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

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

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

      10. *-lowering-*.f64 69.0%

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

   9. Applied egg-rr 69.0%

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

   if 1.02000000000000004e-97 < y < 3.5e37

   1. Initial program 93.2%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 93.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 93.2%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot -1\right) \cdot {z}^{2}}{y} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(-1 \cdot {z}^{2}\right)}{y} \]

      4. neg-mul-1 N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 47.4%

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

   7. Simplified 47.4%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{z \cdot \left(z \cdot \frac{-1}{2}\right)}{y} \]

      2. associate-/l* N/A

         \[\leadsto z \cdot \color{blue}{\frac{z \cdot \frac{-1}{2}}{y}} \]

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

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

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

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

      5. *-lowering-*.f64 47.6%

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

   9. Applied egg-rr 47.6%

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

   if 3.5e37 < y < 9.4999999999999995e106

   1. Initial program 78.5%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 78.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2}\right), y\right), 2\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x\right), y\right), 2\right) \]

      3. *-lowering-*.f64 51.6%

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

   7. Simplified 51.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 61.5%

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

   9. Applied egg-rr 61.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{2} \]

   if 9.4999999999999995e106 < y

   1. Initial program 22.2%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 59.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

   7. Simplified 74.6%

      \[\leadsto \frac{\color{blue}{y}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-220}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot -0.5}{y}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-97}:\\ \;\;\;\;\frac{0.5}{\frac{y}{x \cdot x}}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-220}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot -0.5}{y}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{0.5}{\frac{y}{x \cdot x}}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+53}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8e-220)
   (/ (* (* z z) -0.5) y)
   (if (<= y 8.5e-98)
     (/ 0.5 (/ y (* x x)))
     (if (<= y 5.8e+53) (* z (/ (* z -0.5) y)) (/ y 2.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8e-220) {
		tmp = ((z * z) * -0.5) / y;
	} else if (y <= 8.5e-98) {
		tmp = 0.5 / (y / (x * x));
	} else if (y <= 5.8e+53) {
		tmp = z * ((z * -0.5) / y);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 8d-220) then
        tmp = ((z * z) * (-0.5d0)) / y
    else if (y <= 8.5d-98) then
        tmp = 0.5d0 / (y / (x * x))
    else if (y <= 5.8d+53) then
        tmp = z * ((z * (-0.5d0)) / y)
    else
        tmp = y / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8e-220) {
		tmp = ((z * z) * -0.5) / y;
	} else if (y <= 8.5e-98) {
		tmp = 0.5 / (y / (x * x));
	} else if (y <= 5.8e+53) {
		tmp = z * ((z * -0.5) / y);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8e-220:
		tmp = ((z * z) * -0.5) / y
	elif y <= 8.5e-98:
		tmp = 0.5 / (y / (x * x))
	elif y <= 5.8e+53:
		tmp = z * ((z * -0.5) / y)
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8e-220)
		tmp = Float64(Float64(Float64(z * z) * -0.5) / y);
	elseif (y <= 8.5e-98)
		tmp = Float64(0.5 / Float64(y / Float64(x * x)));
	elseif (y <= 5.8e+53)
		tmp = Float64(z * Float64(Float64(z * -0.5) / y));
	else
		tmp = Float64(y / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8e-220)
		tmp = ((z * z) * -0.5) / y;
	elseif (y <= 8.5e-98)
		tmp = 0.5 / (y / (x * x));
	elseif (y <= 5.8e+53)
		tmp = z * ((z * -0.5) / y);
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8e-220], N[(N[(N[(z * z), $MachinePrecision] * -0.5), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 8.5e-98], N[(0.5 / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e+53], N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 7.99999999999999994e-220

   1. Initial program 72.7%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 84.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 84.0%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot -1\right) \cdot {z}^{2}}{y} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(-1 \cdot {z}^{2}\right)}{y} \]

      4. neg-mul-1 N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 37.0%

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

   7. Simplified 37.0%

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

   if 7.99999999999999994e-220 < y < 8.4999999999999997e-98

   1. Initial program 94.5%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 94.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2}\right), y\right), 2\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x\right), y\right), 2\right) \]

      3. *-lowering-*.f64 68.9%

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

   7. Simplified 68.9%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{1 \cdot \frac{1}{2}}{\frac{y}{x \cdot x}} \]

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

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

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

      10. *-lowering-*.f64 69.0%

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

   9. Applied egg-rr 69.0%

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

   if 8.4999999999999997e-98 < y < 5.8000000000000004e53

   1. Initial program 93.6%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 93.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot -1\right) \cdot {z}^{2}}{y} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(-1 \cdot {z}^{2}\right)}{y} \]

      4. neg-mul-1 N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 44.6%

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

   7. Simplified 44.6%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{z \cdot \left(z \cdot \frac{-1}{2}\right)}{y} \]

      2. associate-/l* N/A

         \[\leadsto z \cdot \color{blue}{\frac{z \cdot \frac{-1}{2}}{y}} \]

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

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

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

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

      5. *-lowering-*.f64 44.7%

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

   9. Applied egg-rr 44.7%

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

   if 5.8000000000000004e53 < y

   1. Initial program 27.9%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

   7. Simplified 68.2%

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

Alternative 4: 42.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.76 \cdot 10^{-217}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{-0.5}{y}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-98}:\\ \;\;\;\;\frac{0.5}{\frac{y}{x \cdot x}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+54}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.76e-217)
   (* (* z z) (/ -0.5 y))
   (if (<= y 9e-98)
     (/ 0.5 (/ y (* x x)))
     (if (<= y 1.35e+54) (* z (/ (* z -0.5) y)) (/ y 2.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.76e-217) {
		tmp = (z * z) * (-0.5 / y);
	} else if (y <= 9e-98) {
		tmp = 0.5 / (y / (x * x));
	} else if (y <= 1.35e+54) {
		tmp = z * ((z * -0.5) / y);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.76d-217) then
        tmp = (z * z) * ((-0.5d0) / y)
    else if (y <= 9d-98) then
        tmp = 0.5d0 / (y / (x * x))
    else if (y <= 1.35d+54) then
        tmp = z * ((z * (-0.5d0)) / y)
    else
        tmp = y / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.76e-217) {
		tmp = (z * z) * (-0.5 / y);
	} else if (y <= 9e-98) {
		tmp = 0.5 / (y / (x * x));
	} else if (y <= 1.35e+54) {
		tmp = z * ((z * -0.5) / y);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.76e-217:
		tmp = (z * z) * (-0.5 / y)
	elif y <= 9e-98:
		tmp = 0.5 / (y / (x * x))
	elif y <= 1.35e+54:
		tmp = z * ((z * -0.5) / y)
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.76e-217)
		tmp = Float64(Float64(z * z) * Float64(-0.5 / y));
	elseif (y <= 9e-98)
		tmp = Float64(0.5 / Float64(y / Float64(x * x)));
	elseif (y <= 1.35e+54)
		tmp = Float64(z * Float64(Float64(z * -0.5) / y));
	else
		tmp = Float64(y / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.76e-217)
		tmp = (z * z) * (-0.5 / y);
	elseif (y <= 9e-98)
		tmp = 0.5 / (y / (x * x));
	elseif (y <= 1.35e+54)
		tmp = z * ((z * -0.5) / y);
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.76e-217], N[(N[(z * z), $MachinePrecision] * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e-98], N[(0.5 / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+54], N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.76000000000000006e-217

   1. Initial program 72.7%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 84.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 84.0%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot -1\right) \cdot {z}^{2}}{y} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(-1 \cdot {z}^{2}\right)}{y} \]

      4. neg-mul-1 N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 37.0%

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

   7. Simplified 37.0%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(z \cdot z\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{y}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{y} \cdot \color{blue}{\left(z \cdot z\right)} \]

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

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

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

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

      5. *-lowering-*.f64 37.0%

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

   9. Applied egg-rr 37.0%

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

   if 1.76000000000000006e-217 < y < 8.99999999999999994e-98

   1. Initial program 94.5%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 94.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{2}\right), y\right), 2\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x\right), y\right), 2\right) \]

      3. *-lowering-*.f64 68.9%

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

   7. Simplified 68.9%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{1 \cdot \frac{1}{2}}{\frac{y}{x \cdot x}} \]

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

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

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

      10. *-lowering-*.f64 69.0%

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

   9. Applied egg-rr 69.0%

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

   if 8.99999999999999994e-98 < y < 1.35000000000000005e54

   1. Initial program 93.6%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 93.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot -1\right) \cdot {z}^{2}}{y} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(-1 \cdot {z}^{2}\right)}{y} \]

      4. neg-mul-1 N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 44.6%

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

   7. Simplified 44.6%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{z \cdot \left(z \cdot \frac{-1}{2}\right)}{y} \]

      2. associate-/l* N/A

         \[\leadsto z \cdot \color{blue}{\frac{z \cdot \frac{-1}{2}}{y}} \]

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

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

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

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

      5. *-lowering-*.f64 44.7%

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

   9. Applied egg-rr 44.7%

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

   if 1.35000000000000005e54 < y

   1. Initial program 27.9%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

   7. Simplified 68.2%

      \[\leadsto \frac{\color{blue}{y}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.76 \cdot 10^{-217}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{-0.5}{y}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-98}:\\ \;\;\;\;\frac{0.5}{\frac{y}{x \cdot x}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+54}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+30}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{x}{y} \cdot \left(x - z\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e+30)
   (/ (- y (* z (/ z y))) 2.0)
   (/ (+ y (* (/ x y) (- x z))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+30) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else {
		tmp = (y + ((x / y) * (x - z))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 1d+30) then
        tmp = (y - (z * (z / y))) / 2.0d0
    else
        tmp = (y + ((x / y) * (x - z))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+30) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else {
		tmp = (y + ((x / y) * (x - z))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e+30:
		tmp = (y - (z * (z / y))) / 2.0
	else:
		tmp = (y + ((x / y) * (x - z))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e+30)
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	else
		tmp = Float64(Float64(y + Float64(Float64(x / y) * Float64(x - z))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e+30)
		tmp = (y - (z * (z / y))) / 2.0;
	else
		tmp = (y + ((x / y) * (x - z))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+30], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y + N[(N[(x / y), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e30

   1. Initial program 67.7%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 86.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 86.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y - \frac{{z}^{2}}{y}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{{z}^{2}}{y}\right)\right), 2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left({z}^{2}\right), y\right)\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z\right), y\right)\right), 2\right) \]

      4. *-lowering-*.f64 79.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), y\right)\right), 2\right) \]

   7. Simplified 79.0%

      \[\leadsto \frac{\color{blue}{y - \frac{z \cdot z}{y}}}{2} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(z \cdot \frac{z}{y}\right)\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z}{y} \cdot z\right)\right), 2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{z}{y}\right), z\right)\right), 2\right) \]

      4. /-lowering-/.f64 90.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, y\right), z\right)\right), 2\right) \]

   9. Applied egg-rr 90.8%

      \[\leadsto \frac{y - \color{blue}{\frac{z}{y} \cdot z}}{2} \]

   if 1e30 < (*.f64 x x)

   1. Initial program 64.7%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 73.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-inv N/A

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

      2. difference-of-squares N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 89.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, x\right), \mathsf{/.f64}\left(x, y\right)\right)\right), 2\right) \]

   9. Simplified 89.2%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+30}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{x}{y} \cdot \left(x - z\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(z + x\right) \cdot \frac{x - z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.8e+75)
   (/ (* (+ z x) (/ (- x z) y)) 2.0)
   (/ (+ y (/ x (/ y x))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.8e+75) {
		tmp = ((z + x) * ((x - z) / y)) / 2.0;
	} else {
		tmp = (y + (x / (y / x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 7.8d+75) then
        tmp = ((z + x) * ((x - z) / y)) / 2.0d0
    else
        tmp = (y + (x / (y / x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.8e+75) {
		tmp = ((z + x) * ((x - z) / y)) / 2.0;
	} else {
		tmp = (y + (x / (y / x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.8e+75:
		tmp = ((z + x) * ((x - z) / y)) / 2.0
	else:
		tmp = (y + (x / (y / x))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.8e+75)
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y)) / 2.0);
	else
		tmp = Float64(Float64(y + Float64(x / Float64(y / x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7.8e+75)
		tmp = ((z + x) * ((x - z) / y)) / 2.0;
	else
		tmp = (y + (x / (y / x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.8e+75], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.80000000000000075e75

   1. Initial program 78.0%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 86.2%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. difference-of-squares N/A

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

      4. associate-/l* N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 78.1%

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

   7. Simplified 78.1%

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

   if 7.80000000000000075e75 < y

   1. Initial program 27.2%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 61.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 68.8%

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

   7. Simplified 68.8%

      \[\leadsto \frac{\color{blue}{y + \frac{x \cdot x}{y}}}{2} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 84.5%

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

   9. Applied egg-rr 84.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(z + x\right) \cdot \frac{x - z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+46}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e+46)
   (/ (- y (* z (/ z y))) 2.0)
   (/ (+ y (/ x (/ y x))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+46) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else {
		tmp = (y + (x / (y / x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 1d+46) then
        tmp = (y - (z * (z / y))) / 2.0d0
    else
        tmp = (y + (x / (y / x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+46) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else {
		tmp = (y + (x / (y / x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e+46:
		tmp = (y - (z * (z / y))) / 2.0
	else:
		tmp = (y + (x / (y / x))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e+46)
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	else
		tmp = Float64(Float64(y + Float64(x / Float64(y / x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e+46)
		tmp = (y - (z * (z / y))) / 2.0;
	else
		tmp = (y + (x / (y / x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+46], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.9999999999999999e45

   1. Initial program 68.6%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 87.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 87.0%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y - \frac{{z}^{2}}{y}\right)}, 2\right) \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{{z}^{2}}{y}\right)\right), 2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left({z}^{2}\right), y\right)\right), 2\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z\right), y\right)\right), 2\right) \]

      4. *-lowering-*.f64 78.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), y\right)\right), 2\right) \]

   7. Simplified 78.9%

      \[\leadsto \frac{\color{blue}{y - \frac{z \cdot z}{y}}}{2} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(z \cdot \frac{z}{y}\right)\right), 2\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{z}{y} \cdot z\right)\right), 2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{z}{y}\right), z\right)\right), 2\right) \]

      4. /-lowering-/.f64 90.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, y\right), z\right)\right), 2\right) \]

   9. Applied egg-rr 90.4%

      \[\leadsto \frac{y - \color{blue}{\frac{z}{y} \cdot z}}{2} \]

   if 9.9999999999999999e45 < (*.f64 x x)

   1. Initial program 63.4%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 72.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 72.3%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 72.4%

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

   7. Simplified 72.4%

      \[\leadsto \frac{\color{blue}{y + \frac{x \cdot x}{y}}}{2} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 86.0%

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

   9. Applied egg-rr 86.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+46}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.5 \cdot 10^{+237}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1.5e+237) (/ (+ y (/ x (/ y x))) 2.0) (* z (/ (* z -0.5) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1.5e+237) {
		tmp = (y + (x / (y / x))) / 2.0;
	} else {
		tmp = z * ((z * -0.5) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 1.5d+237) then
        tmp = (y + (x / (y / x))) / 2.0d0
    else
        tmp = z * ((z * (-0.5d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1.5e+237) {
		tmp = (y + (x / (y / x))) / 2.0;
	} else {
		tmp = z * ((z * -0.5) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1.5e+237:
		tmp = (y + (x / (y / x))) / 2.0
	else:
		tmp = z * ((z * -0.5) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1.5e+237)
		tmp = Float64(Float64(y + Float64(x / Float64(y / x))) / 2.0);
	else
		tmp = Float64(z * Float64(Float64(z * -0.5) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1.5e+237)
		tmp = (y + (x / (y / x))) / 2.0;
	else
		tmp = z * ((z * -0.5) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1.5e+237], N[(N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.5e237

   1. Initial program 74.0%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 91.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 91.7%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 74.4%

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

   7. Simplified 74.4%

      \[\leadsto \frac{\color{blue}{y + \frac{x \cdot x}{y}}}{2} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 82.5%

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

   9. Applied egg-rr 82.5%

      \[\leadsto \frac{y + \color{blue}{\frac{x}{\frac{y}{x}}}}{2} \]

   if 1.5e237 < (*.f64 z z)

   1. Initial program 47.9%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 53.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot -1\right) \cdot {z}^{2}}{y} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(-1 \cdot {z}^{2}\right)}{y} \]

      4. neg-mul-1 N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 57.7%

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

   7. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{z \cdot \left(z \cdot \frac{-1}{2}\right)}{y} \]

      2. associate-/l* N/A

         \[\leadsto z \cdot \color{blue}{\frac{z \cdot \frac{-1}{2}}{y}} \]

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

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

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

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

      5. *-lowering-*.f64 70.1%

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

   9. Applied egg-rr 70.1%

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

Alternative 9: 44.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{+51}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.32e+51) (* z (/ (* z -0.5) y)) (/ y 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.32e+51) {
		tmp = z * ((z * -0.5) / y);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.32d+51) then
        tmp = z * ((z * (-0.5d0)) / y)
    else
        tmp = y / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.32e+51) {
		tmp = z * ((z * -0.5) / y);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.32e+51:
		tmp = z * ((z * -0.5) / y)
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.32e+51)
		tmp = Float64(z * Float64(Float64(z * -0.5) / y));
	else
		tmp = Float64(y / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.32e+51)
		tmp = z * ((z * -0.5) / y);
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.32e+51], N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.32e51

   1. Initial program 78.3%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 86.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 86.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot -1\right) \cdot {z}^{2}}{y} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(-1 \cdot {z}^{2}\right)}{y} \]

      4. neg-mul-1 N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 37.8%

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

   7. Simplified 37.8%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{z \cdot \left(z \cdot \frac{-1}{2}\right)}{y} \]

      2. associate-/l* N/A

         \[\leadsto z \cdot \color{blue}{\frac{z \cdot \frac{-1}{2}}{y}} \]

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

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

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

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

      5. *-lowering-*.f64 39.7%

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

   9. Applied egg-rr 39.7%

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

   if 1.32e51 < y

   1. Initial program 27.9%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

   7. Simplified 68.2%

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

Alternative 10: 44.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+51}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3e+51) (* z (* z (/ -0.5 y))) (/ y 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e+51) {
		tmp = z * (z * (-0.5 / y));
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3d+51) then
        tmp = z * (z * ((-0.5d0) / y))
    else
        tmp = y / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e+51) {
		tmp = z * (z * (-0.5 / y));
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3e+51:
		tmp = z * (z * (-0.5 / y))
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3e+51)
		tmp = Float64(z * Float64(z * Float64(-0.5 / y)));
	else
		tmp = Float64(y / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3e+51)
		tmp = z * (z * (-0.5 / y));
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3e+51], N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3e51

   1. Initial program 78.3%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 86.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 86.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot -1\right) \cdot {z}^{2}}{y} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(-1 \cdot {z}^{2}\right)}{y} \]

      4. neg-mul-1 N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 37.8%

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

   7. Simplified 37.8%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(z \cdot z\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{y}} \]

      2. associate-*l* N/A

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \frac{\frac{-1}{2}}{y}\right)} \]

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

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

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

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

      5. /-lowering-/.f64 39.6%

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

   9. Applied egg-rr 39.6%

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

   if 3e51 < y

   1. Initial program 27.9%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

      15. *-lowering-*.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

   7. Simplified 68.2%

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

Alternative 11: 34.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{y}{2} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ y 2.0))

C

double code(double x, double y, double z) {
	return y / 2.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y / 2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return y / 2.0;
}

Python

def code(x, y, z):
	return y / 2.0

Julia

function code(x, y, z)
	return Float64(y / 2.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = y / 2.0;
end

Wolfram

code[x_, y_, z_] := N[(y / 2.0), $MachinePrecision]

Derivation

1. Initial program 66.3%

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

   1. associate-/r* N/A

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

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

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

   5. associate-+l- N/A

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

   6. div-sub N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. *-inverses N/A

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

   10. *-lft-identity N/A

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

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

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

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot z - x \cdot x\right), y\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(x \cdot x\right)\right), y\right)\right), 2\right) \]

   15. *-lowering-*.f64 80.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(x, x\right)\right), y\right)\right), 2\right) \]

3. Simplified 80.4%

   \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
4. Add Preprocessing

5. Taylor expanded in y around inf

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

7. Simplified 33.9%

   \[\leadsto \frac{\color{blue}{y}}{2} \]
8. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x))))

C

double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

public static double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Python

def code(x, y, z):
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x))

Julia

function code(x, y, z)
	return Float64(Float64(y * 0.5) - Float64(Float64(Float64(0.5 / y) * Float64(z + x)) * Float64(z - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] - N[(N[(N[(0.5 / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024155 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* y 1/2) (* (* (/ 1/2 y) (+ z x)) (- z x))))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))