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

Percentage Accurate: 69.9% → 99.9%

Time: 10.9s

Alternatives: 13

Speedup: 1.2×

• 42.4% 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: 69.9% 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

\[\begin{array}{l} \\ \frac{y + \frac{z + x}{\frac{y}{x - z}}}{2} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (y + ((z + x) / (y / (x - z)))) / 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) / (y / (x - z)))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 82.2%

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

3. Simplified 82.2%

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

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

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

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

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

   8. --lowering--.f64 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \frac{y + \frac{z + x}{\frac{y}{x - z}}}{2} \]
10. Add Preprocessing

Alternative 2: 52.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-316}:\\ \;\;\;\;\frac{z}{\frac{y}{z \cdot -0.5}}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-199}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+67}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+248}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{x}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e-316)
   (/ z (/ y (* z -0.5)))
   (if (<= (* x x) 5e-199)
     (/ y 2.0)
     (if (<= (* x x) 2e+67)
       (* z (/ (* z -0.5) y))
       (if (<= (* x x) 2e+248) (/ y 2.0) (/ (/ x (/ y x)) 2.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e-316) {
		tmp = z / (y / (z * -0.5));
	} else if ((x * x) <= 5e-199) {
		tmp = y / 2.0;
	} else if ((x * x) <= 2e+67) {
		tmp = z * ((z * -0.5) / y);
	} else if ((x * x) <= 2e+248) {
		tmp = y / 2.0;
	} else {
		tmp = (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-316) then
        tmp = z / (y / (z * (-0.5d0)))
    else if ((x * x) <= 5d-199) then
        tmp = y / 2.0d0
    else if ((x * x) <= 2d+67) then
        tmp = z * ((z * (-0.5d0)) / y)
    else if ((x * x) <= 2d+248) then
        tmp = y / 2.0d0
    else
        tmp = (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-316) {
		tmp = z / (y / (z * -0.5));
	} else if ((x * x) <= 5e-199) {
		tmp = y / 2.0;
	} else if ((x * x) <= 2e+67) {
		tmp = z * ((z * -0.5) / y);
	} else if ((x * x) <= 2e+248) {
		tmp = y / 2.0;
	} else {
		tmp = (x / (y / x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e-316:
		tmp = z / (y / (z * -0.5))
	elif (x * x) <= 5e-199:
		tmp = y / 2.0
	elif (x * x) <= 2e+67:
		tmp = z * ((z * -0.5) / y)
	elif (x * x) <= 2e+248:
		tmp = y / 2.0
	else:
		tmp = (x / (y / x)) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e-316)
		tmp = Float64(z / Float64(y / Float64(z * -0.5)));
	elseif (Float64(x * x) <= 5e-199)
		tmp = Float64(y / 2.0);
	elseif (Float64(x * x) <= 2e+67)
		tmp = Float64(z * Float64(Float64(z * -0.5) / y));
	elseif (Float64(x * x) <= 2e+248)
		tmp = Float64(y / 2.0);
	else
		tmp = Float64(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-316)
		tmp = z / (y / (z * -0.5));
	elseif ((x * x) <= 5e-199)
		tmp = y / 2.0;
	elseif ((x * x) <= 2e+67)
		tmp = z * ((z * -0.5) / y);
	elseif ((x * x) <= 2e+248)
		tmp = y / 2.0;
	else
		tmp = (x / (y / x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-316], N[(z / N[(y / N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e-199], N[(y / 2.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+67], N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+248], N[(y / 2.0), $MachinePrecision], N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 9.999999837e-317

   1. Initial program 71.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 90.5%

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

   3. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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. mul-1-neg 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. mul-1-neg 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 50.8%

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

   7. Simplified 50.8%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   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 58.7%

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

   9. Applied egg-rr 58.7%

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

       1. associate-*r/ N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

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

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

       6. *-lowering-*.f64 58.7%

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

   11. Applied egg-rr 58.7%

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

   if 9.999999837e-317 < (*.f64 x x) < 4.9999999999999996e-199 or 1.99999999999999997e67 < (*.f64 x x) < 2.00000000000000009e248

   1. Initial program 66.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 y around inf

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

   7. Simplified 60.3%

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

   if 4.9999999999999996e-199 < (*.f64 x x) < 1.99999999999999997e67

   1. Initial program 75.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 89.7%

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

   3. Simplified 89.7%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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. mul-1-neg 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. mul-1-neg 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 50.3%

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

   7. Simplified 50.3%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   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 56.2%

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

   9. Applied egg-rr 56.2%

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

   if 2.00000000000000009e248 < (*.f64 x x)

   1. Initial program 61.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 62.6%

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

   3. Simplified 62.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 67.8%

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

   7. Simplified 67.8%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot x}{y}}}{2} \]
   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 72.1%

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

   9. Applied egg-rr 72.1%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

       5. /-lowering-/.f64 72.1%

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

   11. Applied egg-rr 72.1%

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

Alternative 3: 52.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-316}:\\ \;\;\;\;\frac{z}{\frac{y}{z \cdot -0.5}}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-199}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+67}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+248}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e-316)
   (/ z (/ y (* z -0.5)))
   (if (<= (* x x) 5e-199)
     (/ y 2.0)
     (if (<= (* x x) 2e+67)
       (* z (/ (* z -0.5) y))
       (if (<= (* x x) 2e+248) (/ y 2.0) (/ (* x (/ x y)) 2.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e-316) {
		tmp = z / (y / (z * -0.5));
	} else if ((x * x) <= 5e-199) {
		tmp = y / 2.0;
	} else if ((x * x) <= 2e+67) {
		tmp = z * ((z * -0.5) / y);
	} else if ((x * x) <= 2e+248) {
		tmp = y / 2.0;
	} else {
		tmp = (x * (x / 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 ((x * x) <= 1d-316) then
        tmp = z / (y / (z * (-0.5d0)))
    else if ((x * x) <= 5d-199) then
        tmp = y / 2.0d0
    else if ((x * x) <= 2d+67) then
        tmp = z * ((z * (-0.5d0)) / y)
    else if ((x * x) <= 2d+248) then
        tmp = y / 2.0d0
    else
        tmp = (x * (x / y)) / 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-316) {
		tmp = z / (y / (z * -0.5));
	} else if ((x * x) <= 5e-199) {
		tmp = y / 2.0;
	} else if ((x * x) <= 2e+67) {
		tmp = z * ((z * -0.5) / y);
	} else if ((x * x) <= 2e+248) {
		tmp = y / 2.0;
	} else {
		tmp = (x * (x / y)) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e-316:
		tmp = z / (y / (z * -0.5))
	elif (x * x) <= 5e-199:
		tmp = y / 2.0
	elif (x * x) <= 2e+67:
		tmp = z * ((z * -0.5) / y)
	elif (x * x) <= 2e+248:
		tmp = y / 2.0
	else:
		tmp = (x * (x / y)) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e-316)
		tmp = Float64(z / Float64(y / Float64(z * -0.5)));
	elseif (Float64(x * x) <= 5e-199)
		tmp = Float64(y / 2.0);
	elseif (Float64(x * x) <= 2e+67)
		tmp = Float64(z * Float64(Float64(z * -0.5) / y));
	elseif (Float64(x * x) <= 2e+248)
		tmp = Float64(y / 2.0);
	else
		tmp = Float64(Float64(x * Float64(x / y)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e-316)
		tmp = z / (y / (z * -0.5));
	elseif ((x * x) <= 5e-199)
		tmp = y / 2.0;
	elseif ((x * x) <= 2e+67)
		tmp = z * ((z * -0.5) / y);
	elseif ((x * x) <= 2e+248)
		tmp = y / 2.0;
	else
		tmp = (x * (x / y)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-316], N[(z / N[(y / N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e-199], N[(y / 2.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+67], N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+248], N[(y / 2.0), $MachinePrecision], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 9.999999837e-317

   1. Initial program 71.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 90.5%

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

   3. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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. mul-1-neg 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. mul-1-neg 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 50.8%

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

   7. Simplified 50.8%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   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 58.7%

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

   9. Applied egg-rr 58.7%

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

       1. associate-*r/ N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

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

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

       6. *-lowering-*.f64 58.7%

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

   11. Applied egg-rr 58.7%

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

   if 9.999999837e-317 < (*.f64 x x) < 4.9999999999999996e-199 or 1.99999999999999997e67 < (*.f64 x x) < 2.00000000000000009e248

   1. Initial program 66.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 y around inf

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

   7. Simplified 60.3%

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

   if 4.9999999999999996e-199 < (*.f64 x x) < 1.99999999999999997e67

   1. Initial program 75.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 89.7%

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

   3. Simplified 89.7%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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. mul-1-neg 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. mul-1-neg 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 50.3%

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

   7. Simplified 50.3%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   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 56.2%

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

   9. Applied egg-rr 56.2%

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

   if 2.00000000000000009e248 < (*.f64 x x)

   1. Initial program 61.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 62.6%

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

   3. Simplified 62.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 67.8%

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

   7. Simplified 67.8%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot x}{y}}}{2} \]
   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 72.1%

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

   9. Applied egg-rr 72.1%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-316}:\\ \;\;\;\;\frac{z}{\frac{y}{z \cdot -0.5}}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-199}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+67}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+248}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-207}:\\ \;\;\;\;\frac{z}{\frac{y}{z \cdot -0.5}}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{0.5}{\frac{y}{x \cdot x}}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+87}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.5e-207)
   (/ z (/ y (* z -0.5)))
   (if (<= y 2.4e-81)
     (/ 0.5 (/ y (* x x)))
     (if (<= y 1.15e+87) (* z (/ (* z -0.5) y)) (/ y 2.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e-207) {
		tmp = z / (y / (z * -0.5));
	} else if (y <= 2.4e-81) {
		tmp = 0.5 / (y / (x * x));
	} else if (y <= 1.15e+87) {
		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 <= 4.5d-207) then
        tmp = z / (y / (z * (-0.5d0)))
    else if (y <= 2.4d-81) then
        tmp = 0.5d0 / (y / (x * x))
    else if (y <= 1.15d+87) 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 <= 4.5e-207) {
		tmp = z / (y / (z * -0.5));
	} else if (y <= 2.4e-81) {
		tmp = 0.5 / (y / (x * x));
	} else if (y <= 1.15e+87) {
		tmp = z * ((z * -0.5) / y);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.5e-207:
		tmp = z / (y / (z * -0.5))
	elif y <= 2.4e-81:
		tmp = 0.5 / (y / (x * x))
	elif y <= 1.15e+87:
		tmp = z * ((z * -0.5) / y)
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.5e-207)
		tmp = Float64(z / Float64(y / Float64(z * -0.5)));
	elseif (y <= 2.4e-81)
		tmp = Float64(0.5 / Float64(y / Float64(x * x)));
	elseif (y <= 1.15e+87)
		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 <= 4.5e-207)
		tmp = z / (y / (z * -0.5));
	elseif (y <= 2.4e-81)
		tmp = 0.5 / (y / (x * x));
	elseif (y <= 1.15e+87)
		tmp = z * ((z * -0.5) / y);
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.5e-207], N[(z / N[(y / N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e-81], N[(0.5 / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+87], 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 < 4.49999999999999992e-207

   1. Initial program 74.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 86.1%

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

   3. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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. mul-1-neg 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. mul-1-neg 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 37.6%

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

   7. Simplified 37.6%

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

      1. associate-/l* N/A

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

      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 40.2%

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

   9. Applied egg-rr 40.2%

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

       1. associate-*r/ N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

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

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

       6. *-lowering-*.f64 40.2%

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

   11. Applied egg-rr 40.2%

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

   if 4.49999999999999992e-207 < y < 2.3999999999999999e-81

   1. Initial program 89.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 89.9%

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

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 45.8%

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

   7. Simplified 45.8%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot x}{y}}}{2} \]
   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 45.7%

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

   9. Applied egg-rr 45.7%

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

   if 2.3999999999999999e-81 < y < 1.1500000000000001e87

   1. Initial program 96.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 96.3%

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

   3. Simplified 96.3%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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. mul-1-neg 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. mul-1-neg 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 41.8%

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

   7. Simplified 41.8%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   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 41.8%

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

   9. Applied egg-rr 41.8%

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

   if 1.1500000000000001e87 < y

   1. Initial program 35.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 66.1%

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

   3. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 67.0%

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

Alternative 5: 42.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{if}\;y \leq 4.2 \cdot 10^{-207}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-81}:\\ \;\;\;\;\frac{0.5}{\frac{y}{x \cdot x}}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ (* z -0.5) y))))
   (if (<= y 4.2e-207)
     t_0
     (if (<= y 2e-81)
       (/ 0.5 (/ y (* x x)))
       (if (<= y 8.2e+86) t_0 (/ y 2.0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * -0.5) / y);
	double tmp;
	if (y <= 4.2e-207) {
		tmp = t_0;
	} else if (y <= 2e-81) {
		tmp = 0.5 / (y / (x * x));
	} else if (y <= 8.2e+86) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = z * ((z * (-0.5d0)) / y)
    if (y <= 4.2d-207) then
        tmp = t_0
    else if (y <= 2d-81) then
        tmp = 0.5d0 / (y / (x * x))
    else if (y <= 8.2d+86) then
        tmp = t_0
    else
        tmp = y / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z * -0.5) / y);
	double tmp;
	if (y <= 4.2e-207) {
		tmp = t_0;
	} else if (y <= 2e-81) {
		tmp = 0.5 / (y / (x * x));
	} else if (y <= 8.2e+86) {
		tmp = t_0;
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * -0.5) / y)
	tmp = 0
	if y <= 4.2e-207:
		tmp = t_0
	elif y <= 2e-81:
		tmp = 0.5 / (y / (x * x))
	elif y <= 8.2e+86:
		tmp = t_0
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * -0.5) / y))
	tmp = 0.0
	if (y <= 4.2e-207)
		tmp = t_0;
	elseif (y <= 2e-81)
		tmp = Float64(0.5 / Float64(y / Float64(x * x)));
	elseif (y <= 8.2e+86)
		tmp = t_0;
	else
		tmp = Float64(y / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * -0.5) / y);
	tmp = 0.0;
	if (y <= 4.2e-207)
		tmp = t_0;
	elseif (y <= 2e-81)
		tmp = 0.5 / (y / (x * x));
	elseif (y <= 8.2e+86)
		tmp = t_0;
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.2e-207], t$95$0, If[LessEqual[y, 2e-81], N[(0.5 / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e+86], t$95$0, N[(y / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.20000000000000007e-207 or 1.9999999999999999e-81 < y < 8.1999999999999998e86

   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 87.7%

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

   3. Simplified 87.7%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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. mul-1-neg 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. mul-1-neg 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 38.3%

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

   7. Simplified 38.3%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   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 40.4%

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

   9. Applied egg-rr 40.4%

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

   if 4.20000000000000007e-207 < y < 1.9999999999999999e-81

   1. Initial program 89.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 89.9%

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

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 45.8%

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

   7. Simplified 45.8%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot x}{y}}}{2} \]
   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 45.7%

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

   9. Applied egg-rr 45.7%

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

   if 8.1999999999999998e86 < y

   1. Initial program 35.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 66.1%

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

   3. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 67.0%

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

Alternative 6: 88.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+67}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x - z}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+67)
   (/ (- 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) <= 2e+67) {
		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) <= 2d+67) 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) <= 2e+67) {
		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) <= 2e+67:
		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) <= 2e+67)
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	else
		tmp = Float64(Float64(y + Float64(x / Float64(y / Float64(x - z)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+67)
		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], 2e+67], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y + N[(x / N[(y / N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999997e67

   1. Initial program 71.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 89.9%

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

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 84.2%

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

   7. Simplified 84.2%

      \[\leadsto \frac{\color{blue}{y - \frac{z \cdot z}{y}}}{2} \]
   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 94.1%

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

   9. Applied egg-rr 94.1%

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

   if 1.99999999999999997e67 < (*.f64 x x)

   1. Initial program 62.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 72.6%

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

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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. Taylor expanded in z around 0

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

   9. Simplified 92.4%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

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

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

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

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

       5. --lowering--.f64 92.4%

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

   11. Applied egg-rr 92.4%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+67}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x - z}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+67}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x \cdot \frac{x - z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+67)
   (/ (- y (* z (/ z y))) 2.0)
   (/ (+ y (* x (/ (- x z) y))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+67) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else {
		tmp = (y + (x * ((x - z) / 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 ((x * x) <= 2d+67) then
        tmp = (y - (z * (z / y))) / 2.0d0
    else
        tmp = (y + (x * ((x - z) / y))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+67) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else {
		tmp = (y + (x * ((x - z) / y))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+67:
		tmp = (y - (z * (z / y))) / 2.0
	else:
		tmp = (y + (x * ((x - z) / y))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+67)
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	else
		tmp = Float64(Float64(y + Float64(x * Float64(Float64(x - z) / y))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+67)
		tmp = (y - (z * (z / y))) / 2.0;
	else
		tmp = (y + (x * ((x - z) / y))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+67], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y + N[(x * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999997e67

   1. Initial program 71.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 89.9%

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

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 84.2%

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

   7. Simplified 84.2%

      \[\leadsto \frac{\color{blue}{y - \frac{z \cdot z}{y}}}{2} \]
   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 94.1%

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

   9. Applied egg-rr 94.1%

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

   if 1.99999999999999997e67 < (*.f64 x x)

   1. Initial program 62.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 72.6%

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

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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. Taylor expanded in z around 0

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

   9. Simplified 92.4%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+67}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x \cdot \frac{x - z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+75}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x \cdot \frac{x}{y}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 5e+75)
   (/ (- y (* z (/ z y))) 2.0)
   (/ (+ y (* x (/ x y))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e+75) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else {
		tmp = (y + (x * (x / 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 ((x * x) <= 5d+75) then
        tmp = (y - (z * (z / y))) / 2.0d0
    else
        tmp = (y + (x * (x / y))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e+75) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else {
		tmp = (y + (x * (x / y))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 5e+75:
		tmp = (y - (z * (z / y))) / 2.0
	else:
		tmp = (y + (x * (x / y))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 5e+75)
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	else
		tmp = Float64(Float64(y + Float64(x * Float64(x / y))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 5e+75)
		tmp = (y - (z * (z / y))) / 2.0;
	else
		tmp = (y + (x * (x / y))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+75], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.0000000000000002e75

   1. Initial program 71.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 90.1%

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

   3. Simplified 90.1%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 84.5%

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

   7. Simplified 84.5%

      \[\leadsto \frac{\color{blue}{y - \frac{z \cdot z}{y}}}{2} \]
   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 94.2%

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

   9. Applied egg-rr 94.2%

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

   if 5.0000000000000002e75 < (*.f64 x x)

   1. Initial program 62.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 71.9%

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

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 72.2%

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

   7. Simplified 72.2%

      \[\leadsto \frac{\color{blue}{y + \frac{x \cdot x}{y}}}{2} \]
   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. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 84.7%

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

   9. Applied egg-rr 84.7%

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

4. Final simplification 90.1%

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

Alternative 9: 72.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+153}:\\ \;\;\;\;\frac{y + x \cdot \frac{x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{z \cdot -0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.45e+153) (/ (+ y (* x (/ x y))) 2.0) (/ z (/ y (* z -0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.45e+153) {
		tmp = (y + (x * (x / y))) / 2.0;
	} else {
		tmp = z / (y / (z * -0.5));
	}
	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 <= 1.45d+153) then
        tmp = (y + (x * (x / y))) / 2.0d0
    else
        tmp = z / (y / (z * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.45e+153) {
		tmp = (y + (x * (x / y))) / 2.0;
	} else {
		tmp = z / (y / (z * -0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.45e+153:
		tmp = (y + (x * (x / y))) / 2.0
	else:
		tmp = z / (y / (z * -0.5))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.45e+153)
		tmp = Float64(Float64(y + Float64(x * Float64(x / y))) / 2.0);
	else
		tmp = Float64(z / Float64(y / Float64(z * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.45e+153)
		tmp = (y + (x * (x / y))) / 2.0;
	else
		tmp = z / (y / (z * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.45e+153], N[(N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(z / N[(y / N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.45000000000000001e153

   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 89.0%

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

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 65.4%

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

   7. Simplified 65.4%

      \[\leadsto \frac{\color{blue}{y + \frac{x \cdot x}{y}}}{2} \]
   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. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 71.3%

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

   9. Applied egg-rr 71.3%

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

   if 1.45000000000000001e153 < z

   1. Initial program 32.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 43.5%

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

   3. Simplified 43.5%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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. mul-1-neg 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. mul-1-neg 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 54.2%

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

   7. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   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 71.0%

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

   9. Applied egg-rr 71.0%

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

       1. associate-*r/ N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

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

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

       6. *-lowering-*.f64 71.1%

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

   11. Applied egg-rr 71.1%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+153}:\\ \;\;\;\;\frac{y + x \cdot \frac{x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{z \cdot -0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 67.9%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 82.2%

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

3. Simplified 82.2%

   \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
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 11: 43.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{+86}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.8e+86) (* z (/ (* z -0.5) y)) (/ y 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.8e+86) {
		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 <= 4.8d+86) 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 <= 4.8e+86) {
		tmp = z * ((z * -0.5) / y);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.8e+86:
		tmp = z * ((z * -0.5) / y)
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.8e+86)
		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 <= 4.8e+86)
		tmp = z * ((z * -0.5) / y);
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.8e+86], 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 < 4.8000000000000001e86

   1. Initial program 79.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 88.0%

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

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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. mul-1-neg 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. mul-1-neg 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 39.6%

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

   7. Simplified 39.6%

      \[\leadsto \color{blue}{\frac{\left(z \cdot z\right) \cdot -0.5}{y}} \]
   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 41.5%

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

   9. Applied egg-rr 41.5%

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

   if 4.8000000000000001e86 < y

   1. Initial program 35.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 66.1%

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

   3. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 67.0%

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

Alternative 12: 43.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.6 \cdot 10^{+86}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.6e+86) (* z (* z (/ -0.5 y))) (/ y 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.6e+86) {
		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 <= 9.6d+86) 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 <= 9.6e+86) {
		tmp = z * (z * (-0.5 / y));
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9.6e+86:
		tmp = z * (z * (-0.5 / y))
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.6e+86)
		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 <= 9.6e+86)
		tmp = z * (z * (-0.5 / y));
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.6e+86], 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 < 9.6000000000000001e86

   1. Initial program 79.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 88.0%

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

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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. mul-1-neg 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. mul-1-neg 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 39.6%

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

   7. Simplified 39.6%

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

      1. associate-/l* N/A

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

      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 41.5%

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

   9. Applied egg-rr 41.5%

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

   if 9.6000000000000001e86 < y

   1. Initial program 35.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 66.1%

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

   3. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 67.0%

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

Alternative 13: 33.4% 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 67.9%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 82.2%

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

3. Simplified 82.2%

   \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
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 37.0%

   \[\leadsto \frac{\color{blue}{y}}{2} \]
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 2024160 
(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)))