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

Percentage Accurate: 68.6% → 99.9%

Time: 10.2s

Alternatives: 9

Speedup: 1.2×

• 41.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.6% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 82.9%

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

3. Simplified 82.9%

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

   1. clear-num N/A

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

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

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

   3. difference-of-squares N/A

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

   4. associate-/r* N/A

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

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

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

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

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

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

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

   8. --lowering--.f64 99.8%

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

6. Applied egg-rr 99.8%

   \[\leadsto \frac{y - \color{blue}{\frac{1}{\frac{\frac{y}{z + x}}{z - x}}}}{2} \]
7. Step-by-step derivation

   1. clear-num N/A

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

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

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

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

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

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

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

   5. +-lowering-+.f64 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 73.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + \frac{x \cdot x}{y}}{2}\\ \mathbf{if}\;z \cdot z \leq 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\frac{z \cdot z}{y} \cdot -0.5\\ \mathbf{elif}\;z \cdot z \leq 10^{+268}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot \left(z \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y (/ (* x x) y)) 2.0)))
   (if (<= (* z z) 1e+53)
     t_0
     (if (<= (* z z) 2e+173)
       (* (/ (* z z) y) -0.5)
       (if (<= (* z z) 1e+268) t_0 (* (/ z y) (* z -0.5)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y + ((x * x) / y)) / 2.0;
	double tmp;
	if ((z * z) <= 1e+53) {
		tmp = t_0;
	} else if ((z * z) <= 2e+173) {
		tmp = ((z * z) / y) * -0.5;
	} else if ((z * z) <= 1e+268) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (y + ((x * x) / y)) / 2.0d0
    if ((z * z) <= 1d+53) then
        tmp = t_0
    else if ((z * z) <= 2d+173) then
        tmp = ((z * z) / y) * (-0.5d0)
    else if ((z * z) <= 1d+268) then
        tmp = t_0
    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 t_0 = (y + ((x * x) / y)) / 2.0;
	double tmp;
	if ((z * z) <= 1e+53) {
		tmp = t_0;
	} else if ((z * z) <= 2e+173) {
		tmp = ((z * z) / y) * -0.5;
	} else if ((z * z) <= 1e+268) {
		tmp = t_0;
	} else {
		tmp = (z / y) * (z * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + ((x * x) / y)) / 2.0
	tmp = 0
	if (z * z) <= 1e+53:
		tmp = t_0
	elif (z * z) <= 2e+173:
		tmp = ((z * z) / y) * -0.5
	elif (z * z) <= 1e+268:
		tmp = t_0
	else:
		tmp = (z / y) * (z * -0.5)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + Float64(Float64(x * x) / y)) / 2.0)
	tmp = 0.0
	if (Float64(z * z) <= 1e+53)
		tmp = t_0;
	elseif (Float64(z * z) <= 2e+173)
		tmp = Float64(Float64(Float64(z * z) / y) * -0.5);
	elseif (Float64(z * z) <= 1e+268)
		tmp = t_0;
	else
		tmp = Float64(Float64(z / y) * Float64(z * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + ((x * x) / y)) / 2.0;
	tmp = 0.0;
	if ((z * z) <= 1e+53)
		tmp = t_0;
	elseif ((z * z) <= 2e+173)
		tmp = ((z * z) / y) * -0.5;
	elseif ((z * z) <= 1e+268)
		tmp = t_0;
	else
		tmp = (z / y) * (z * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 1e+53], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 2e+173], N[(N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+268], t$95$0, N[(N[(z / y), $MachinePrecision] * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 9.9999999999999999e52 or 2e173 < (*.f64 z z) < 9.9999999999999997e267

   1. Initial program 73.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in z around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 80.6%

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

   7. Simplified 80.6%

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

   if 9.9999999999999999e52 < (*.f64 z z) < 2e173

   1. Initial program 91.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 95.8%

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

   3. Simplified 95.8%

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

      1. clear-num N/A

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

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

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

      3. difference-of-squares N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 70.6%

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

   9. Simplified 70.6%

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

   if 9.9999999999999997e267 < (*.f64 z z)

   1. Initial program 60.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 66.9%

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

   3. Simplified 66.9%

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

      1. clear-num N/A

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

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

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

      3. difference-of-squares N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 73.5%

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

   9. Simplified 73.5%

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

       1. associate-*l/ N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. *-lowering-*.f64 79.8%

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

   11. Applied egg-rr 79.8%

       \[\leadsto \color{blue}{\frac{z}{y} \cdot \left(z \cdot -0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 43.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7.6 \cdot 10^{-237}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{elif}\;z \leq 1.62 \cdot 10^{-77}:\\ \;\;\;\;\frac{0.5}{\frac{y}{x \cdot x}}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot \left(z \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 7.6e-237)
   (/ y 2.0)
   (if (<= z 1.62e-77)
     (/ 0.5 (/ y (* x x)))
     (if (<= z 7.8e+21) (/ y 2.0) (* (/ z y) (* z -0.5))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 7.6e-237) {
		tmp = y / 2.0;
	} else if (z <= 1.62e-77) {
		tmp = 0.5 / (y / (x * x));
	} else if (z <= 7.8e+21) {
		tmp = 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 <= 7.6d-237) then
        tmp = y / 2.0d0
    else if (z <= 1.62d-77) then
        tmp = 0.5d0 / (y / (x * x))
    else if (z <= 7.8d+21) then
        tmp = 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 <= 7.6e-237) {
		tmp = y / 2.0;
	} else if (z <= 1.62e-77) {
		tmp = 0.5 / (y / (x * x));
	} else if (z <= 7.8e+21) {
		tmp = y / 2.0;
	} else {
		tmp = (z / y) * (z * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 7.6e-237:
		tmp = y / 2.0
	elif z <= 1.62e-77:
		tmp = 0.5 / (y / (x * x))
	elif z <= 7.8e+21:
		tmp = y / 2.0
	else:
		tmp = (z / y) * (z * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 7.6e-237)
		tmp = Float64(y / 2.0);
	elseif (z <= 1.62e-77)
		tmp = Float64(0.5 / Float64(y / Float64(x * x)));
	elseif (z <= 7.8e+21)
		tmp = Float64(y / 2.0);
	else
		tmp = Float64(Float64(z / y) * Float64(z * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 7.6e-237)
		tmp = y / 2.0;
	elseif (z <= 1.62e-77)
		tmp = 0.5 / (y / (x * x));
	elseif (z <= 7.8e+21)
		tmp = y / 2.0;
	else
		tmp = (z / y) * (z * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 7.6e-237], N[(y / 2.0), $MachinePrecision], If[LessEqual[z, 1.62e-77], N[(0.5 / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.8e+21], N[(y / 2.0), $MachinePrecision], N[(N[(z / y), $MachinePrecision] * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 7.60000000000000047e-237 or 1.62000000000000006e-77 < z < 7.8e21

   1. Initial program 72.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 85.1%

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

   3. Simplified 85.1%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 32.5%

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

   if 7.60000000000000047e-237 < z < 1.62000000000000006e-77

   1. Initial program 87.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 90.8%

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

   3. Simplified 90.8%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 53.9%

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

   7. Simplified 53.9%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

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

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

      10. *-lowering-*.f64 54.0%

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

   9. Applied egg-rr 54.0%

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

   if 7.8e21 < z

   1. Initial program 59.7%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 73.1%

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

   3. Simplified 73.1%

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

      1. clear-num N/A

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

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

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

      3. difference-of-squares N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 60.3%

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

   9. Simplified 60.3%

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

       1. associate-*l/ N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. *-lowering-*.f64 66.7%

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

   11. Applied egg-rr 66.7%

       \[\leadsto \color{blue}{\frac{z}{y} \cdot \left(z \cdot -0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 43.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-237}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-74}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+21}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot \left(z \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 7.2e-237)
   (/ y 2.0)
   (if (<= z 5.4e-74)
     (* (* x x) (/ 0.5 y))
     (if (<= z 8e+21) (/ y 2.0) (* (/ z y) (* z -0.5))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 7.2e-237) {
		tmp = y / 2.0;
	} else if (z <= 5.4e-74) {
		tmp = (x * x) * (0.5 / y);
	} else if (z <= 8e+21) {
		tmp = 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 <= 7.2d-237) then
        tmp = y / 2.0d0
    else if (z <= 5.4d-74) then
        tmp = (x * x) * (0.5d0 / y)
    else if (z <= 8d+21) then
        tmp = 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 <= 7.2e-237) {
		tmp = y / 2.0;
	} else if (z <= 5.4e-74) {
		tmp = (x * x) * (0.5 / y);
	} else if (z <= 8e+21) {
		tmp = y / 2.0;
	} else {
		tmp = (z / y) * (z * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 7.2e-237:
		tmp = y / 2.0
	elif z <= 5.4e-74:
		tmp = (x * x) * (0.5 / y)
	elif z <= 8e+21:
		tmp = y / 2.0
	else:
		tmp = (z / y) * (z * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 7.2e-237)
		tmp = Float64(y / 2.0);
	elseif (z <= 5.4e-74)
		tmp = Float64(Float64(x * x) * Float64(0.5 / y));
	elseif (z <= 8e+21)
		tmp = Float64(y / 2.0);
	else
		tmp = Float64(Float64(z / y) * Float64(z * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 7.2e-237)
		tmp = y / 2.0;
	elseif (z <= 5.4e-74)
		tmp = (x * x) * (0.5 / y);
	elseif (z <= 8e+21)
		tmp = y / 2.0;
	else
		tmp = (z / y) * (z * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 7.2e-237], N[(y / 2.0), $MachinePrecision], If[LessEqual[z, 5.4e-74], N[(N[(x * x), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8e+21], N[(y / 2.0), $MachinePrecision], N[(N[(z / y), $MachinePrecision] * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 7.19999999999999993e-237 or 5.40000000000000036e-74 < z < 8e21

   1. Initial program 72.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 85.1%

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

   3. Simplified 85.1%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 32.5%

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

   if 7.19999999999999993e-237 < z < 5.40000000000000036e-74

   1. Initial program 87.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 90.8%

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

   3. Simplified 90.8%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 53.9%

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

   7. Simplified 53.9%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

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

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

      8. metadata-eval 54.0%

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

   9. Applied egg-rr 54.0%

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

   if 8e21 < z

   1. Initial program 59.7%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 73.1%

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

   3. Simplified 73.1%

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

      1. clear-num N/A

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

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

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

      3. difference-of-squares N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 60.3%

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

   9. Simplified 60.3%

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

       1. associate-*l/ N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. *-lowering-*.f64 66.7%

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

   11. Applied egg-rr 66.7%

       \[\leadsto \color{blue}{\frac{z}{y} \cdot \left(z \cdot -0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 52.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-156}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{2}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot \left(z \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-156)
   (/ (* x (/ x y)) 2.0)
   (if (<= (* z z) 5e+43) (/ y 2.0) (* (/ z y) (* z -0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-156) {
		tmp = (x * (x / y)) / 2.0;
	} else if ((z * z) <= 5e+43) {
		tmp = 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 * z) <= 1d-156) then
        tmp = (x * (x / y)) / 2.0d0
    else if ((z * z) <= 5d+43) then
        tmp = 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 * z) <= 1e-156) {
		tmp = (x * (x / y)) / 2.0;
	} else if ((z * z) <= 5e+43) {
		tmp = y / 2.0;
	} else {
		tmp = (z / y) * (z * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e-156:
		tmp = (x * (x / y)) / 2.0
	elif (z * z) <= 5e+43:
		tmp = y / 2.0
	else:
		tmp = (z / y) * (z * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-156)
		tmp = Float64(Float64(x * Float64(x / y)) / 2.0);
	elseif (Float64(z * z) <= 5e+43)
		tmp = Float64(y / 2.0);
	else
		tmp = Float64(Float64(z / y) * Float64(z * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e-156)
		tmp = (x * (x / y)) / 2.0;
	elseif ((z * z) <= 5e+43)
		tmp = y / 2.0;
	else
		tmp = (z / y) * (z * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-156], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+43], N[(y / 2.0), $MachinePrecision], N[(N[(z / y), $MachinePrecision] * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1.00000000000000004e-156

   1. Initial program 80.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 90.9%

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

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 52.4%

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

   7. Simplified 52.4%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot x}{y}}}{2} \]
   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 55.5%

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

   9. Applied egg-rr 55.5%

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

   if 1.00000000000000004e-156 < (*.f64 z z) < 5.0000000000000004e43

   1. Initial program 66.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 88.3%

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

   3. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 52.8%

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

   if 5.0000000000000004e43 < (*.f64 z z)

   1. Initial program 65.6%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 76.0%

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

   3. Simplified 76.0%

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

      1. clear-num N/A

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

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

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

      3. difference-of-squares N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 64.5%

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

   9. Simplified 64.5%

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

       1. associate-*l/ N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. *-lowering-*.f64 68.7%

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

   11. Applied egg-rr 68.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-156}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{2}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot \left(z \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+56}:\\ \;\;\;\;\frac{\left(z + x\right) \cdot \frac{x - z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9e+56)
   (/ (* (+ z x) (/ (- x z) y)) 2.0)
   (/ (- y (* z (/ z y))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e+56) {
		tmp = ((z + x) * ((x - z) / y)) / 2.0;
	} else {
		tmp = (y - (z * (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 (y <= 9d+56) then
        tmp = ((z + x) * ((x - z) / y)) / 2.0d0
    else
        tmp = (y - (z * (z / y))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e+56) {
		tmp = ((z + x) * ((x - z) / y)) / 2.0;
	} else {
		tmp = (y - (z * (z / y))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9e+56:
		tmp = ((z + x) * ((x - z) / y)) / 2.0
	else:
		tmp = (y - (z * (z / y))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9e+56)
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y)) / 2.0);
	else
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9e+56)
		tmp = ((z + x) * ((x - z) / y)) / 2.0;
	else
		tmp = (y - (z * (z / y))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9e+56], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.0000000000000006e56

   1. Initial program 75.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. difference-of-squares N/A

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

      4. associate-/l* N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 79.6%

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

   7. Simplified 79.6%

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

   if 9.0000000000000006e56 < y

   1. Initial program 44.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 79.3%

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

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 75.4%

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

   7. Simplified 75.4%

      \[\leadsto \frac{\color{blue}{y - \frac{z \cdot z}{y}}}{2} \]
   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 84.6%

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

   9. Applied egg-rr 84.6%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+56}:\\ \;\;\;\;\frac{\left(z + x\right) \cdot \frac{x - z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{y + \frac{x \cdot x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e-11)
   (/ (+ y (/ (* x x) y)) 2.0)
   (/ (- y (* z (/ z y))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e-11) {
		tmp = (y + ((x * x) / y)) / 2.0;
	} else {
		tmp = (y - (z * (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 ((z * z) <= 4d-11) then
        tmp = (y + ((x * x) / y)) / 2.0d0
    else
        tmp = (y - (z * (z / y))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e-11) {
		tmp = (y + ((x * x) / y)) / 2.0;
	} else {
		tmp = (y - (z * (z / y))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 4e-11:
		tmp = (y + ((x * x) / y)) / 2.0
	else:
		tmp = (y - (z * (z / y))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e-11)
		tmp = Float64(Float64(y + Float64(Float64(x * x) / y)) / 2.0);
	else
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 4e-11)
		tmp = (y + ((x * x) / y)) / 2.0;
	else
		tmp = (y - (z * (z / y))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e-11], N[(N[(y + N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.99999999999999976e-11

   1. Initial program 77.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in z around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 85.3%

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

   7. Simplified 85.3%

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

   if 3.99999999999999976e-11 < (*.f64 z z)

   1. Initial program 65.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 75.5%

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

   3. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 73.6%

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

   7. Simplified 73.6%

      \[\leadsto \frac{\color{blue}{y - \frac{z \cdot z}{y}}}{2} \]
   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 82.5%

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

   9. Applied egg-rr 82.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{y + \frac{x \cdot x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{+59}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.65e+59) (* (* x x) (/ 0.5 y)) (/ y 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.65e+59) {
		tmp = (x * x) * (0.5 / y);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.65d+59) then
        tmp = (x * x) * (0.5d0 / y)
    else
        tmp = y / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.65e+59) {
		tmp = (x * x) * (0.5 / y);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.65e+59:
		tmp = (x * x) * (0.5 / y)
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.65e+59)
		tmp = Float64(Float64(x * x) * 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 <= 1.65e+59)
		tmp = (x * x) * (0.5 / y);
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.65e+59], N[(N[(x * x), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], N[(y / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.65e59

   1. Initial program 75.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 83.6%

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

   3. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 33.5%

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

   7. Simplified 33.5%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

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

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

      8. metadata-eval 33.5%

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

   9. Applied egg-rr 33.5%

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

   if 1.65e59 < y

   1. Initial program 44.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 79.3%

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

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 70.2%

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

Alternative 9: 34.9% 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 70.7%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 82.9%

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

3. Simplified 82.9%

   \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
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 29.9%

   \[\leadsto \frac{\color{blue}{y}}{2} \]
8. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x))))

C

double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

public static double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Python

def code(x, y, z):
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x))

Julia

function code(x, y, z)
	return Float64(Float64(y * 0.5) - Float64(Float64(Float64(0.5 / y) * Float64(z + x)) * Float64(z - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] - N[(N[(N[(0.5 / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024161 
(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)))