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

Percentage Accurate: 69.1% → 99.8%

Time: 10.9s

Alternatives: 10

Speedup: 1.2×

• 41.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% 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: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.9%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 85.8%

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

3. Simplified 85.8%

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

   1. difference-of-squares N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

   6. --lowering--.f64 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 43.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{if}\;z \leq 2.8 \cdot 10^{-264}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z}{\frac{y}{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* x (/ 0.5 y)))))
   (if (<= z 2.8e-264)
     t_0
     (if (<= z 1.16e-41)
       (/ y 2.0)
       (if (<= z 3.8e+56) t_0 (* z (/ z (/ y -0.5))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if (z <= 2.8e-264) {
		tmp = t_0;
	} else if (z <= 1.16e-41) {
		tmp = y / 2.0;
	} else if (z <= 3.8e+56) {
		tmp = t_0;
	} else {
		tmp = z * (z / (y / -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 = x * (x * (0.5d0 / y))
    if (z <= 2.8d-264) then
        tmp = t_0
    else if (z <= 1.16d-41) then
        tmp = y / 2.0d0
    else if (z <= 3.8d+56) then
        tmp = t_0
    else
        tmp = z * (z / (y / (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if (z <= 2.8e-264) {
		tmp = t_0;
	} else if (z <= 1.16e-41) {
		tmp = y / 2.0;
	} else if (z <= 3.8e+56) {
		tmp = t_0;
	} else {
		tmp = z * (z / (y / -0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (x * (0.5 / y))
	tmp = 0
	if z <= 2.8e-264:
		tmp = t_0
	elif z <= 1.16e-41:
		tmp = y / 2.0
	elif z <= 3.8e+56:
		tmp = t_0
	else:
		tmp = z * (z / (y / -0.5))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(x * Float64(0.5 / y)))
	tmp = 0.0
	if (z <= 2.8e-264)
		tmp = t_0;
	elseif (z <= 1.16e-41)
		tmp = Float64(y / 2.0);
	elseif (z <= 3.8e+56)
		tmp = t_0;
	else
		tmp = Float64(z * Float64(z / Float64(y / -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (x * (0.5 / y));
	tmp = 0.0;
	if (z <= 2.8e-264)
		tmp = t_0;
	elseif (z <= 1.16e-41)
		tmp = y / 2.0;
	elseif (z <= 3.8e+56)
		tmp = t_0;
	else
		tmp = z * (z / (y / -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 2.8e-264], t$95$0, If[LessEqual[z, 1.16e-41], N[(y / 2.0), $MachinePrecision], If[LessEqual[z, 3.8e+56], t$95$0, N[(z * N[(z / N[(y / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.80000000000000012e-264 or 1.1600000000000001e-41 < z < 3.79999999999999996e56

   1. Initial program 68.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 87.3%

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

   3. Simplified 87.3%

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

      1. difference-of-squares N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. *-commutative N/A

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

      2. associate-/r/ N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      9. +-lowering-+.f64 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, x\right), y\right), \color{blue}{\left(\frac{1}{x}\right)}\right)\right), 2\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 79.8%

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

   11. Simplified 79.8%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-/l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*l/ N/A

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

       9. associate-*r/ N/A

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

       10. metadata-eval N/A

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

       11. associate-*r/ N/A

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

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

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. /-lowering-/.f64 39.5%

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

   14. Simplified 39.5%

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

   if 2.80000000000000012e-264 < z < 1.1600000000000001e-41

   1. Initial program 62.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 95.3%

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

   3. Simplified 95.3%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 58.2%

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

   if 3.79999999999999996e56 < 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 69.7%

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

   3. Simplified 69.7%

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

      1. difference-of-squares N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. associate-*r/ N/A

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

   9. Simplified 71.1%

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

       1. *-commutative N/A

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

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

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

       3. clear-num N/A

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

       4. un-div-inv N/A

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

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

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

       6. /-lowering-/.f64 71.1%

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

   11. Applied egg-rr 71.1%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{-264}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z}{\frac{y}{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{if}\;z \leq 2.9 \cdot 10^{-264}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-42}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{elif}\;z \leq 7.3 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* x (/ 0.5 y)))))
   (if (<= z 2.9e-264)
     t_0
     (if (<= z 3e-42)
       (/ y 2.0)
       (if (<= z 7.3e+56) t_0 (* z (* z (/ -0.5 y))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if (z <= 2.9e-264) {
		tmp = t_0;
	} else if (z <= 3e-42) {
		tmp = y / 2.0;
	} else if (z <= 7.3e+56) {
		tmp = t_0;
	} else {
		tmp = z * (z * (-0.5 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (0.5d0 / y))
    if (z <= 2.9d-264) then
        tmp = t_0
    else if (z <= 3d-42) then
        tmp = y / 2.0d0
    else if (z <= 7.3d+56) then
        tmp = t_0
    else
        tmp = z * (z * ((-0.5d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (x * (0.5 / y));
	double tmp;
	if (z <= 2.9e-264) {
		tmp = t_0;
	} else if (z <= 3e-42) {
		tmp = y / 2.0;
	} else if (z <= 7.3e+56) {
		tmp = t_0;
	} else {
		tmp = z * (z * (-0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (x * (0.5 / y))
	tmp = 0
	if z <= 2.9e-264:
		tmp = t_0
	elif z <= 3e-42:
		tmp = y / 2.0
	elif z <= 7.3e+56:
		tmp = t_0
	else:
		tmp = z * (z * (-0.5 / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(x * Float64(0.5 / y)))
	tmp = 0.0
	if (z <= 2.9e-264)
		tmp = t_0;
	elseif (z <= 3e-42)
		tmp = Float64(y / 2.0);
	elseif (z <= 7.3e+56)
		tmp = t_0;
	else
		tmp = Float64(z * Float64(z * Float64(-0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (x * (0.5 / y));
	tmp = 0.0;
	if (z <= 2.9e-264)
		tmp = t_0;
	elseif (z <= 3e-42)
		tmp = y / 2.0;
	elseif (z <= 7.3e+56)
		tmp = t_0;
	else
		tmp = z * (z * (-0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 2.9e-264], t$95$0, If[LessEqual[z, 3e-42], N[(y / 2.0), $MachinePrecision], If[LessEqual[z, 7.3e+56], t$95$0, N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.8999999999999999e-264 or 3.00000000000000027e-42 < z < 7.3e56

   1. Initial program 68.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 87.3%

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

   3. Simplified 87.3%

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

      1. difference-of-squares N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. *-commutative N/A

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

      2. associate-/r/ N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      9. +-lowering-+.f64 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, x\right), y\right), \color{blue}{\left(\frac{1}{x}\right)}\right)\right), 2\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 79.8%

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

   11. Simplified 79.8%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-/l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*l/ N/A

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

       9. associate-*r/ N/A

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

       10. metadata-eval N/A

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

       11. associate-*r/ N/A

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

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

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. /-lowering-/.f64 39.5%

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

   14. Simplified 39.5%

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

   if 2.8999999999999999e-264 < z < 3.00000000000000027e-42

   1. Initial program 62.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 95.3%

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

   3. Simplified 95.3%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 58.2%

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

   if 7.3e56 < 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 69.7%

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

   3. Simplified 69.7%

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

      1. difference-of-squares N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. associate-*r/ N/A

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

   9. Simplified 71.1%

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

Alternative 4: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z + x\right) \cdot \frac{x - z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+65)
   (/ (+ y (/ x (/ y x))) 2.0)
   (/ (* (+ z x) (/ (- x z) y)) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+65) {
		tmp = (y + (x / (y / x))) / 2.0;
	} else {
		tmp = ((z + x) * ((x - z) / y)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+65) then
        tmp = (y + (x / (y / x))) / 2.0d0
    else
        tmp = ((z + x) * ((x - z) / y)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+65) {
		tmp = (y + (x / (y / x))) / 2.0;
	} else {
		tmp = ((z + x) * ((x - z) / y)) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+65:
		tmp = (y + (x / (y / x))) / 2.0
	else:
		tmp = ((z + x) * ((x - z) / y)) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+65)
		tmp = Float64(Float64(y + Float64(x / Float64(y / x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+65)
		tmp = (y + (x / (y / x))) / 2.0;
	else
		tmp = ((z + x) * ((x - z) / y)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+65], N[(N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e65

   1. Initial program 65.4%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 92.5%

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

   3. Simplified 92.5%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 86.6%

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

   7. Simplified 86.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 92.4%

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

   9. Applied egg-rr 92.4%

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

       1. +-commutative N/A

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

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

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

       3. *-commutative N/A

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

       4. remove-double-div N/A

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

       5. times-frac N/A

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

       6. metadata-eval N/A

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

       7. associate-/r/ N/A

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

       8. remove-double-div N/A

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

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

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

       10. *-commutative N/A

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

       11. div-inv N/A

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

       12. /-lowering-/.f64 92.4%

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

   11. Applied egg-rr 92.4%

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

   if 2e65 < (*.f64 z z)

   1. Initial program 69.4%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 74.6%

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

   3. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 83.4%

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

   7. Simplified 83.4%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z + x\right) \cdot \frac{x - z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+106)
   (/ (+ y (/ x (/ y x))) 2.0)
   (/ (- y (* z (/ z y))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+106) {
		tmp = (y + (x / (y / x))) / 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) <= 5d+106) then
        tmp = (y + (x / (y / x))) / 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) <= 5e+106) {
		tmp = (y + (x / (y / x))) / 2.0;
	} else {
		tmp = (y - (z * (z / y))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+106:
		tmp = (y + (x / (y / x))) / 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) <= 5e+106)
		tmp = Float64(Float64(y + Float64(x / Float64(y / x))) / 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) <= 5e+106)
		tmp = (y + (x / (y / x))) / 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], 5e+106], N[(N[(y + N[(x / N[(y / x), $MachinePrecision]), $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) < 4.9999999999999998e106

   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 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in z around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 86.1%

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

   7. Simplified 86.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 91.6%

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

   9. Applied egg-rr 91.6%

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

       1. +-commutative N/A

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

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

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

       3. *-commutative N/A

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

       4. remove-double-div N/A

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

       5. times-frac N/A

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

       6. metadata-eval N/A

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

       7. associate-/r/ N/A

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

       8. remove-double-div N/A

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

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

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

       10. *-commutative N/A

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

       11. div-inv N/A

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

       12. /-lowering-/.f64 91.6%

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

   11. Applied egg-rr 91.6%

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

   if 4.9999999999999998e106 < (*.f64 z z)

   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 71.7%

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

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 70.5%

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

   7. Simplified 70.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 80.7%

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

   9. Applied egg-rr 80.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{x - z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+126)
   (/ (+ y (/ x (/ y x))) 2.0)
   (/ (* z (/ (- x z) y)) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+126) {
		tmp = (y + (x / (y / x))) / 2.0;
	} else {
		tmp = (z * ((x - z) / y)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+126) then
        tmp = (y + (x / (y / x))) / 2.0d0
    else
        tmp = (z * ((x - z) / y)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+126) {
		tmp = (y + (x / (y / x))) / 2.0;
	} else {
		tmp = (z * ((x - z) / y)) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+126:
		tmp = (y + (x / (y / x))) / 2.0
	else:
		tmp = (z * ((x - z) / y)) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+126)
		tmp = Float64(Float64(y + Float64(x / Float64(y / x))) / 2.0);
	else
		tmp = Float64(Float64(z * Float64(Float64(x - z) / y)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+126)
		tmp = (y + (x / (y / x))) / 2.0;
	else
		tmp = (z * ((x - z) / y)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+126], N[(N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(z * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999985e126

   1. Initial program 67.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 92.6%

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

   3. Simplified 92.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 84.7%

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

   7. Simplified 84.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 90.6%

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

   9. Applied egg-rr 90.6%

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

       1. +-commutative N/A

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

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

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

       3. *-commutative N/A

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

       4. remove-double-div N/A

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

       5. times-frac N/A

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

       6. metadata-eval N/A

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

       7. associate-/r/ N/A

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

       8. remove-double-div N/A

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

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

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

       10. *-commutative N/A

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

       11. div-inv N/A

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

       12. /-lowering-/.f64 90.7%

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

   11. Applied egg-rr 90.7%

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

   if 1.99999999999999985e126 < (*.f64 z z)

   1. Initial program 66.5%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 71.5%

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

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 85.4%

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

   7. Simplified 85.4%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 72.5%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{x - z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{y + x \cdot \frac{x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{x - z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+126)
   (/ (+ y (* x (/ x y))) 2.0)
   (/ (* z (/ (- x z) y)) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+126) {
		tmp = (y + (x * (x / y))) / 2.0;
	} else {
		tmp = (z * ((x - z) / y)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+126) then
        tmp = (y + (x * (x / y))) / 2.0d0
    else
        tmp = (z * ((x - z) / y)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+126) {
		tmp = (y + (x * (x / y))) / 2.0;
	} else {
		tmp = (z * ((x - z) / y)) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+126:
		tmp = (y + (x * (x / y))) / 2.0
	else:
		tmp = (z * ((x - z) / y)) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+126)
		tmp = Float64(Float64(y + Float64(x * Float64(x / y))) / 2.0);
	else
		tmp = Float64(Float64(z * Float64(Float64(x - z) / y)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+126)
		tmp = (y + (x * (x / y))) / 2.0;
	else
		tmp = (z * ((x - z) / y)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+126], N[(N[(y + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(z * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999985e126

   1. Initial program 67.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 92.6%

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

   3. Simplified 92.6%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 84.7%

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

   7. Simplified 84.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 90.6%

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

   9. Applied egg-rr 90.6%

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

   if 1.99999999999999985e126 < (*.f64 z z)

   1. Initial program 66.5%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 71.5%

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

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 85.4%

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

   7. Simplified 85.4%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 72.5%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{y + x \cdot \frac{x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{x - z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+98}:\\ \;\;\;\;\frac{z \cdot \frac{x - z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.25e+98) (/ (* z (/ (- x z) y)) 2.0) (/ y 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.25e+98) {
		tmp = (z * ((x - z) / y)) / 2.0;
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.25d+98) then
        tmp = (z * ((x - z) / y)) / 2.0d0
    else
        tmp = y / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.25e+98) {
		tmp = (z * ((x - z) / y)) / 2.0;
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.25e+98:
		tmp = (z * ((x - z) / y)) / 2.0
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.25e+98)
		tmp = Float64(Float64(z * Float64(Float64(x - z) / y)) / 2.0);
	else
		tmp = Float64(y / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.25e+98)
		tmp = (z * ((x - z) / y)) / 2.0;
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.25e+98], N[(N[(z * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(y / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.25e98

   1. Initial program 72.7%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 86.8%

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

   3. Simplified 86.8%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 69.0%

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

   7. Simplified 69.0%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 39.5%

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

   if 1.25e98 < y

   1. Initial program 26.4%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 79.0%

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

   3. Simplified 79.0%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 84.4%

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

Alternative 9: 43.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+81}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.05e+81) (* x (* x (/ 0.5 y))) (/ y 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.05e+81) {
		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 <= 1.05d+81) 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 <= 1.05e+81) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.05e+81:
		tmp = x * (x * (0.5 / y))
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.05e+81)
		tmp = Float64(x * Float64(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 <= 1.05e+81)
		tmp = x * (x * (0.5 / y));
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.05e+81], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0499999999999999e81

   1. Initial program 72.8%

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

      1. associate-/r* N/A

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

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

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-inverses N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 87.3%

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

   3. Simplified 87.3%

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

      1. difference-of-squares N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. *-commutative N/A

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

      2. associate-/r/ N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      9. +-lowering-+.f64 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, x\right), y\right), \color{blue}{\left(\frac{1}{x}\right)}\right)\right), 2\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 74.8%

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

   11. Simplified 74.8%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-/l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*l/ N/A

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

       9. associate-*r/ N/A

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

       10. metadata-eval N/A

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

       11. associate-*r/ N/A

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

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

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. /-lowering-/.f64 39.6%

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

   14. Simplified 39.6%

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

   if 1.0499999999999999e81 < y

   1. Initial program 33.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 77.5%

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

   3. Simplified 77.5%

      \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
   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 76.5%

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

Alternative 10: 33.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{y}{2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.9%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 85.8%

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

3. Simplified 85.8%

   \[\leadsto \color{blue}{\frac{y - \frac{z \cdot z - x \cdot x}{y}}{2}} \]
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 38.9%

   \[\leadsto \frac{\color{blue}{y}}{2} \]
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 2024152 
(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)))