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

Percentage Accurate: 69.2% → 99.9%

Time: 10.0s

Alternatives: 13

Speedup: 1.2×

• 42.2% 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: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.5%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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. Simplified 84.2%

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

   1. +-commutative N/A

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

   2. difference-of-squares N/A

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

   3. associate-/l* N/A

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

   4. fma-define N/A

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

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

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

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

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

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

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

   8. --lowering--.f64 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x + z, \frac{x - z}{y}, y\right)}}{2} \]
7. Add Preprocessing

Alternative 2: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\frac{\left(z + y\right) \cdot \frac{y - z}{y}}{2}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(x \cdot x - z \cdot z\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+73)
   (/ (* (+ z y) (/ (- y z) y)) 2.0)
   (if (<= (* x x) 2e+307)
     (* (- (* x x) (* z z)) (/ 0.5 y))
     (/ (+ y (/ x (/ y x))) 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+73) {
		tmp = ((z + y) * ((y - z) / y)) / 2.0;
	} else if ((x * x) <= 2e+307) {
		tmp = ((x * x) - (z * z)) * (0.5 / y);
	} else {
		tmp = (y + (x / (y / x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 2d+73) then
        tmp = ((z + y) * ((y - z) / y)) / 2.0d0
    else if ((x * x) <= 2d+307) then
        tmp = ((x * x) - (z * z)) * (0.5d0 / y)
    else
        tmp = (y + (x / (y / x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+73) {
		tmp = ((z + y) * ((y - z) / y)) / 2.0;
	} else if ((x * x) <= 2e+307) {
		tmp = ((x * x) - (z * z)) * (0.5 / y);
	} else {
		tmp = (y + (x / (y / x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+73:
		tmp = ((z + y) * ((y - z) / y)) / 2.0
	elif (x * x) <= 2e+307:
		tmp = ((x * x) - (z * z)) * (0.5 / y)
	else:
		tmp = (y + (x / (y / x))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+73)
		tmp = Float64(Float64(Float64(z + y) * Float64(Float64(y - z) / y)) / 2.0);
	elseif (Float64(x * x) <= 2e+307)
		tmp = Float64(Float64(Float64(x * x) - Float64(z * z)) * Float64(0.5 / y));
	else
		tmp = Float64(Float64(y + Float64(x / Float64(y / x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+73)
		tmp = ((z + y) * ((y - z) / y)) / 2.0;
	elseif ((x * x) <= 2e+307)
		tmp = ((x * x) - (z * z)) * (0.5 / y);
	else
		tmp = (y + (x / (y / x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+73], N[(N[(N[(z + y), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+307], N[(N[(N[(x * x), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], N[(N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.99999999999999997e73

   1. Initial program 66.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 58.6%

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

   5. Simplified 58.6%

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

      1. associate-/r* N/A

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

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

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

      3. difference-of-squares N/A

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

      4. associate-/l* N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. --lowering--.f64 90.9%

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

   7. Applied egg-rr 90.9%

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

   if 1.99999999999999997e73 < (*.f64 x x) < 1.99999999999999997e307

   1. Initial program 90.6%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 87.3%

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

   7. Simplified 87.3%

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

   if 1.99999999999999997e307 < (*.f64 x x)

   1. Initial program 66.8%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 66.8%

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

      1. +-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-/l* N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

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

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 78.5%

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

   11. Simplified 78.5%

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

       1. associate-*l/ N/A

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

       2. associate-/r/ N/A

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

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

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

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

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

       5. /-lowering-/.f64 94.2%

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

   13. Applied egg-rr 94.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\frac{\left(z + y\right) \cdot \frac{y - z}{y}}{2}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(x \cdot x - z \cdot z\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(x \cdot x - z \cdot z\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+73)
   (/ (- y (* z (/ z y))) 2.0)
   (if (<= (* x x) 2e+307)
     (* (- (* x x) (* z z)) (/ 0.5 y))
     (/ (+ y (/ x (/ y x))) 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+73) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else if ((x * x) <= 2e+307) {
		tmp = ((x * x) - (z * z)) * (0.5 / y);
	} else {
		tmp = (y + (x / (y / x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 2d+73) then
        tmp = (y - (z * (z / y))) / 2.0d0
    else if ((x * x) <= 2d+307) then
        tmp = ((x * x) - (z * z)) * (0.5d0 / y)
    else
        tmp = (y + (x / (y / x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+73) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else if ((x * x) <= 2e+307) {
		tmp = ((x * x) - (z * z)) * (0.5 / y);
	} else {
		tmp = (y + (x / (y / x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+73:
		tmp = (y - (z * (z / y))) / 2.0
	elif (x * x) <= 2e+307:
		tmp = ((x * x) - (z * z)) * (0.5 / y)
	else:
		tmp = (y + (x / (y / x))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+73)
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	elseif (Float64(x * x) <= 2e+307)
		tmp = Float64(Float64(Float64(x * x) - Float64(z * z)) * Float64(0.5 / y));
	else
		tmp = Float64(Float64(y + Float64(x / Float64(y / x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+73)
		tmp = (y - (z * (z / y))) / 2.0;
	elseif ((x * x) <= 2e+307)
		tmp = ((x * x) - (z * z)) * (0.5 / y);
	else
		tmp = (y + (x / (y / x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+73], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+307], N[(N[(N[(x * x), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], N[(N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.99999999999999997e73

   1. Initial program 66.8%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 80.1%

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

   7. Simplified 80.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 90.9%

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

   9. Applied egg-rr 90.9%

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

   if 1.99999999999999997e73 < (*.f64 x x) < 1.99999999999999997e307

   1. Initial program 90.6%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 87.3%

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

   7. Simplified 87.3%

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

   if 1.99999999999999997e307 < (*.f64 x x)

   1. Initial program 66.8%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 66.8%

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

      1. +-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-/l* N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

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

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 78.5%

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

   11. Simplified 78.5%

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

       1. associate-*l/ N/A

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

       2. associate-/r/ N/A

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

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

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

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

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

       5. /-lowering-/.f64 94.2%

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

   13. Applied egg-rr 94.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(x \cdot x - z \cdot z\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\frac{\left(z + y\right) \cdot \frac{y - z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - z\right) \cdot \frac{x + z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+73)
   (/ (* (+ z y) (/ (- y z) y)) 2.0)
   (/ (* (- x z) (/ (+ x z) y)) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+73) {
		tmp = ((z + y) * ((y - z) / y)) / 2.0;
	} else {
		tmp = ((x - 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 ((x * x) <= 2d+73) then
        tmp = ((z + y) * ((y - z) / y)) / 2.0d0
    else
        tmp = ((x - 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 ((x * x) <= 2e+73) {
		tmp = ((z + y) * ((y - z) / y)) / 2.0;
	} else {
		tmp = ((x - z) * ((x + z) / y)) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+73:
		tmp = ((z + y) * ((y - z) / y)) / 2.0
	else:
		tmp = ((x - z) * ((x + z) / y)) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+73)
		tmp = Float64(Float64(Float64(z + y) * Float64(Float64(y - z) / y)) / 2.0);
	else
		tmp = Float64(Float64(Float64(x - z) * Float64(Float64(x + z) / y)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+73)
		tmp = ((z + y) * ((y - z) / y)) / 2.0;
	else
		tmp = ((x - z) * ((x + z) / y)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+73], N[(N[(N[(z + y), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x - z), $MachinePrecision] * N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999997e73

   1. Initial program 66.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 58.6%

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

   5. Simplified 58.6%

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

      1. associate-/r* N/A

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

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

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

      3. difference-of-squares N/A

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

      4. associate-/l* N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. --lowering--.f64 90.9%

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

   7. Applied egg-rr 90.9%

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

   if 1.99999999999999997e73 < (*.f64 x x)

   1. Initial program 75.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Simplified 78.8%

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

      1. +-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-/l* N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 94.0%

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

   9. Simplified 94.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\frac{\left(z + y\right) \cdot \frac{y - z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - z\right) \cdot \frac{x + z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e+153)
   (/ (- y (* z (/ z y))) 2.0)
   (/ (+ y (/ x (/ y x))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+153) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else {
		tmp = (y + (x / (y / x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 1d+153) then
        tmp = (y - (z * (z / y))) / 2.0d0
    else
        tmp = (y + (x / (y / x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+153) {
		tmp = (y - (z * (z / y))) / 2.0;
	} else {
		tmp = (y + (x / (y / x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e+153:
		tmp = (y - (z * (z / y))) / 2.0
	else:
		tmp = (y + (x / (y / x))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e+153)
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	else
		tmp = Float64(Float64(y + Float64(x / Float64(y / x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e+153)
		tmp = (y - (z * (z / y))) / 2.0;
	else
		tmp = (y + (x / (y / x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+153], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e153

   1. Initial program 69.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Simplified 89.3%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 78.3%

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

   7. Simplified 78.3%

      \[\leadsto \frac{\color{blue}{y - \frac{z \cdot z}{y}}}{2} \]
   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 88.1%

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

   9. Applied egg-rr 88.1%

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

   if 1e153 < (*.f64 x x)

   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. Simplified 75.5%

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

      1. +-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-/l* N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

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

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 77.8%

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

   11. Simplified 77.8%

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

       1. associate-*l/ N/A

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

       2. associate-/r/ N/A

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

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

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

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

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

       5. /-lowering-/.f64 89.4%

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

   13. Applied egg-rr 89.4%

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

4. Final simplification 88.6%

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

Alternative 6: 73.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.75 \cdot 10^{+113}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{z}} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.75e+113) (/ (+ y (/ x (/ y x))) 2.0) (* (/ z (/ y z)) -0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.75e+113) {
		tmp = (y + (x / (y / x))) / 2.0;
	} else {
		tmp = (z / (y / z)) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.75d+113) then
        tmp = (y + (x / (y / x))) / 2.0d0
    else
        tmp = (z / (y / z)) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.75e+113) {
		tmp = (y + (x / (y / x))) / 2.0;
	} else {
		tmp = (z / (y / z)) * -0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.75e+113:
		tmp = (y + (x / (y / x))) / 2.0
	else:
		tmp = (z / (y / z)) * -0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.75e+113)
		tmp = Float64(Float64(y + Float64(x / Float64(y / x))) / 2.0);
	else
		tmp = Float64(Float64(z / Float64(y / z)) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.75e+113)
		tmp = (y + (x / (y / x))) / 2.0;
	else
		tmp = (z / (y / z)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.75e+113], N[(N[(y + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.75e113

   1. Initial program 73.0%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 87.4%

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

      1. +-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-/l* N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

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

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 69.1%

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

   11. Simplified 69.1%

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

       1. associate-*l/ N/A

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

       2. associate-/r/ N/A

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

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

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

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

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

       5. /-lowering-/.f64 73.9%

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

   13. Applied egg-rr 73.9%

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

   if 1.75e113 < z

   1. Initial program 50.6%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 58.0%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. /-lowering-/.f64 63.4%

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

   7. Simplified 63.4%

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

      1. associate-*r* N/A

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

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

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

      6. /-lowering-/.f64 63.5%

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

   9. Applied egg-rr 63.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.75 \cdot 10^{+113}:\\ \;\;\;\;\frac{y + \frac{x}{\frac{y}{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{z}} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.55 \cdot 10^{+113}:\\ \;\;\;\;\frac{y + \frac{x \cdot x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y}{z}} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.55e+113) (/ (+ y (/ (* x x) y)) 2.0) (* (/ z (/ y z)) -0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.55e+113) {
		tmp = (y + ((x * x) / y)) / 2.0;
	} else {
		tmp = (z / (y / z)) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.55d+113) then
        tmp = (y + ((x * x) / y)) / 2.0d0
    else
        tmp = (z / (y / z)) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.55e+113) {
		tmp = (y + ((x * x) / y)) / 2.0;
	} else {
		tmp = (z / (y / z)) * -0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.55e+113:
		tmp = (y + ((x * x) / y)) / 2.0
	else:
		tmp = (z / (y / z)) * -0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.55e+113)
		tmp = Float64(Float64(y + Float64(Float64(x * x) / y)) / 2.0);
	else
		tmp = Float64(Float64(z / Float64(y / z)) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.55e+113)
		tmp = (y + ((x * x) / y)) / 2.0;
	else
		tmp = (z / (y / z)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.55e+113], N[(N[(y + N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.54999999999999997e113

   1. Initial program 73.0%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 69.1%

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

   7. Simplified 69.1%

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

   if 2.54999999999999997e113 < z

   1. Initial program 50.6%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 58.0%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. /-lowering-/.f64 63.4%

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

   7. Simplified 63.4%

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

      1. associate-*r* N/A

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

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

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

      6. /-lowering-/.f64 63.5%

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

   9. Applied egg-rr 63.5%

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

Alternative 8: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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. Simplified 84.2%

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

   1. difference-of-squares N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

   6. --lowering--.f64 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 9: 43.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{x}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 9.5e+37) (/ y 2.0) (/ (/ x (/ y x)) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 9.5e+37) {
		tmp = y / 2.0;
	} else {
		tmp = (x / (y / x)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 9.5d+37) then
        tmp = y / 2.0d0
    else
        tmp = (x / (y / x)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 9.5e+37) {
		tmp = y / 2.0;
	} else {
		tmp = (x / (y / x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 9.5e+37:
		tmp = y / 2.0
	else:
		tmp = (x / (y / x)) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 9.5e+37)
		tmp = Float64(y / 2.0);
	else
		tmp = Float64(Float64(x / Float64(y / x)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 9.5e+37)
		tmp = y / 2.0;
	else
		tmp = (x / (y / x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 9.5e+37], N[(y / 2.0), $MachinePrecision], N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.4999999999999995e37

   1. Initial program 68.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 38.8%

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

   if 9.4999999999999995e37 < x

   1. Initial program 79.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Simplified 80.9%

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

   5. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 65.0%

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

   7. Simplified 65.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 71.4%

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

   9. Applied egg-rr 71.4%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

       5. /-lowering-/.f64 71.4%

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

   11. Applied egg-rr 71.4%

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

Alternative 10: 43.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{2}{\frac{x}{y}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.4e+38) (/ y 2.0) (/ x (/ 2.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4e+38) {
		tmp = y / 2.0;
	} else {
		tmp = x / (2.0 / (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.4d+38) then
        tmp = y / 2.0d0
    else
        tmp = x / (2.0d0 / (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.4e+38) {
		tmp = y / 2.0;
	} else {
		tmp = x / (2.0 / (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.4e+38:
		tmp = y / 2.0
	else:
		tmp = x / (2.0 / (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.4e+38)
		tmp = Float64(y / 2.0);
	else
		tmp = Float64(x / Float64(2.0 / Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.4e+38)
		tmp = y / 2.0;
	else
		tmp = x / (2.0 / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.4e+38], N[(y / 2.0), $MachinePrecision], N[(x / N[(2.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e38

   1. Initial program 68.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 38.8%

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

   if 1.4e38 < x

   1. Initial program 79.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Simplified 80.9%

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

   5. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 65.0%

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

   7. Simplified 65.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 71.4%

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

   9. Applied egg-rr 71.4%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

       5. /-lowering-/.f64 71.4%

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

   11. Applied egg-rr 71.4%

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

       1. associate-/l/ N/A

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

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

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

       3. clear-num N/A

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

       4. un-div-inv N/A

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

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

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

       6. /-lowering-/.f64 71.4%

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

   13. Applied egg-rr 71.4%

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

Alternative 11: 42.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8.8e+39) (/ y 2.0) (* (* x x) (/ 0.5 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8.8e+39) {
		tmp = y / 2.0;
	} else {
		tmp = (x * x) * (0.5 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 8.8d+39) then
        tmp = y / 2.0d0
    else
        tmp = (x * x) * (0.5d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 8.8e+39) {
		tmp = y / 2.0;
	} else {
		tmp = (x * x) * (0.5 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 8.8e+39:
		tmp = y / 2.0
	else:
		tmp = (x * x) * (0.5 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8.8e+39)
		tmp = Float64(y / 2.0);
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 8.8e+39)
		tmp = y / 2.0;
	else
		tmp = (x * x) * (0.5 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8.8e+39], N[(y / 2.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.8000000000000006e39

   1. Initial program 68.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 38.8%

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

   if 8.8000000000000006e39 < x

   1. Initial program 79.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Simplified 80.9%

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

   5. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 65.0%

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

   7. Simplified 65.0%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

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

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

      8. metadata-eval 65.0%

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

   9. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \frac{0.5}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 44.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{+82}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.1e+82) (/ y 2.0) (* z (* (/ z y) -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.1e+82) {
		tmp = y / 2.0;
	} 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) :: tmp
    if (z <= 1.1d+82) then
        tmp = y / 2.0d0
    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 tmp;
	if (z <= 1.1e+82) {
		tmp = y / 2.0;
	} else {
		tmp = z * ((z / y) * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.1e+82:
		tmp = y / 2.0
	else:
		tmp = z * ((z / y) * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.1e+82)
		tmp = Float64(y / 2.0);
	else
		tmp = Float64(z * Float64(Float64(z / y) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.1e+82)
		tmp = y / 2.0;
	else
		tmp = z * ((z / y) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.1e+82], N[(y / 2.0), $MachinePrecision], N[(z * N[(N[(z / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.1000000000000001e82

   1. Initial program 72.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Simplified 87.0%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 34.7%

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

   if 1.1000000000000001e82 < z

   1. Initial program 57.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Simplified 66.4%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. /-lowering-/.f64 56.4%

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

   7. Simplified 56.4%

      \[\leadsto \color{blue}{z \cdot \left(\frac{z}{y} \cdot -0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 34.0% 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 70.5%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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. Simplified 84.2%

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

5. Taylor expanded in y around inf

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

7. Simplified 32.4%

   \[\leadsto \frac{\color{blue}{y}}{2} \]
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 2024138 
(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)))