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

Percentage Accurate: 68.9% → 99.9%

Time: 10.7s

Alternatives: 13

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

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 72.0%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 83.0%

   \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
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 2: 43.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\frac{y}{x \cdot x}}\\ \mathbf{if}\;z \leq 6 \cdot 10^{-245}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-144}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{2}\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ 0.5 (/ y (* x x)))))
   (if (<= z 6e-245)
     (/ y 2.0)
     (if (<= z 7e-144)
       t_0
       (if (<= z 1.6e-69)
         (/ y 2.0)
         (if (<= z 8.4e+45) t_0 (* z (* (/ z y) -0.5))))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 / (y / (x * x));
	double tmp;
	if (z <= 6e-245) {
		tmp = y / 2.0;
	} else if (z <= 7e-144) {
		tmp = t_0;
	} else if (z <= 1.6e-69) {
		tmp = y / 2.0;
	} else if (z <= 8.4e+45) {
		tmp = t_0;
	} else {
		tmp = z * ((z / y) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 / (y / (x * x))
    if (z <= 6d-245) then
        tmp = y / 2.0d0
    else if (z <= 7d-144) then
        tmp = t_0
    else if (z <= 1.6d-69) then
        tmp = y / 2.0d0
    else if (z <= 8.4d+45) then
        tmp = t_0
    else
        tmp = z * ((z / y) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.5 / (y / (x * x));
	double tmp;
	if (z <= 6e-245) {
		tmp = y / 2.0;
	} else if (z <= 7e-144) {
		tmp = t_0;
	} else if (z <= 1.6e-69) {
		tmp = y / 2.0;
	} else if (z <= 8.4e+45) {
		tmp = t_0;
	} else {
		tmp = z * ((z / y) * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 / (y / (x * x))
	tmp = 0
	if z <= 6e-245:
		tmp = y / 2.0
	elif z <= 7e-144:
		tmp = t_0
	elif z <= 1.6e-69:
		tmp = y / 2.0
	elif z <= 8.4e+45:
		tmp = t_0
	else:
		tmp = z * ((z / y) * -0.5)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 / Float64(y / Float64(x * x)))
	tmp = 0.0
	if (z <= 6e-245)
		tmp = Float64(y / 2.0);
	elseif (z <= 7e-144)
		tmp = t_0;
	elseif (z <= 1.6e-69)
		tmp = Float64(y / 2.0);
	elseif (z <= 8.4e+45)
		tmp = t_0;
	else
		tmp = Float64(z * Float64(Float64(z / y) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.5 / (y / (x * x));
	tmp = 0.0;
	if (z <= 6e-245)
		tmp = y / 2.0;
	elseif (z <= 7e-144)
		tmp = t_0;
	elseif (z <= 1.6e-69)
		tmp = y / 2.0;
	elseif (z <= 8.4e+45)
		tmp = t_0;
	else
		tmp = z * ((z / y) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 6e-245], N[(y / 2.0), $MachinePrecision], If[LessEqual[z, 7e-144], t$95$0, If[LessEqual[z, 1.6e-69], N[(y / 2.0), $MachinePrecision], If[LessEqual[z, 8.4e+45], t$95$0, N[(z * N[(N[(z / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 6.0000000000000004e-245 or 6.9999999999999997e-144 < z < 1.59999999999999999e-69

   1. Initial program 71.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 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 32.6%

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

   if 6.0000000000000004e-245 < z < 6.9999999999999997e-144 or 1.59999999999999999e-69 < z < 8.39999999999999979e45

   1. Initial program 76.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 86.0%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 47.1%

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

   7. Simplified 47.1%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. *-lowering-*.f64 47.1%

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

   9. Applied egg-rr 47.1%

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

   if 8.39999999999999979e45 < z

   1. Initial program 70.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 76.4%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 65.5%

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

   7. Simplified 65.5%

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

Alternative 3: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{\left(x + z\right) \cdot \frac{x - z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{-1}{\frac{\frac{y}{z}}{z}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.5e+82)
   (/ (* (+ x z) (/ (- x z) y)) 2.0)
   (/ (+ y (/ -1.0 (/ (/ y z) z))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.5e+82) {
		tmp = ((x + z) * ((x - z) / y)) / 2.0;
	} else {
		tmp = (y + (-1.0 / ((y / z) / z))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 9.5d+82) then
        tmp = ((x + z) * ((x - z) / y)) / 2.0d0
    else
        tmp = (y + ((-1.0d0) / ((y / z) / z))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.5e+82) {
		tmp = ((x + z) * ((x - z) / y)) / 2.0;
	} else {
		tmp = (y + (-1.0 / ((y / z) / z))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9.5e+82:
		tmp = ((x + z) * ((x - z) / y)) / 2.0
	else:
		tmp = (y + (-1.0 / ((y / z) / z))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.5e+82)
		tmp = Float64(Float64(Float64(x + z) * Float64(Float64(x - z) / y)) / 2.0);
	else
		tmp = Float64(Float64(y + Float64(-1.0 / Float64(Float64(y / z) / z))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9.5e+82)
		tmp = ((x + z) * ((x - z) / y)) / 2.0;
	else
		tmp = (y + (-1.0 / ((y / z) / z))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.5e+82], N[(N[(N[(x + z), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y + N[(-1.0 / N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.50000000000000049e82

   1. Initial program 79.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 83.9%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. clear-num N/A

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

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

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

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

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

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

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

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

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

      8. +-lowering-+.f64 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around 0

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

       1. associate-/l* N/A

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

       2. div-sub N/A

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

       3. unsub-neg N/A

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

       4. mul-1-neg N/A

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

       5. +-commutative N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. mul-1-neg N/A

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

       10. unsub-neg N/A

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

       11. div-sub N/A

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

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

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

       13. --lowering--.f64 80.6%

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

   11. Simplified 80.6%

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

   if 9.50000000000000049e82 < y

   1. Initial program 40.3%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 86.4%

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

   7. Simplified 86.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 90.6%

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

   9. Applied egg-rr 90.6%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

       4. clear-num N/A

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

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

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

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

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

       7. /-lowering-/.f64 90.7%

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

   11. Applied egg-rr 90.7%

       \[\leadsto \frac{y - \color{blue}{\frac{1}{\frac{\frac{y}{z}}{z}}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{\left(x + z\right) \cdot \frac{x - z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \frac{-1}{\frac{\frac{y}{z}}{z}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{\left(x + z\right) \cdot \frac{x - z}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{z}{\frac{y}{z}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.8e+78)
   (/ (* (+ x z) (/ (- x z) y)) 2.0)
   (/ (- y (/ z (/ y z))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.8e+78) {
		tmp = ((x + z) * ((x - z) / y)) / 2.0;
	} else {
		tmp = (y - (z / (y / z))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 9.8d+78) then
        tmp = ((x + z) * ((x - z) / y)) / 2.0d0
    else
        tmp = (y - (z / (y / z))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.8e+78) {
		tmp = ((x + z) * ((x - z) / y)) / 2.0;
	} else {
		tmp = (y - (z / (y / z))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9.8e+78:
		tmp = ((x + z) * ((x - z) / y)) / 2.0
	else:
		tmp = (y - (z / (y / z))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.8e+78)
		tmp = Float64(Float64(Float64(x + z) * Float64(Float64(x - z) / y)) / 2.0);
	else
		tmp = Float64(Float64(y - Float64(z / Float64(y / z))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9.8e+78)
		tmp = ((x + z) * ((x - z) / y)) / 2.0;
	else
		tmp = (y - (z / (y / z))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.8e+78], N[(N[(N[(x + z), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.8000000000000004e78

   1. Initial program 78.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 83.8%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. clear-num N/A

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

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

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

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

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

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

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

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

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

      8. +-lowering-+.f64 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around 0

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

       1. associate-/l* N/A

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

       2. div-sub N/A

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

       3. unsub-neg N/A

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

       4. mul-1-neg N/A

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

       5. +-commutative N/A

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

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

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

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

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

       8. +-commutative N/A

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

       9. mul-1-neg N/A

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

       10. unsub-neg N/A

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

       11. div-sub N/A

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

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

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

       13. --lowering--.f64 80.5%

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

   11. Simplified 80.5%

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

   if 9.8000000000000004e78 < y

   1. Initial program 41.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 79.5%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 86.7%

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

   7. Simplified 86.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 90.8%

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

   9. Applied egg-rr 90.8%

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

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

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

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

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

       3. *-commutative N/A

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

       4. clear-num N/A

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

       5. un-div-inv N/A

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

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

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

       7. /-lowering-/.f64 90.8%

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

   11. Applied egg-rr 90.8%

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

Alternative 5: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{x + z}{\frac{y}{x - z}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{z}{\frac{y}{z}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.2e+72)
   (* (/ (+ x z) (/ y (- x z))) 0.5)
   (/ (- y (/ z (/ y z))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.2e+72) {
		tmp = ((x + z) / (y / (x - z))) * 0.5;
	} else {
		tmp = (y - (z / (y / z))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.2d+72) then
        tmp = ((x + z) / (y / (x - z))) * 0.5d0
    else
        tmp = (y - (z / (y / z))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.2e+72) {
		tmp = ((x + z) / (y / (x - z))) * 0.5;
	} else {
		tmp = (y - (z / (y / z))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.2e+72:
		tmp = ((x + z) / (y / (x - z))) * 0.5
	else:
		tmp = (y - (z / (y / z))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.2e+72)
		tmp = Float64(Float64(Float64(x + z) / Float64(y / Float64(x - z))) * 0.5);
	else
		tmp = Float64(Float64(y - Float64(z / Float64(y / z))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.2e+72)
		tmp = ((x + z) / (y / (x - z))) * 0.5;
	else
		tmp = (y - (z / (y / z))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.2e+72], N[(N[(N[(x + z), $MachinePrecision] / N[(y / N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2000000000000001e72

   1. Initial program 78.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 83.8%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 68.3%

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

   7. Simplified 68.3%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

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

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

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

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

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

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

      15. --lowering--.f64 80.5%

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

   9. Applied egg-rr 80.5%

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

   if 3.2000000000000001e72 < y

   1. Initial program 41.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 79.5%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 86.7%

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

   7. Simplified 86.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 90.8%

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

   9. Applied egg-rr 90.8%

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

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

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

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

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

       3. *-commutative N/A

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

       4. clear-num N/A

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

       5. un-div-inv N/A

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

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

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

       7. /-lowering-/.f64 90.8%

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

   11. Applied egg-rr 90.8%

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

Alternative 6: 69.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+44}:\\ \;\;\;\;\left(x \cdot x - z \cdot z\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.5e+44)
   (* (- (* x x) (* z z)) (/ 0.5 y))
   (/ (- y (* z (/ z y))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.5e+44) {
		tmp = ((x * x) - (z * z)) * (0.5 / y);
	} else {
		tmp = (y - (z * (z / y))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 9.5d+44) then
        tmp = ((x * x) - (z * z)) * (0.5d0 / y)
    else
        tmp = (y - (z * (z / y))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.5e+44) {
		tmp = ((x * x) - (z * z)) * (0.5 / y);
	} else {
		tmp = (y - (z * (z / y))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9.5e+44:
		tmp = ((x * x) - (z * z)) * (0.5 / y)
	else:
		tmp = (y - (z * (z / y))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.5e+44)
		tmp = Float64(Float64(Float64(x * x) - Float64(z * z)) * Float64(0.5 / y));
	else
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9.5e+44)
		tmp = ((x * x) - (z * z)) * (0.5 / y);
	else
		tmp = (y - (z * (z / y))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.5e+44], N[(N[(N[(x * x), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.5000000000000004e44

   1. Initial program 78.5%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 83.5%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 67.7%

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

   7. Simplified 67.7%

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

   if 9.5000000000000004e44 < y

   1. Initial program 46.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 81.1%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 84.1%

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

   7. Simplified 84.1%

      \[\leadsto \frac{\color{blue}{y - \frac{z \cdot z}{y}}}{2} \]
   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 87.9%

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

   9. Applied egg-rr 87.9%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+44}:\\ \;\;\;\;\left(x \cdot x - z \cdot z\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{y + \frac{x \cdot x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+76)
   (/ (+ y (/ (* x x) y)) 2.0)
   (/ (- y (* z (/ z y))) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+76) {
		tmp = (y + ((x * x) / y)) / 2.0;
	} else {
		tmp = (y - (z * (z / y))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+76) then
        tmp = (y + ((x * x) / y)) / 2.0d0
    else
        tmp = (y - (z * (z / y))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+76) {
		tmp = (y + ((x * x) / y)) / 2.0;
	} else {
		tmp = (y - (z * (z / y))) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+76:
		tmp = (y + ((x * x) / y)) / 2.0
	else:
		tmp = (y - (z * (z / y))) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+76)
		tmp = Float64(Float64(y + Float64(Float64(x * x) / y)) / 2.0);
	else
		tmp = Float64(Float64(y - Float64(z * Float64(z / y))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+76)
		tmp = (y + ((x * x) / y)) / 2.0;
	else
		tmp = (y - (z * (z / y))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+76], N[(N[(y + N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999991e76

   1. Initial program 76.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 92.4%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 85.2%

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

   7. Simplified 85.2%

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

   if 4.99999999999999991e76 < (*.f64 z z)

   1. Initial program 67.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 72.4%

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

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

   7. Simplified 72.1%

      \[\leadsto \frac{\color{blue}{y - \frac{z \cdot z}{y}}}{2} \]
   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 81.7%

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

   9. Applied egg-rr 81.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{y + \frac{x \cdot x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z \cdot \frac{z}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+183}:\\ \;\;\;\;\frac{y + \frac{x \cdot x}{y}}{2}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+183) (/ (+ y (/ (* x x) y)) 2.0) (* z (* (/ z y) -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+183) {
		tmp = (y + ((x * x) / 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 * z) <= 5d+183) then
        tmp = (y + ((x * x) / 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 * z) <= 5e+183) {
		tmp = (y + ((x * x) / y)) / 2.0;
	} else {
		tmp = z * ((z / y) * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+183:
		tmp = (y + ((x * x) / y)) / 2.0
	else:
		tmp = z * ((z / y) * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+183)
		tmp = Float64(Float64(y + Float64(Float64(x * x) / 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 * z) <= 5e+183)
		tmp = (y + ((x * x) / y)) / 2.0;
	else
		tmp = z * ((z / y) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+183], N[(N[(y + N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(z * N[(N[(z / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000009e183

   1. Initial program 76.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 91.4%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 79.2%

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

   7. Simplified 79.2%

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

   if 5.00000000000000009e183 < (*.f64 z z)

   1. Initial program 64.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 68.3%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 72.5%

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

   7. Simplified 72.5%

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

Alternative 9: 52.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-20}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e-20) (* z (* (/ z y) -0.5)) (/ (* x (/ x y)) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e-20) {
		tmp = z * ((z / y) * -0.5);
	} else {
		tmp = (x * (x / y)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 1d-20) then
        tmp = z * ((z / y) * (-0.5d0))
    else
        tmp = (x * (x / y)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e-20) {
		tmp = z * ((z / y) * -0.5);
	} else {
		tmp = (x * (x / y)) / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e-20:
		tmp = z * ((z / y) * -0.5)
	else:
		tmp = (x * (x / y)) / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e-20)
		tmp = Float64(z * Float64(Float64(z / y) * -0.5));
	else
		tmp = Float64(Float64(x * Float64(x / y)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e-20)
		tmp = z * ((z / y) * -0.5);
	else
		tmp = (x * (x / y)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-20], N[(z * N[(N[(z / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.99999999999999945e-21

   1. Initial program 76.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 93.1%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 56.3%

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

   7. Simplified 56.3%

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

   if 9.99999999999999945e-21 < (*.f64 x x)

   1. Initial program 67.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 73.8%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 59.3%

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

   7. Simplified 59.3%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot x}{y}}}{2} \]
   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 62.0%

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

   9. Applied egg-rr 62.0%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-20}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-20}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{y}{x}}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e-20) (* z (* (/ z y) -0.5)) (/ 0.5 (/ (/ y x) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e-20) {
		tmp = z * ((z / y) * -0.5);
	} else {
		tmp = 0.5 / ((y / x) / x);
	}
	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-20) then
        tmp = z * ((z / y) * (-0.5d0))
    else
        tmp = 0.5d0 / ((y / x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e-20) {
		tmp = z * ((z / y) * -0.5);
	} else {
		tmp = 0.5 / ((y / x) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e-20:
		tmp = z * ((z / y) * -0.5)
	else:
		tmp = 0.5 / ((y / x) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e-20)
		tmp = Float64(z * Float64(Float64(z / y) * -0.5));
	else
		tmp = Float64(0.5 / Float64(Float64(y / x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e-20)
		tmp = z * ((z / y) * -0.5);
	else
		tmp = 0.5 / ((y / x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-20], N[(z * N[(N[(z / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.99999999999999945e-21

   1. Initial program 76.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 93.1%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 56.3%

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

   7. Simplified 56.3%

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

   if 9.99999999999999945e-21 < (*.f64 x x)

   1. Initial program 67.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 73.8%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 59.3%

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

   7. Simplified 59.3%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. *-lowering-*.f64 59.3%

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

   9. Applied egg-rr 59.3%

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

       1. associate-/r* N/A

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

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

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

       3. /-lowering-/.f64 61.9%

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

   11. Applied egg-rr 61.9%

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

Alternative 11: 43.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+110}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.5e+110) (* z (* (/ z y) -0.5)) (/ y 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.5e+110) {
		tmp = z * ((z / y) * -0.5);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 7.5d+110) then
        tmp = z * ((z / y) * (-0.5d0))
    else
        tmp = y / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.5e+110) {
		tmp = z * ((z / y) * -0.5);
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.5e+110:
		tmp = z * ((z / y) * -0.5)
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.5e+110)
		tmp = Float64(z * Float64(Float64(z / y) * -0.5));
	else
		tmp = Float64(y / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7.5e+110)
		tmp = z * ((z / y) * -0.5);
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.5e+110], N[(z * N[(N[(z / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(y / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5e110

   1. Initial program 78.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 83.4%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 40.6%

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

   7. Simplified 40.6%

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

   if 7.5e110 < y

   1. Initial program 36.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 81.0%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 79.3%

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

Alternative 12: 43.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+110}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e+110) (* z (* z (/ -0.5 y))) (/ y 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e+110) {
		tmp = z * (z * (-0.5 / y));
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 6.5d+110) then
        tmp = z * (z * ((-0.5d0) / y))
    else
        tmp = y / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e+110) {
		tmp = z * (z * (-0.5 / y));
	} else {
		tmp = y / 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.5e+110:
		tmp = z * (z * (-0.5 / y))
	else:
		tmp = y / 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.5e+110)
		tmp = Float64(z * Float64(z * Float64(-0.5 / y)));
	else
		tmp = Float64(y / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.5e+110)
		tmp = z * (z * (-0.5 / y));
	else
		tmp = y / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.5e+110], N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4999999999999997e110

   1. Initial program 78.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 83.4%

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

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

      1. *-commutative N/A

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

      2. 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-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 40.6%

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

   9. Simplified 40.6%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. /-lowering-/.f64 40.5%

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

   11. Applied egg-rr 40.5%

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

   if 6.4999999999999997e110 < y

   1. Initial program 36.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 81.0%

      \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
   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 79.3%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+110}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 83.0%

   \[\leadsto \color{blue}{\frac{y + \frac{x \cdot x - z \cdot z}{y}}{2}} \]
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 30.2%

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

Developer Target 1: 99.8% 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 2024145 
(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)))