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

Percentage Accurate: 68.9% → 99.9%

Time: 9.7s

Alternatives: 7

Speedup: 1.4×

• 40.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in y around 0 84.4%

   \[\leadsto \color{blue}{0.5 \cdot y + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
4. Step-by-step derivation

   1. distribute-lft-out 84.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]

5. Simplified 84.4%

   \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
6. Step-by-step derivation

   1. div-inv 84.4%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left({x}^{2} - {z}^{2}\right) \cdot \frac{1}{y}}\right) \]

7. Applied egg-rr 84.4%

   \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left({x}^{2} - {z}^{2}\right) \cdot \frac{1}{y}}\right) \]
8. Step-by-step derivation

   1. div-inv 84.4%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   2. unpow2 84.4%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

   3. unpow2 84.4%

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

   4. difference-of-squares 89.5%

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

   5. associate-/l* 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

    \[\leadsto 0.5 \cdot \left(y + \frac{x + z}{\frac{y}{x - z}}\right) \]
11. Add Preprocessing

Alternative 2: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+104}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x + z}{\frac{-y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x + z}{\frac{y}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.7e+104)
   (* 0.5 (+ y (/ (+ x z) (/ (- y) z))))
   (* 0.5 (+ y (/ (+ x z) (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.7e+104) {
		tmp = 0.5 * (y + ((x + z) / (-y / z)));
	} else {
		tmp = 0.5 * (y + ((x + z) / (y / 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 <= 2.7d+104) then
        tmp = 0.5d0 * (y + ((x + z) / (-y / z)))
    else
        tmp = 0.5d0 * (y + ((x + z) / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.7e+104) {
		tmp = 0.5 * (y + ((x + z) / (-y / z)));
	} else {
		tmp = 0.5 * (y + ((x + z) / (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.7e+104:
		tmp = 0.5 * (y + ((x + z) / (-y / z)))
	else:
		tmp = 0.5 * (y + ((x + z) / (y / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.7e+104)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x + z) / Float64(Float64(-y) / z))));
	else
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x + z) / Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.7e+104)
		tmp = 0.5 * (y + ((x + z) / (-y / z)));
	else
		tmp = 0.5 * (y + ((x + z) / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.7e+104], N[(0.5 * N[(y + N[(N[(x + z), $MachinePrecision] / N[((-y) / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(N[(x + z), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.69999999999999985e104

   1. Initial program 63.2%

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

   3. Taylor expanded in y around 0 86.5%

      \[\leadsto \color{blue}{0.5 \cdot y + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. distribute-lft-out 86.5%

         \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]

   5. Simplified 86.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. div-inv 86.5%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left({x}^{2} - {z}^{2}\right) \cdot \frac{1}{y}}\right) \]

   7. Applied egg-rr 86.5%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left({x}^{2} - {z}^{2}\right) \cdot \frac{1}{y}}\right) \]
   8. Step-by-step derivation

      1. div-inv 86.5%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

      2. unpow2 86.5%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      3. unpow2 86.5%

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

      4. difference-of-squares 89.3%

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

      5. associate-/l* 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around 0 82.3%

       \[\leadsto 0.5 \cdot \left(y + \frac{x + z}{\color{blue}{-1 \cdot \frac{y}{z}}}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 82.3%

          \[\leadsto 0.5 \cdot \left(y + \frac{x + z}{\color{blue}{\frac{-1 \cdot y}{z}}}\right) \]

       2. neg-mul-1 82.3%

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

   12. Simplified 82.3%

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

   if 2.69999999999999985e104 < x

   1. Initial program 68.4%

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

   3. Taylor expanded in y around 0 73.1%

      \[\leadsto \color{blue}{0.5 \cdot y + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. distribute-lft-out 73.1%

         \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]

   5. Simplified 73.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. div-inv 73.0%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left({x}^{2} - {z}^{2}\right) \cdot \frac{1}{y}}\right) \]

   7. Applied egg-rr 73.0%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left({x}^{2} - {z}^{2}\right) \cdot \frac{1}{y}}\right) \]
   8. Step-by-step derivation

      1. div-inv 73.1%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

      2. unpow2 73.1%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      3. unpow2 73.1%

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

      4. difference-of-squares 90.6%

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

      5. associate-/l* 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around inf 89.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+104}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x + z}{\frac{-y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x + z}{\frac{y}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{+38}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x + z}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\left(x + z\right) \cdot \left(x - z\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 3.1e+38)
   (* 0.5 (+ y (/ (+ x z) (/ y x))))
   (* 0.5 (/ (* (+ x z) (- x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.1e+38) {
		tmp = 0.5 * (y + ((x + z) / (y / x)));
	} else {
		tmp = 0.5 * (((x + z) * (x - z)) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 3.1d+38) then
        tmp = 0.5d0 * (y + ((x + z) / (y / x)))
    else
        tmp = 0.5d0 * (((x + z) * (x - z)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.1e+38) {
		tmp = 0.5 * (y + ((x + z) / (y / x)));
	} else {
		tmp = 0.5 * (((x + z) * (x - z)) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 3.1e+38:
		tmp = 0.5 * (y + ((x + z) / (y / x)))
	else:
		tmp = 0.5 * (((x + z) * (x - z)) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.1e+38)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x + z) / Float64(y / x))));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(x + z) * Float64(x - z)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 3.1e+38)
		tmp = 0.5 * (y + ((x + z) / (y / x)));
	else
		tmp = 0.5 * (((x + z) * (x - z)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.1e+38], N[(0.5 * N[(y + N[(N[(x + z), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(x + z), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.10000000000000018e38

   1. Initial program 64.3%

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

   3. Taylor expanded in y around 0 87.6%

      \[\leadsto \color{blue}{0.5 \cdot y + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. distribute-lft-out 87.6%

         \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]

   5. Simplified 87.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. div-inv 87.6%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left({x}^{2} - {z}^{2}\right) \cdot \frac{1}{y}}\right) \]

   7. Applied egg-rr 87.6%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left({x}^{2} - {z}^{2}\right) \cdot \frac{1}{y}}\right) \]
   8. Step-by-step derivation

      1. div-inv 87.6%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

      2. unpow2 87.6%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      3. unpow2 87.6%

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

      4. difference-of-squares 90.5%

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

      5. associate-/l* 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around inf 75.1%

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

   if 3.10000000000000018e38 < z

   1. Initial program 63.0%

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

   3. Taylor expanded in y around 0 72.4%

      \[\leadsto \color{blue}{0.5 \cdot y + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. distribute-lft-out 72.4%

         \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]

   5. Simplified 72.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. unpow2 72.4%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      2. unpow2 72.4%

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

      3. difference-of-squares 85.6%

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

   7. Applied egg-rr 85.6%

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

   8. Taylor expanded in y around 0 76.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{+38}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x + z}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\left(x + z\right) \cdot \left(x - z\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in y around 0 84.4%

   \[\leadsto \color{blue}{0.5 \cdot y + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
4. Step-by-step derivation

   1. distribute-lft-out 84.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]

5. Simplified 84.4%

   \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
6. Step-by-step derivation

   1. div-inv 84.4%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left({x}^{2} - {z}^{2}\right) \cdot \frac{1}{y}}\right) \]

7. Applied egg-rr 84.4%

   \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left({x}^{2} - {z}^{2}\right) \cdot \frac{1}{y}}\right) \]
8. Step-by-step derivation

   1. div-inv 84.4%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   2. unpow2 84.4%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

   3. unpow2 84.4%

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

   4. difference-of-squares 89.5%

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

   5. associate-/l* 99.9%

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

9. Applied egg-rr 99.9%

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

    1. associate-/r/ 99.9%

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

    \[\leadsto 0.5 \cdot \left(y + \left(x - z\right) \cdot \frac{x + z}{y}\right) \]
13. Add Preprocessing

Alternative 5: 44.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.2e+43) (* 0.5 y) (* (/ x y) (/ x 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.2e+43) {
		tmp = 0.5 * y;
	} else {
		tmp = (x / y) * (x / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 6.2d+43) then
        tmp = 0.5d0 * y
    else
        tmp = (x / y) * (x / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.2e+43) {
		tmp = 0.5 * y;
	} else {
		tmp = (x / y) * (x / 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 6.2e+43:
		tmp = 0.5 * y
	else:
		tmp = (x / y) * (x / 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.2e+43)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 6.2e+43)
		tmp = 0.5 * y;
	else
		tmp = (x / y) * (x / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.2e+43], N[(0.5 * y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.2000000000000003e43

   1. Initial program 62.5%

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

   3. Taylor expanded in y around inf 43.8%

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

   if 6.2000000000000003e43 < x

   1. Initial program 70.4%

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

   3. Taylor expanded in x around inf 66.5%

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

      1. unpow2 66.5%

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

      2. times-frac 66.5%

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

   5. Applied egg-rr 66.5%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 0.5 * (y + ((x + z) / (y / 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 = 0.5d0 * (y + ((x + z) / (y / x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in y around 0 84.4%

   \[\leadsto \color{blue}{0.5 \cdot y + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
4. Step-by-step derivation

   1. distribute-lft-out 84.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]

5. Simplified 84.4%

   \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
6. Step-by-step derivation

   1. div-inv 84.4%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left({x}^{2} - {z}^{2}\right) \cdot \frac{1}{y}}\right) \]

7. Applied egg-rr 84.4%

   \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left({x}^{2} - {z}^{2}\right) \cdot \frac{1}{y}}\right) \]
8. Step-by-step derivation

   1. div-inv 84.4%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   2. unpow2 84.4%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

   3. unpow2 84.4%

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

   4. difference-of-squares 89.5%

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

   5. associate-/l* 99.9%

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

9. Applied egg-rr 99.9%

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

10. Taylor expanded in x around inf 69.1%

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

11. Final simplification 69.1%

    \[\leadsto 0.5 \cdot \left(y + \frac{x + z}{\frac{y}{x}}\right) \]
12. Add Preprocessing

Alternative 7: 34.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 y))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in y around inf 38.9%

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

4. Final simplification 38.9%

   \[\leadsto 0.5 \cdot y \]
5. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024034 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))