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

Percentage Accurate: 69.6% → 99.9%

Time: 9.6s

Alternatives: 6

Speedup: 1.2×

• 41.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

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 71.5%

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

3. Taylor expanded in y around 0 86.7%

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

   1. distribute-lft-out 86.7%

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

5. Simplified 86.7%

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

   1. unpow2 86.7%

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

   2. unpow2 86.7%

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

   3. difference-of-squares 90.0%

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

7. Applied egg-rr 90.0%

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

   1. div-inv 89.9%

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

   2. associate-*l* 99.9%

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

9. Applied egg-rr 99.9%

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

10. Taylor expanded in x around 0 98.4%

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

    1. +-commutative 98.4%

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

    2. mul-1-neg 98.4%

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

    3. sub-neg 98.4%

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

    4. div-sub 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{+108} \lor \neg \left(y \leq 7.8 \cdot 10^{+150}\right) \land y \leq 3.2 \cdot 10^{+166}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y 6.6e+108) (and (not (<= y 7.8e+150)) (<= y 3.2e+166)))
   (* 0.5 (* (+ x z) (/ (- x z) y)))
   (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= 6.6e+108) || (!(y <= 7.8e+150) && (y <= 3.2e+166))) {
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= 6.6d+108) .or. (.not. (y <= 7.8d+150)) .and. (y <= 3.2d+166)) then
        tmp = 0.5d0 * ((x + z) * ((x - z) / y))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= 6.6e+108) || (!(y <= 7.8e+150) && (y <= 3.2e+166))) {
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= 6.6e+108) or (not (y <= 7.8e+150) and (y <= 3.2e+166)):
		tmp = 0.5 * ((x + z) * ((x - z) / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= 6.6e+108) || (!(y <= 7.8e+150) && (y <= 3.2e+166)))
		tmp = Float64(0.5 * Float64(Float64(x + z) * Float64(Float64(x - z) / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= 6.6e+108) || (~((y <= 7.8e+150)) && (y <= 3.2e+166)))
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, 6.6e+108], And[N[Not[LessEqual[y, 7.8e+150]], $MachinePrecision], LessEqual[y, 3.2e+166]]], N[(0.5 * N[(N[(x + z), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.60000000000000038e108 or 7.79999999999999981e150 < y < 3.19999999999999968e166

   1. Initial program 78.1%

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

   3. Taylor expanded in y around 0 89.0%

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

      1. distribute-lft-out 89.0%

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

   5. Simplified 89.0%

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

      1. unpow2 89.0%

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

      2. unpow2 89.0%

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

      3. difference-of-squares 92.7%

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

   7. Applied egg-rr 92.7%

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

   8. Taylor expanded in y around 0 69.3%

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

      1. associate-*r/ 73.2%

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

      2. +-commutative 73.2%

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

   10. Simplified 73.2%

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

   if 6.60000000000000038e108 < y < 7.79999999999999981e150 or 3.19999999999999968e166 < y

   1. Initial program 28.0%

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

   3. Taylor expanded in y around inf 77.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{+108} \lor \neg \left(y \leq 7.8 \cdot 10^{+150}\right) \land y \leq 3.2 \cdot 10^{+166}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \left(x + z\right) \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.5e+70)
   (* 0.5 (* (+ x z) (/ (- x z) y)))
   (* 0.5 (+ y (* (+ x z) (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5e+70) {
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	} else {
		tmp = 0.5 * (y + ((x + z) * (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.5d+70) then
        tmp = 0.5d0 * ((x + z) * ((x - z) / y))
    else
        tmp = 0.5d0 * (y + ((x + z) * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5e+70) {
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	} else {
		tmp = 0.5 * (y + ((x + z) * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.5e+70:
		tmp = 0.5 * ((x + z) * ((x - z) / y))
	else:
		tmp = 0.5 * (y + ((x + z) * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.5e+70)
		tmp = Float64(0.5 * Float64(Float64(x + z) * Float64(Float64(x - z) / y)));
	else
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x + z) * Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.5e+70)
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	else
		tmp = 0.5 * (y + ((x + z) * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.5e+70], N[(0.5 * N[(N[(x + z), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(N[(x + z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.49999999999999988e70

   1. Initial program 78.7%

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

   3. Taylor expanded in y around 0 89.3%

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

      1. distribute-lft-out 89.3%

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

   5. Simplified 89.3%

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

      1. unpow2 89.3%

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

      2. unpow2 89.3%

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

      3. difference-of-squares 92.7%

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

   7. Applied egg-rr 92.7%

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

   8. Taylor expanded in y around 0 71.2%

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

      1. associate-*r/ 74.9%

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

      2. +-commutative 74.9%

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

   10. Simplified 74.9%

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

   if 1.49999999999999988e70 < y

   1. Initial program 38.3%

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

   3. Taylor expanded in y around 0 75.3%

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

      1. distribute-lft-out 75.3%

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

   5. Simplified 75.3%

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

      1. unpow2 75.3%

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

      2. unpow2 75.3%

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

      3. difference-of-squares 77.4%

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

   7. Applied egg-rr 77.4%

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

      1. div-inv 77.4%

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

      2. associate-*l* 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around inf 82.7%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \left(x + z\right) \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.18 \cdot 10^{+76}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.18e+76) (* x (* x (/ 0.5 y))) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.18e+76) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.18d+76) then
        tmp = x * (x * (0.5d0 / y))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.18e+76) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.18e+76:
		tmp = x * (x * (0.5 / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.18e+76)
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.18e+76)
		tmp = x * (x * (0.5 / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.18e+76], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.17999999999999997e76

   1. Initial program 78.8%

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

   3. Taylor expanded in x around inf 35.6%

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

      1. div-inv 35.5%

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

      2. *-commutative 35.5%

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

      3. associate-/r* 35.5%

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

      4. metadata-eval 35.5%

         \[\leadsto {x}^{2} \cdot \frac{\color{blue}{0.5}}{y} \]

      5. unpow2 35.5%

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

      6. associate-*l* 37.4%

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

   5. Applied egg-rr 37.4%

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

   if 1.17999999999999997e76 < y

   1. Initial program 36.9%

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

   3. Taylor expanded in y around inf 71.3%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.18 \cdot 10^{+76}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+76}:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6e+76) (/ (* 0.5 x) (/ y x)) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+76) {
		tmp = (0.5 * x) / (y / x);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 6d+76) then
        tmp = (0.5d0 * x) / (y / x)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+76) {
		tmp = (0.5 * x) / (y / x);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6e+76:
		tmp = (0.5 * x) / (y / x)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6e+76)
		tmp = Float64(Float64(0.5 * x) / Float64(y / x));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6e+76)
		tmp = (0.5 * x) / (y / x);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6e+76], N[(N[(0.5 * x), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.9999999999999996e76

   1. Initial program 78.8%

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

   3. Taylor expanded in x around inf 35.6%

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

      1. div-inv 35.5%

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

      2. *-commutative 35.5%

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

      3. associate-/r* 35.5%

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

      4. metadata-eval 35.5%

         \[\leadsto {x}^{2} \cdot \frac{\color{blue}{0.5}}{y} \]

      5. unpow2 35.5%

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

      6. associate-*l* 37.4%

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

   5. Applied egg-rr 37.4%

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

      1. associate-*r/ 37.4%

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

      2. metadata-eval 37.4%

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

      3. div-inv 37.4%

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

      4. associate-*r/ 35.5%

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

      5. *-commutative 35.5%

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

      6. associate-/l* 37.5%

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

      7. div-inv 37.5%

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

      8. metadata-eval 37.5%

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

   7. Applied egg-rr 37.5%

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

   if 5.9999999999999996e76 < y

   1. Initial program 36.9%

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

   3. Taylor expanded in y around inf 71.3%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+76}:\\ \;\;\;\;\frac{0.5 \cdot x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.3% 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 71.5%

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

3. Taylor expanded in y around inf 34.7%

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

4. Final simplification 34.7%

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

Developer target: 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 2024027 
(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)))