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

Percentage Accurate: 68.6% → 99.2%

Time: 11.9s

Alternatives: 6

Speedup: 0.7×

• 40.7% 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: 68.6% 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: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\frac{\frac{y}{x - z}}{x + z}} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* y (- (cbrt -0.125))) (/ (cbrt -0.125) (/ (/ y (- x z)) (+ x z)))))

C

double code(double x, double y, double z) {
	return (y * -cbrt(-0.125)) - (cbrt(-0.125) / ((y / (x - z)) / (x + z)));
}

Java

public static double code(double x, double y, double z) {
	return (y * -Math.cbrt(-0.125)) - (Math.cbrt(-0.125) / ((y / (x - z)) / (x + z)));
}

Julia

function code(x, y, z)
	return Float64(Float64(y * Float64(-cbrt(-0.125))) - Float64(cbrt(-0.125) / Float64(Float64(y / Float64(x - z)) / Float64(x + z))))
end

Wolfram

code[x_, y_, z_] := N[(N[(y * (-N[Power[-0.125, 1/3], $MachinePrecision])), $MachinePrecision] - N[(N[Power[-0.125, 1/3], $MachinePrecision] / N[(N[(y / N[(x - z), $MachinePrecision]), $MachinePrecision] / N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

   1. add-cbrt-cube 53.5%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\right) \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}}} \]

   2. pow3 53.6%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\right)}^{3}}} \]

   3. div-inv 53.6%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(\left(x \cdot x + y \cdot y\right) - z \cdot z\right) \cdot \frac{1}{y \cdot 2}\right)}}^{3}} \]

   4. add-sqr-sqrt 53.6%

      \[\leadsto \sqrt[3]{{\left(\left(\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}} - z \cdot z\right) \cdot \frac{1}{y \cdot 2}\right)}^{3}} \]

   5. pow2 53.6%

      \[\leadsto \sqrt[3]{{\left(\left(\color{blue}{{\left(\sqrt{x \cdot x + y \cdot y}\right)}^{2}} - z \cdot z\right) \cdot \frac{1}{y \cdot 2}\right)}^{3}} \]

   6. hypot-def 53.6%

      \[\leadsto \sqrt[3]{{\left(\left({\color{blue}{\left(\mathsf{hypot}\left(x, y\right)\right)}}^{2} - z \cdot z\right) \cdot \frac{1}{y \cdot 2}\right)}^{3}} \]

   7. pow2 53.6%

      \[\leadsto \sqrt[3]{{\left(\left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - \color{blue}{{z}^{2}}\right) \cdot \frac{1}{y \cdot 2}\right)}^{3}} \]

   8. *-commutative 53.6%

      \[\leadsto \sqrt[3]{{\left(\left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}\right) \cdot \frac{1}{\color{blue}{2 \cdot y}}\right)}^{3}} \]

   9. associate-/r* 53.6%

      \[\leadsto \sqrt[3]{{\left(\left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}\right) \cdot \color{blue}{\frac{\frac{1}{2}}{y}}\right)}^{3}} \]

   10. metadata-eval 53.6%

       \[\leadsto \sqrt[3]{{\left(\left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}\right) \cdot \frac{\color{blue}{0.5}}{y}\right)}^{3}} \]

4. Applied egg-rr 53.6%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}\right) \cdot \frac{0.5}{y}\right)}^{3}}} \]

5. Taylor expanded in y around -inf 84.5%

   \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \sqrt[3]{-0.125}\right) + -1 \cdot \frac{\sqrt[3]{-0.125} \cdot \left({x}^{2} - {z}^{2}\right)}{y}} \]
6. Step-by-step derivation

   1. mul-1-neg 84.5%

      \[\leadsto -1 \cdot \left(y \cdot \sqrt[3]{-0.125}\right) + \color{blue}{\left(-\frac{\sqrt[3]{-0.125} \cdot \left({x}^{2} - {z}^{2}\right)}{y}\right)} \]

   2. unsub-neg 84.5%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125} \cdot \left({x}^{2} - {z}^{2}\right)}{y}} \]

   3. mul-1-neg 84.5%

      \[\leadsto \color{blue}{\left(-y \cdot \sqrt[3]{-0.125}\right)} - \frac{\sqrt[3]{-0.125} \cdot \left({x}^{2} - {z}^{2}\right)}{y} \]

   4. distribute-rgt-neg-in 84.5%

      \[\leadsto \color{blue}{y \cdot \left(-\sqrt[3]{-0.125}\right)} - \frac{\sqrt[3]{-0.125} \cdot \left({x}^{2} - {z}^{2}\right)}{y} \]

   5. associate-/l* 84.6%

      \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \color{blue}{\frac{\sqrt[3]{-0.125}}{\frac{y}{{x}^{2} - {z}^{2}}}} \]

7. Simplified 84.6%

   \[\leadsto \color{blue}{y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\frac{y}{{x}^{2} - {z}^{2}}}} \]
8. Step-by-step derivation

   1. unpow2 84.6%

      \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\frac{y}{\color{blue}{x \cdot x} - {z}^{2}}} \]

   2. unpow2 84.6%

      \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\frac{y}{x \cdot x - \color{blue}{z \cdot z}}} \]

   3. difference-of-squares 90.1%

      \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\frac{y}{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}} \]

9. Applied egg-rr 90.1%

   \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\frac{y}{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 77.7%

       \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\left(x + z\right) \cdot \left(x - z\right)}\right)\right)}} \]

    2. expm1-udef 34.8%

       \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{y}{\left(x + z\right) \cdot \left(x - z\right)}\right)} - 1}} \]

11. Applied egg-rr 34.8%

    \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{y}{\left(x + z\right) \cdot \left(x - z\right)}\right)} - 1}} \]
12. Step-by-step derivation

    1. expm1-def 77.7%

       \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\left(x + z\right) \cdot \left(x - z\right)}\right)\right)}} \]

    2. expm1-log1p 90.1%

       \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\color{blue}{\frac{y}{\left(x + z\right) \cdot \left(x - z\right)}}} \]

    3. *-lft-identity 90.1%

       \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\frac{\color{blue}{1 \cdot y}}{\left(x + z\right) \cdot \left(x - z\right)}} \]

    4. times-frac 99.2%

       \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\color{blue}{\frac{1}{x + z} \cdot \frac{y}{x - z}}} \]

    5. associate-*l/ 99.2%

       \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\color{blue}{\frac{1 \cdot \frac{y}{x - z}}{x + z}}} \]

    6. *-lft-identity 99.2%

       \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\frac{\color{blue}{\frac{y}{x - z}}}{x + z}} \]

    7. +-commutative 99.2%

       \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\frac{\frac{y}{x - z}}{\color{blue}{z + x}}} \]

13. Simplified 99.2%

    \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\color{blue}{\frac{\frac{y}{x - z}}{z + x}}} \]

14. Final simplification 99.2%

    \[\leadsto y \cdot \left(-\sqrt[3]{-0.125}\right) - \frac{\sqrt[3]{-0.125}}{\frac{\frac{y}{x - z}}{x + z}} \]
15. Add Preprocessing

Alternative 2: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(y - z\right) \cdot \left(\left(y + z\right) \cdot \frac{1}{y}\right)\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ (* 0.5 (* (- y z) (* (+ y z) (/ 1.0 y)))) (* 0.5 (* x (/ x y)))))

C

double code(double x, double y, double z) {
	return (0.5 * ((y - z) * ((y + z) * (1.0 / y)))) + (0.5 * (x * (x / 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 - z) * ((y + z) * (1.0d0 / y)))) + (0.5d0 * (x * (x / y)))
end function

Java

public static double code(double x, double y, double z) {
	return (0.5 * ((y - z) * ((y + z) * (1.0 / y)))) + (0.5 * (x * (x / y)));
}

Python

def code(x, y, z):
	return (0.5 * ((y - z) * ((y + z) * (1.0 / y)))) + (0.5 * (x * (x / y)))

Julia

function code(x, y, z)
	return Float64(Float64(0.5 * Float64(Float64(y - z) * Float64(Float64(y + z) * Float64(1.0 / y)))) + Float64(0.5 * Float64(x * Float64(x / y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (0.5 * ((y - z) * ((y + z) * (1.0 / y)))) + (0.5 * (x * (x / y)));
end

Wolfram

code[x_, y_, z_] := N[(N[(0.5 * N[(N[(y - z), $MachinePrecision] * N[(N[(y + z), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

   1. associate--l+ 71.3%

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

   2. +-commutative 71.3%

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

   3. sqr-neg 71.3%

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

   4. difference-of-squares 72.3%

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

   5. fma-def 77.0%

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

   6. sub-neg 77.0%

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

   7. sub-neg 77.0%

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

   8. remove-double-neg 77.0%

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

3. Simplified 77.0%

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

5. Taylor expanded in x around 0 67.3%

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

   1. div-inv 67.2%

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

   2. *-commutative 67.2%

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

   3. associate-*l* 86.0%

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

7. Applied egg-rr 86.0%

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

   1. unpow2 86.0%

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

   2. *-un-lft-identity 86.0%

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

   3. times-frac 90.8%

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

9. Applied egg-rr 90.8%

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

10. Final simplification 90.8%

    \[\leadsto 0.5 \cdot \left(\left(y - z\right) \cdot \left(\left(y + z\right) \cdot \frac{1}{y}\right)\right) + 0.5 \cdot \left(x \cdot \frac{x}{y}\right) \]
11. Add Preprocessing

Alternative 3: 66.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

   1. associate--l+ 71.3%

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

   2. +-commutative 71.3%

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

   3. sqr-neg 71.3%

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

   4. difference-of-squares 72.3%

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

   5. fma-def 77.0%

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

   6. sub-neg 77.0%

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

   7. sub-neg 77.0%

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

   8. remove-double-neg 77.0%

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

3. Simplified 77.0%

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

5. Taylor expanded in x around 0 67.3%

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

6. Taylor expanded in x around 0 45.6%

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

   1. associate-*r/ 63.5%

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

8. Simplified 63.5%

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

9. Final simplification 63.5%

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

Alternative 4: 34.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{y}{x}}{x \cdot 0.5}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 1.0 (/ (/ y x) (* x 0.5))))

C

double code(double x, double y, double z) {
	return 1.0 / ((y / x) / (x * 0.5));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 / ((y / x) / (x * 0.5d0))
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 / ((y / x) / (x * 0.5));
}

Python

def code(x, y, z):
	return 1.0 / ((y / x) / (x * 0.5))

Julia

function code(x, y, z)
	return Float64(1.0 / Float64(Float64(y / x) / Float64(x * 0.5)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 / ((y / x) / (x * 0.5));
end

Wolfram

code[x_, y_, z_] := N[(1.0 / N[(N[(y / x), $MachinePrecision] / N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

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

3. Taylor expanded in x around inf 34.9%

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

   1. div-inv 34.9%

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

   2. *-commutative 34.9%

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

   3. associate-/r* 34.9%

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

   4. metadata-eval 34.9%

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

   5. unpow2 34.9%

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

   6. associate-*l* 37.3%

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

5. Applied egg-rr 37.3%

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

   1. associate-*r/ 37.3%

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

   2. metadata-eval 37.3%

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

   3. div-inv 37.3%

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

   4. div-inv 37.3%

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

   5. associate-*l* 34.9%

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

   6. *-commutative 34.9%

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

   7. div-inv 34.9%

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

   8. associate-/l* 37.4%

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

   9. clear-num 37.3%

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

   10. div-inv 37.3%

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

   11. metadata-eval 37.3%

       \[\leadsto \frac{1}{\frac{\frac{y}{x}}{x \cdot \color{blue}{0.5}}} \]

7. Applied egg-rr 37.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{x}}{x \cdot 0.5}}} \]

8. Final simplification 37.3%

   \[\leadsto \frac{1}{\frac{\frac{y}{x}}{x \cdot 0.5}} \]
9. Add Preprocessing

Alternative 5: 34.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

3. Taylor expanded in x around inf 34.9%

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

   1. div-inv 34.9%

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

   2. *-commutative 34.9%

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

   3. associate-/r* 34.9%

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

   4. metadata-eval 34.9%

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

   5. unpow2 34.9%

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

   6. associate-*l* 37.3%

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

5. Applied egg-rr 37.3%

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

6. Final simplification 37.3%

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

Alternative 6: 34.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.3%

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

3. Taylor expanded in y around inf 30.1%

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

4. Final simplification 30.1%

   \[\leadsto y \cdot 0.5 \]
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 2024033 
(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)))