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

Percentage Accurate: 69.0% → 99.9%

Time: 10.8s

Precision: binary64

Cost: 832

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

↓

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

Target

Original	69.0%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Initial program 73.4%

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

2. Simplified 99.9%

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

   [Start] 73.4	\[ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   sub-neg [=>] 73.4	\[ \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) + \left(-z \cdot z\right)}}{y \cdot 2} \]
   +-commutative [=>] 73.4	\[ \frac{\color{blue}{\left(-z \cdot z\right) + \left(x \cdot x + y \cdot y\right)}}{y \cdot 2} \]
   neg-sub0 [=>] 73.4	\[ \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]
   associate-+l- [=>] 73.4	\[ \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   sub0-neg [=>] 73.4	\[ \frac{\color{blue}{-\left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   neg-mul-1 [=>] 73.4	\[ \frac{\color{blue}{-1 \cdot \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   *-commutative [=>] 73.4	\[ \frac{\color{blue}{\left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right) \cdot -1}}{y \cdot 2} \]
   times-frac [=>] 73.4	\[ \color{blue}{\frac{z \cdot z - \left(x \cdot x + y \cdot y\right)}{y} \cdot \frac{-1}{2}} \]
   associate--r+ [=>] 73.4	\[ \frac{\color{blue}{\left(z \cdot z - x \cdot x\right) - y \cdot y}}{y} \cdot \frac{-1}{2} \]
   div-sub [=>] 73.4	\[ \color{blue}{\left(\frac{z \cdot z - x \cdot x}{y} - \frac{y \cdot y}{y}\right)} \cdot \frac{-1}{2} \]
   difference-of-squares [=>] 79.6	\[ \left(\frac{\color{blue}{\left(z + x\right) \cdot \left(z - x\right)}}{y} - \frac{y \cdot y}{y}\right) \cdot \frac{-1}{2} \]
   +-commutative [<=] 79.6	\[ \left(\frac{\color{blue}{\left(x + z\right)} \cdot \left(z - x\right)}{y} - \frac{y \cdot y}{y}\right) \cdot \frac{-1}{2} \]
   associate-*l/ [<=] 80.5	\[ \left(\color{blue}{\frac{x + z}{y} \cdot \left(z - x\right)} - \frac{y \cdot y}{y}\right) \cdot \frac{-1}{2} \]
   *-commutative [=>] 80.5	\[ \left(\color{blue}{\left(z - x\right) \cdot \frac{x + z}{y}} - \frac{y \cdot y}{y}\right) \cdot \frac{-1}{2} \]
   associate-/l* [=>] 99.9	\[ \left(\left(z - x\right) \cdot \frac{x + z}{y} - \color{blue}{\frac{y}{\frac{y}{y}}}\right) \cdot \frac{-1}{2} \]
   *-inverses [=>] 99.9	\[ \left(\left(z - x\right) \cdot \frac{x + z}{y} - \frac{y}{\color{blue}{1}}\right) \cdot \frac{-1}{2} \]
   /-rgt-identity [=>] 99.9	\[ \left(\left(z - x\right) \cdot \frac{x + z}{y} - \color{blue}{y}\right) \cdot \frac{-1}{2} \]
   metadata-eval [=>] 99.9	\[ \left(\left(z - x\right) \cdot \frac{x + z}{y} - y\right) \cdot \color{blue}{-0.5} \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	52.5%
Cost	2396

\[\begin{array}{l} t_0 := \frac{-0.5 \cdot \left(z \cdot z\right)}{y}\\ \mathbf{if}\;x \cdot x \leq 10^{-161}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-86}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-55}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+46}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \cdot x \leq 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+181}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{2 \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 2
Accuracy	52.5%
Cost	2268

\[\begin{array}{l} t_0 := \frac{-0.5 \cdot \left(z \cdot z\right)}{y}\\ \mathbf{if}\;x \cdot x \leq 10^{-161}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-86}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-55}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+46}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \cdot x \leq 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+181}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 3
Accuracy	86.8%
Cost	1485

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-86}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+49} \lor \neg \left(x \cdot x \leq 10^{+139}\right):\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - z \cdot z}{y \cdot 2}\\ \end{array} \]

Alternative 4
Accuracy	53.7%
Cost	1372

\[\begin{array}{l} t_0 := -0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-20}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-219}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-302}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+22}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \end{array} \]

Alternative 5
Accuracy	79.1%
Cost	1356

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-55}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;x \cdot x \leq 10^{+285}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{2 \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 6
Accuracy	80.0%
Cost	1356

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-55}:\\ \;\;\;\;-0.5 \cdot \left(\left(-y\right) - \frac{x \cdot x}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{+285}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{2 \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 7
Accuracy	87.7%
Cost	1232

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ t_1 := -0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \mathbf{if}\;x \leq -2.85 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-65}:\\ \;\;\;\;-0.5 \cdot \left(\left(-y\right) - \frac{x \cdot x}{y}\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	53.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+20} \lor \neg \left(x \leq 6.5 \cdot 10^{+32}\right):\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 9
Accuracy	53.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{+20}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+34}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \end{array} \]

Alternative 10
Accuracy	34.6%
Cost	192

\[y \cdot 0.5 \]

Reproduce

herbie shell --seed 2023160 
(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)))