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

Average Accuracy: 55.5% → 99.8%

Time: 9.4s

Precision: binary64

Cost: 7168

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (* (fma (/ (+ z x) y) (- z x) (- 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 fma(((z + x) / y), (z - x), -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(fma(Float64(Float64(z + x) / y), Float64(z - x), Float64(-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] / y), $MachinePrecision] * N[(z - x), $MachinePrecision] + (-y)), $MachinePrecision] * -0.5), $MachinePrecision]

Target

Original	55.5%
Target	99.7%
Herbie	99.8%

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

Derivation

1. Initial program 55.5%

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

2. Simplified 99.8%

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

   [Start] 55.5	\[ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   sub-neg [=>] 55.5	\[ \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) + \left(-z \cdot z\right)}}{y \cdot 2} \]
   +-commutative [=>] 55.5	\[ \frac{\color{blue}{\left(-z \cdot z\right) + \left(x \cdot x + y \cdot y\right)}}{y \cdot 2} \]
   neg-sub0 [=>] 55.5	\[ \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]
   associate-+l- [=>] 55.5	\[ \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   sub0-neg [=>] 55.5	\[ \frac{\color{blue}{-\left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   neg-mul-1 [=>] 55.5	\[ \frac{\color{blue}{-1 \cdot \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   *-commutative [=>] 55.5	\[ \frac{\color{blue}{\left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right) \cdot -1}}{y \cdot 2} \]
   times-frac [=>] 55.5	\[ \color{blue}{\frac{z \cdot z - \left(x \cdot x + y \cdot y\right)}{y} \cdot \frac{-1}{2}} \]
   associate--r+ [=>] 55.5	\[ \frac{\color{blue}{\left(z \cdot z - x \cdot x\right) - y \cdot y}}{y} \cdot \frac{-1}{2} \]
   div-sub [=>] 55.5	\[ \color{blue}{\left(\frac{z \cdot z - x \cdot x}{y} - \frac{y \cdot y}{y}\right)} \cdot \frac{-1}{2} \]
   difference-of-squares [=>] 55.5	\[ \left(\frac{\color{blue}{\left(z + x\right) \cdot \left(z - x\right)}}{y} - \frac{y \cdot y}{y}\right) \cdot \frac{-1}{2} \]
   +-commutative [<=] 55.5	\[ \left(\frac{\color{blue}{\left(x + z\right)} \cdot \left(z - x\right)}{y} - \frac{y \cdot y}{y}\right) \cdot \frac{-1}{2} \]
   associate-*l/ [<=] 61.2	\[ \left(\color{blue}{\frac{x + z}{y} \cdot \left(z - x\right)} - \frac{y \cdot y}{y}\right) \cdot \frac{-1}{2} \]
   *-commutative [=>] 61.2	\[ \left(\color{blue}{\left(z - x\right) \cdot \frac{x + z}{y}} - \frac{y \cdot y}{y}\right) \cdot \frac{-1}{2} \]
   associate-/l* [=>] 99.8	\[ \left(\left(z - x\right) \cdot \frac{x + z}{y} - \color{blue}{\frac{y}{\frac{y}{y}}}\right) \cdot \frac{-1}{2} \]
   *-inverses [=>] 99.8	\[ \left(\left(z - x\right) \cdot \frac{x + z}{y} - \frac{y}{\color{blue}{1}}\right) \cdot \frac{-1}{2} \]
   /-rgt-identity [=>] 99.8	\[ \left(\left(z - x\right) \cdot \frac{x + z}{y} - \color{blue}{y}\right) \cdot \frac{-1}{2} \]
   metadata-eval [=>] 99.8	\[ \left(\left(z - x\right) \cdot \frac{x + z}{y} - y\right) \cdot \color{blue}{-0.5} \]

3. Applied egg-rr 99.8%

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

   [Start] 99.8	\[ \left(\left(z - x\right) \cdot \frac{x + z}{y} - y\right) \cdot -0.5 \]
   *-commutative [=>] 99.8	\[ \left(\color{blue}{\frac{x + z}{y} \cdot \left(z - x\right)} - y\right) \cdot -0.5 \]
   fma-neg [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(\frac{x + z}{y}, z - x, -y\right)} \cdot -0.5 \]
   +-commutative [=>] 99.8	\[ \mathsf{fma}\left(\frac{\color{blue}{z + x}}{y}, z - x, -y\right) \cdot -0.5 \]

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	74.8%
Cost	1105

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{-171}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-246}:\\ \;\;\;\;x \cdot \frac{x \cdot 0.5}{y}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-72} \lor \neg \left(y \leq 1.7 \cdot 10^{+58}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.5\right)}{y}\\ \end{array} \]

Alternative 2
Accuracy	74.8%
Cost	1105

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{-171}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-237}:\\ \;\;\;\;x \cdot \frac{x \cdot 0.5}{y}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-70} \lor \neg \left(y \leq 1.7 \cdot 10^{+58}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{x \cdot \left(z - x\right)}{y}\\ \end{array} \]

Alternative 3
Accuracy	88.7%
Cost	905

\[\begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-24} \lor \neg \left(x \leq 1.15 \cdot 10^{+34}\right):\\ \;\;\;\;-0.5 \cdot \left(\left(-y\right) - \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \]

Alternative 4
Accuracy	88.7%
Cost	904

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-30}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+34}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(-y\right) - \frac{x}{\frac{y}{x}}\right)\\ \end{array} \]

Alternative 5
Accuracy	63.2%
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{-101}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-72}:\\ \;\;\;\;z \cdot \frac{z}{\frac{y}{-0.5}}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{\frac{y}{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 6
Accuracy	63.0%
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{-101}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-74}:\\ \;\;\;\;z \cdot \frac{z}{\frac{y}{-0.5}}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+58}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7
Accuracy	99.8%
Cost	832

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

Alternative 8
Accuracy	62.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-168}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{\frac{y}{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 9
Accuracy	58.0%
Cost	192

\[y \cdot 0.5 \]

Reproduce

herbie shell --seed 2023147 
(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)))