Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G

Average Accuracy: 100.0% → 100.0%

Time: 7.6s

Precision: binary64

Cost: 13248

Math

\[\left(x + y\right) \cdot \left(z + 1\right) \]

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (+ x (fma z y (fma z x y))))

C

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

↓

double code(double x, double y, double z) {
	return x + fma(z, y, fma(z, x, y));
}

Julia

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

↓

function code(x, y, z)
	return Float64(x + fma(z, y, fma(z, x, y)))
end

Wolfram

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

↓

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

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(z + 1\right) \]

2. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \left(x + y\right) \cdot \left(z + 1\right) \]
   distribute-rgt-in [=>] 100.0	\[ \color{blue}{z \cdot \left(x + y\right) + 1 \cdot \left(x + y\right)} \]
   *-un-lft-identity [<=] 100.0	\[ z \cdot \left(x + y\right) + \color{blue}{\left(x + y\right)} \]
   +-commutative [=>] 100.0	\[ z \cdot \left(x + y\right) + \color{blue}{\left(y + x\right)} \]
   associate-+r+ [=>] 100.0	\[ \color{blue}{\left(z \cdot \left(x + y\right) + y\right) + x} \]
   +-commutative [=>] 100.0	\[ \color{blue}{\left(y + z \cdot \left(x + y\right)\right)} + x \]
   *-commutative [=>] 100.0	\[ \left(y + \color{blue}{\left(x + y\right) \cdot z}\right) + x \]

3. Applied egg-rr 29.6%

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

   [Start] 100.0	\[ \left(y + \left(x + y\right) \cdot z\right) + x \]
   flip-+ [=>] 59.4	\[ \color{blue}{\frac{y \cdot y - \left(\left(x + y\right) \cdot z\right) \cdot \left(\left(x + y\right) \cdot z\right)}{y - \left(x + y\right) \cdot z}} + x \]
   flip-- [=>] 32.7	\[ \frac{\color{blue}{\frac{\left(y \cdot y\right) \cdot \left(y \cdot y\right) - \left(\left(\left(x + y\right) \cdot z\right) \cdot \left(\left(x + y\right) \cdot z\right)\right) \cdot \left(\left(\left(x + y\right) \cdot z\right) \cdot \left(\left(x + y\right) \cdot z\right)\right)}{y \cdot y + \left(\left(x + y\right) \cdot z\right) \cdot \left(\left(x + y\right) \cdot z\right)}}}{y - \left(x + y\right) \cdot z} + x \]
   associate-/l/ [=>] 29.6	\[ \color{blue}{\frac{\left(y \cdot y\right) \cdot \left(y \cdot y\right) - \left(\left(\left(x + y\right) \cdot z\right) \cdot \left(\left(x + y\right) \cdot z\right)\right) \cdot \left(\left(\left(x + y\right) \cdot z\right) \cdot \left(\left(x + y\right) \cdot z\right)\right)}{\left(y - \left(x + y\right) \cdot z\right) \cdot \left(y \cdot y + \left(\left(x + y\right) \cdot z\right) \cdot \left(\left(x + y\right) \cdot z\right)\right)}} + x \]
   pow2 [=>] 29.6	\[ \frac{\left(y \cdot y\right) \cdot \left(y \cdot y\right) - \color{blue}{{\left(\left(x + y\right) \cdot z\right)}^{2}} \cdot \left(\left(\left(x + y\right) \cdot z\right) \cdot \left(\left(x + y\right) \cdot z\right)\right)}{\left(y - \left(x + y\right) \cdot z\right) \cdot \left(y \cdot y + \left(\left(x + y\right) \cdot z\right) \cdot \left(\left(x + y\right) \cdot z\right)\right)} + x \]
   pow2 [=>] 29.6	\[ \frac{\left(y \cdot y\right) \cdot \left(y \cdot y\right) - {\left(\left(x + y\right) \cdot z\right)}^{2} \cdot \color{blue}{{\left(\left(x + y\right) \cdot z\right)}^{2}}}{\left(y - \left(x + y\right) \cdot z\right) \cdot \left(y \cdot y + \left(\left(x + y\right) \cdot z\right) \cdot \left(\left(x + y\right) \cdot z\right)\right)} + x \]
   +-commutative [=>] 29.6	\[ \frac{\left(y \cdot y\right) \cdot \left(y \cdot y\right) - {\left(\left(x + y\right) \cdot z\right)}^{2} \cdot {\left(\left(x + y\right) \cdot z\right)}^{2}}{\left(y - \left(x + y\right) \cdot z\right) \cdot \color{blue}{\left(\left(\left(x + y\right) \cdot z\right) \cdot \left(\left(x + y\right) \cdot z\right) + y \cdot y\right)}} + x \]
   pow2 [=>] 29.6	\[ \frac{\left(y \cdot y\right) \cdot \left(y \cdot y\right) - {\left(\left(x + y\right) \cdot z\right)}^{2} \cdot {\left(\left(x + y\right) \cdot z\right)}^{2}}{\left(y - \left(x + y\right) \cdot z\right) \cdot \left(\color{blue}{{\left(\left(x + y\right) \cdot z\right)}^{2}} + y \cdot y\right)} + x \]

4. Applied egg-rr 100.0%

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

   [Start] 29.6	\[ \frac{\left(y \cdot y\right) \cdot \left(y \cdot y\right) - {\left(\left(x + y\right) \cdot z\right)}^{2} \cdot {\left(\left(x + y\right) \cdot z\right)}^{2}}{\left(y - \left(x + y\right) \cdot z\right) \cdot \left({\left(\left(x + y\right) \cdot z\right)}^{2} + y \cdot y\right)} + x \]
   *-commutative [=>] 29.6	\[ \frac{\left(y \cdot y\right) \cdot \left(y \cdot y\right) - {\left(\left(x + y\right) \cdot z\right)}^{2} \cdot {\left(\left(x + y\right) \cdot z\right)}^{2}}{\color{blue}{\left({\left(\left(x + y\right) \cdot z\right)}^{2} + y \cdot y\right) \cdot \left(y - \left(x + y\right) \cdot z\right)}} + x \]
   associate-/r* [=>] 32.7	\[ \color{blue}{\frac{\frac{\left(y \cdot y\right) \cdot \left(y \cdot y\right) - {\left(\left(x + y\right) \cdot z\right)}^{2} \cdot {\left(\left(x + y\right) \cdot z\right)}^{2}}{{\left(\left(x + y\right) \cdot z\right)}^{2} + y \cdot y}}{y - \left(x + y\right) \cdot z}} + x \]
   +-commutative [=>] 32.7	\[ \frac{\frac{\left(y \cdot y\right) \cdot \left(y \cdot y\right) - {\left(\left(x + y\right) \cdot z\right)}^{2} \cdot {\left(\left(x + y\right) \cdot z\right)}^{2}}{\color{blue}{y \cdot y + {\left(\left(x + y\right) \cdot z\right)}^{2}}}}{y - \left(x + y\right) \cdot z} + x \]
   flip-- [<=] 59.4	\[ \frac{\color{blue}{y \cdot y - {\left(\left(x + y\right) \cdot z\right)}^{2}}}{y - \left(x + y\right) \cdot z} + x \]
   unpow2 [=>] 59.4	\[ \frac{y \cdot y - \color{blue}{\left(\left(x + y\right) \cdot z\right) \cdot \left(\left(x + y\right) \cdot z\right)}}{y - \left(x + y\right) \cdot z} + x \]
   flip-+ [<=] 100.0	\[ \color{blue}{\left(y + \left(x + y\right) \cdot z\right)} + x \]
   +-commutative [=>] 100.0	\[ \color{blue}{\left(\left(x + y\right) \cdot z + y\right)} + x \]
   *-commutative [=>] 100.0	\[ \left(\color{blue}{z \cdot \left(x + y\right)} + y\right) + x \]
   +-commutative [=>] 100.0	\[ \left(z \cdot \color{blue}{\left(y + x\right)} + y\right) + x \]
   distribute-rgt-in [=>] 100.0	\[ \left(\color{blue}{\left(y \cdot z + x \cdot z\right)} + y\right) + x \]
   associate-+l+ [=>] 100.0	\[ \color{blue}{\left(y \cdot z + \left(x \cdot z + y\right)\right)} + x \]

5. Simplified 100.0%

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

   [Start] 100.0	\[ \left(y \cdot z + \left(x \cdot z + y\right)\right) + x \]
   +-commutative [<=] 100.0	\[ \left(y \cdot z + \color{blue}{\left(y + x \cdot z\right)}\right) + x \]
   *-commutative [<=] 100.0	\[ \left(\color{blue}{z \cdot y} + \left(y + x \cdot z\right)\right) + x \]
   *-commutative [<=] 100.0	\[ \left(z \cdot y + \left(y + \color{blue}{z \cdot x}\right)\right) + x \]
   +-commutative [<=] 100.0	\[ \left(z \cdot y + \color{blue}{\left(z \cdot x + y\right)}\right) + x \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(z, y, z \cdot x + y\right)} + x \]
   fma-def [=>] 100.0	\[ \mathsf{fma}\left(z, y, \color{blue}{\mathsf{fma}\left(z, x, y\right)}\right) + x \]

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	33.1%
Cost	1380

\[\begin{array}{l} \mathbf{if}\;y \leq -4600000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-124}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;y \leq -2.45 \cdot 10^{-267}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{-169}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-47}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-19}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 16000:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+157}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+222}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Accuracy	49.0%
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -1.36 \cdot 10^{-162}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-150}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 0.0034:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]

Alternative 3
Accuracy	59.2%
Cost	717

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-60} \lor \neg \left(x \leq -1.95 \cdot 10^{-117}\right) \land x \leq -1.45 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \]

Alternative 4
Accuracy	97.3%
Cost	716

\[\begin{array}{l} t_0 := z \cdot \left(y + x\right)\\ \mathbf{if}\;z \leq -195000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	97.3%
Cost	716

\[\begin{array}{l} t_0 := z \cdot \left(y + x\right)\\ \mathbf{if}\;z \leq -3900000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{-14}:\\ \;\;\;\;x + z \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	79.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -0.125 \lor \neg \left(z \leq 1.9 \cdot 10^{-27}\right):\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 7
Accuracy	100.0%
Cost	576

\[x + \left(y + z \cdot \left(y + x\right)\right) \]

Alternative 8
Accuracy	79.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq 55:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]

Alternative 9
Accuracy	100.0%
Cost	448

\[\left(y + x\right) \cdot \left(z + 1\right) \]

Alternative 10
Accuracy	39.4%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-56}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 11
Accuracy	32.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))