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

Average Accuracy: 100.0% → 100.0%

Time: 5.6s

Precision: binary64

Cost: 6976

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

\[\mathsf{fma}\left(z + 1, y, \left(z + 1\right) \cdot x\right) \]

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 100.0%

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

2. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z + 1, y, \left(z + 1\right) \cdot x\right)} \]
   Proof

   [Start] 100.0	\[ \left(x + y\right) \cdot \left(z + 1\right) \]
   *-commutative [=>] 100.0	\[ \color{blue}{\left(z + 1\right) \cdot \left(x + y\right)} \]
   +-commutative [=>] 100.0	\[ \left(z + 1\right) \cdot \color{blue}{\left(y + x\right)} \]
   distribute-lft-in [=>] 100.0	\[ \color{blue}{\left(z + 1\right) \cdot y + \left(z + 1\right) \cdot x} \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(z + 1, y, \left(z + 1\right) \cdot x\right)} \]

3. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(z + 1, y, \left(z + 1\right) \cdot x\right) \]

Alternatives

Alternative 1
Accuracy	49.5%
Cost	1248

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+100}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{+84}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-205}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-278}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-127}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-61}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]

Alternative 2
Accuracy	79.3%
Cost	1246

\[\begin{array}{l} t_0 := \left(z + 1\right) \cdot x\\ \mathbf{if}\;y \leq 2.7 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-107}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-94} \lor \neg \left(y \leq 3 \cdot 10^{-57}\right) \land \left(y \leq 2.9 \cdot 10^{-37} \lor \neg \left(y \leq 2.1 \cdot 10^{+23}\right) \land y \leq 3.3 \cdot 10^{+48}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(z + 1\right) \cdot y\\ \end{array} \]

Alternative 3
Accuracy	49.9%
Cost	984

\[\begin{array}{l} \mathbf{if}\;z \leq -0.116:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-278}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-64}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]

Alternative 4
Accuracy	79.0%
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+100}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{+84}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 62:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]

Alternative 5
Accuracy	79.2%
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+98}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{+84}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -0.0074:\\ \;\;\;\;\left(z + 1\right) \cdot y\\ \mathbf{elif}\;z \leq 170:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]

Alternative 6
Accuracy	97.4%
Cost	585

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

Alternative 7
Accuracy	51.1%
Cost	460

\[\begin{array}{l} \mathbf{if}\;x \leq -70:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-58}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-155}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8
Accuracy	100.0%
Cost	448

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

Alternative 9
Accuracy	32.2%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))