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

Average Error: 0.0 → 0.0

Time: 6.8s

Precision: binary64

Cost: 13248

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (fma y z (+ y (fma z x 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(y, z, (y + fma(z, x, x)));
}

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Taylor expanded in x around 0 0.0

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

4. Simplified 0.0

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

   [Start] 0.0	\[ y \cdot z + \left(\left(1 + z\right) \cdot x + y\right) \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(y, z, \left(1 + z\right) \cdot x + y\right)} \]
   +-commutative [=>] 0.0	\[ \mathsf{fma}\left(y, z, \color{blue}{y + \left(1 + z\right) \cdot x}\right) \]
   +-commutative [=>] 0.0	\[ \mathsf{fma}\left(y, z, y + \color{blue}{\left(z + 1\right)} \cdot x\right) \]
   distribute-lft1-in [<=] 0.0	\[ \mathsf{fma}\left(y, z, y + \color{blue}{\left(z \cdot x + x\right)}\right) \]
   fma-def [=>] 0.0	\[ \mathsf{fma}\left(y, z, y + \color{blue}{\mathsf{fma}\left(z, x, x\right)}\right) \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	32.6
Cost	1248

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-163}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-296}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-245}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-178}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-121}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-68}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 28000:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]

Alternative 2
Error	12.4
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq 0.0075:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+19}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]

Alternative 3
Error	12.1
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-5} \lor \neg \left(z \leq 4.4 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 4
Error	1.6
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 5
Error	12.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq 32000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]

Alternative 6
Error	0.0
Cost	448

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

Alternative 7
Error	38.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{-113}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8
Error	42.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022354 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))