Linear.V2:$cdot from linear-1.19.1.3, A

Average Accuracy: 99.2% → 99.6%

Time: 2.1s

Precision: binary64

Cost: 6720

• Unsound rule application detected in e-graph. Results may not be sound.

• 22.5% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[x \cdot y + z \cdot t \]

↓

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

FPCore

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

↓

(FPCore (x y z t) :precision binary64 (fma z t (* y x)))

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 99.2%

   \[x \cdot y + z \cdot t \]

2. Simplified 99.2%

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

   [Start] 99.2	\[ x \cdot y + z \cdot t \]
   fma-def [=>] 99.2	\[ \color{blue}{\mathsf{fma}\left(x, y, z \cdot t\right)} \]

3. Taylor expanded in x around 0 99.2%

   \[\leadsto \color{blue}{y \cdot x + t \cdot z} \]

4. Simplified 100.0%

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

   [Start] 99.2	\[ y \cdot x + t \cdot z \]
   +-commutative [=>] 99.2	\[ \color{blue}{t \cdot z + y \cdot x} \]
   *-commutative [=>] 99.2	\[ \color{blue}{z \cdot t} + y \cdot x \]
   *-commutative [=>] 99.2	\[ z \cdot t + \color{blue}{x \cdot y} \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(z, t, x \cdot y\right)} \]
   *-commutative [<=] 100.0	\[ \mathsf{fma}\left(z, t, \color{blue}{y \cdot x}\right) \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	6720

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

Alternative 2
Accuracy	69.0%
Cost	457

\[\begin{array}{l} \mathbf{if}\;t \leq -0.012 \lor \neg \left(t \leq 1.12 \cdot 10^{-17}\right):\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Accuracy	99.2%
Cost	448

\[y \cdot x + z \cdot t \]

Alternative 4
Accuracy	52.3%
Cost	192

\[z \cdot t \]

Reproduce

herbie shell --seed 2023160 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))