Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.3% → 99.6%

Time: 3.4s

Precision: binary64

Cost: 6784

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

• 22.7% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

FPCore

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

↓

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

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(x, y, (t * -z));
}

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 99.2%

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

2. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(-t\right)\right)} \]
   Step-by-step derivation

   [Start] 99.2%	\[ x \cdot y - z \cdot t \]
   fma-neg [=>] 99.6%	\[ \color{blue}{\mathsf{fma}\left(x, y, -z \cdot t\right)} \]
   distribute-rgt-neg-in [=>] 99.6%	\[ \mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(-t\right)}\right) \]

3. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	6784

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

Alternative 2
Accuracy	68.4%
Cost	520

\[\begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-97}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{+30}:\\ \;\;\;\;t \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	448

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

Alternative 4
Accuracy	53.1%
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2023272 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))