Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.3% → 99.7%

Time: 4.1s

Precision: binary64

Cost: 6784

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

• 23.3% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

FPCore

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

↓

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

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, (x * y));
}

Julia

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

↓

function code(x, y, z, t)
	return fma(Float64(-z), t, Float64(x * y))
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[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

2. Applied egg-rr 99.6%

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

   [Start] 98.8%	\[ x \cdot y - z \cdot t \]
   sub-neg [=>] 98.8%	\[ \color{blue}{x \cdot y + \left(-z \cdot t\right)} \]
   distribute-rgt-neg-out [<=] 98.8%	\[ x \cdot y + \color{blue}{z \cdot \left(-t\right)} \]
   +-commutative [=>] 98.8%	\[ \color{blue}{z \cdot \left(-t\right) + x \cdot y} \]
   distribute-rgt-neg-out [=>] 98.8%	\[ \color{blue}{\left(-z \cdot t\right)} + x \cdot y \]
   distribute-lft-neg-in [=>] 98.8%	\[ \color{blue}{\left(-z\right) \cdot t} + x \cdot y \]
   fma-def [=>] 99.6%	\[ \color{blue}{\mathsf{fma}\left(-z, t, x \cdot y\right)} \]

3. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	6784

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

Alternative 2
Accuracy	99.6%
Cost	6784

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

Alternative 3
Accuracy	67.7%
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -0.024 \lor \neg \left(z \leq 5.2 \cdot 10^{-105}\right):\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	448

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

Alternative 5
Accuracy	52.2%
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))