Linear.V3:cross from linear-1.19.1.3

Average Accuracy: 100.0% → 100.0%

Time: 2.5s

Precision: binary64

Cost: 6784

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 100.0%

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

2. Applied egg-rr 100.0%

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

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

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	65.0%
Cost	1051

\[\begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-103} \lor \neg \left(t \leq 6.2 \cdot 10^{-144}\right) \land \left(t \leq 2.4 \cdot 10^{-133} \lor \neg \left(t \leq 1.75 \cdot 10^{-46}\right) \land \left(t \leq 4 \cdot 10^{-17} \lor \neg \left(t \leq 8 \cdot 10^{+47}\right)\right)\right):\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	448

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

Alternative 3
Accuracy	52.2%
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))