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

Average Error: 0.02% → 0.01%

Time: 2.4s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

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, (z * t));
}

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(z * t))
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[(z * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.02

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

2. Simplified 0.01

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

   [Start] 0.02	\[ x \cdot y + z \cdot t \]
   fma-def [=>] 0.01	\[ \color{blue}{\mathsf{fma}\left(x, y, z \cdot t\right)} \]

3. Final simplification 0.01

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

Alternatives

Alternative 1
Error	35.4%
Cost	721

\[\begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+54}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-36} \lor \neg \left(x \leq -7.2 \cdot 10^{-103}\right) \land x \leq 7.2 \cdot 10^{-152}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2
Error	0.02%
Cost	448

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

Alternative 3
Error	47.6%
Cost	192

\[z \cdot t \]

Reproduce

herbie shell --seed 2023102 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))