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

Average Error: 0.01% → 0.01%

Time: 3.1s

Precision: binary64

Cost: 6720

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 0.01

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

2. Simplified 0.01

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

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

3. Taylor expanded in x around 0 0.01

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

4. Simplified 0.01

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

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

5. Final simplification 0.01

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

Alternatives

Alternative 1
Error	0.01%
Cost	6720

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

Alternative 2
Error	35.02%
Cost	986

\[\begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+120}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{+88} \lor \neg \left(x \leq -4.3 \cdot 10^{-16}\right) \land \left(x \leq -5.5 \cdot 10^{-91} \lor \neg \left(x \leq -4 \cdot 10^{-111}\right) \land x \leq 1.7 \cdot 10^{-135}\right):\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Error	0.01%
Cost	448

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

Alternative 4
Error	48.43%
Cost	192

\[z \cdot t \]

Reproduce

herbie shell --seed 2023089 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))