Average Error: 0.0 → 0.0
Time: 2.1s
Precision: binary64
\[x \cdot y + z \cdot t \]
↓
\[\mathsf{fma}\left(t, z, y \cdot x\right) \]
x \cdot y + z \cdot t
↓
\mathsf{fma}\left(t, z, y \cdot x\right)

(FPCore (x y z t) :precision binary64 (+ (* x y) (* z t)))
↓
(FPCore (x y z t) :precision binary64 (fma t z (* y x)))
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(t, z, (y * x));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Taylor expanded in x around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022077 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))