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

(FPCore (x y z) :precision binary64 (* (+ x y) z))
↓
(FPCore (x y z) :precision binary64 (fma y z (* z x)))
double code(double x, double y, double z) {
	return (x + y) * z;
}
↓
double code(double x, double y, double z) {
	return fma(y, z, (z * x));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[\left(x + y\right) \cdot z \]

  2. Taylor expanded in x around 0 0.0

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

  3. Applied fma-def_binary640.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2021307 
(FPCore (x y z)
  :name "Text.Parsec.Token:makeTokenParser from parsec-3.1.9, B"
  :precision binary64
  (* (+ x y) z))