Average Error: 0.0 → 0.0
Time: 1.5s
Precision: 64
\[\left(x + y\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, z, z \cdot y\right)\]
\left(x + y\right) \cdot z
↓
\mathsf{fma}\left(x, z, z \cdot y\right)
double f(double x, double y, double z) {
        double r2608 = x;
        double r2609 = y;
        double r2610 = r2608 + r2609;
        double r2611 = z;
        double r2612 = r2610 * r2611;
        return r2612;
}
↓
double f(double x, double y, double z) {
        double r2613 = x;
        double r2614 = z;
        double r2615 = y;
        double r2616 = r2614 * r2615;
        double r2617 = fma(r2613, r2614, r2616);
        return r2617;
}

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. Using strategy rm

  3. Applied *-un-lft-identity0.0

    \[\leadsto \color{blue}{\left(1 \cdot \left(x + y\right)\right)} \cdot z\]

  4. Applied associate-*l*0.0

    \[\leadsto \color{blue}{1 \cdot \left(\left(x + y\right) \cdot z\right)}\]

  5. Simplified0.0

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(x, z, z \cdot y\right)}\]

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x y z)
  :name "Text.Parsec.Token:makeTokenParser from parsec-3.1.9, B"
  :precision binary64
  (* (+ x y) z))