Average Error: 0.0 → 0.0
Time: 2.3s
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 r7581 = x;
        double r7582 = y;
        double r7583 = r7581 + r7582;
        double r7584 = z;
        double r7585 = r7583 * r7584;
        return r7585;
}
↓
double f(double x, double y, double z) {
        double r7586 = x;
        double r7587 = z;
        double r7588 = y;
        double r7589 = r7587 * r7588;
        double r7590 = fma(r7586, r7587, r7589);
        return r7590;
}

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 around inf 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z)
  :name "Text.Parsec.Token:makeTokenParser from parsec-3.1.9, B"
  :precision binary64
  (* (+ x y) z))