Average Error: 0.0 → 0.0
Time: 2.8s
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 r64 = x;
        double r65 = y;
        double r66 = r64 + r65;
        double r67 = z;
        double r68 = r66 * r67;
        return r68;
}
↓
double f(double x, double y, double z) {
        double r69 = x;
        double r70 = z;
        double r71 = y;
        double r72 = r70 * r71;
        double r73 = fma(r69, r70, r72);
        return r73;
}

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 2020025 +o rules:numerics
(FPCore (x y z)
  :name "Text.Parsec.Token:makeTokenParser from parsec-3.1.9, B"
  :precision binary64
  (* (+ x y) z))