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 r6930 = x;
        double r6931 = y;
        double r6932 = r6930 + r6931;
        double r6933 = z;
        double r6934 = r6932 * r6933;
        return r6934;
}
↓
double f(double x, double y, double z) {
        double r6935 = x;
        double r6936 = z;
        double r6937 = y;
        double r6938 = r6936 * r6937;
        double r6939 = fma(r6935, r6936, r6938);
        return r6939;
}

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 2020002 +o rules:numerics
(FPCore (x y z)
  :name "Text.Parsec.Token:makeTokenParser from parsec-3.1.9, B"
  :precision binary64
  (* (+ x y) z))