Average Error: 0.0 → 0.0
Time: 7.6s
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 r10357 = x;
        double r10358 = y;
        double r10359 = r10357 + r10358;
        double r10360 = z;
        double r10361 = r10359 * r10360;
        return r10361;
}
↓
double f(double x, double y, double z) {
        double r10362 = x;
        double r10363 = z;
        double r10364 = y;
        double r10365 = r10363 * r10364;
        double r10366 = fma(r10362, r10363, r10365);
        return r10366;
}

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 add-sqr-sqrt32.2

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

  4. Applied associate-*r*32.2

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

  5. Simplified32.2

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

  6. Taylor expanded around 0 0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x y z)
  :name "Text.Parsec.Token:makeTokenParser from parsec-3.1.9, B"
  :precision binary64
  (* (+ x y) z))