Average Error: 0.1 → 0.1
Time: 3.6s
Precision: 64
\[\left(x \cdot y + z\right) \cdot y + t\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, t\right)\]
\left(x \cdot y + z\right) \cdot y + t
↓
\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, t\right)
double f(double x, double y, double z, double t) {
        double r238494 = x;
        double r238495 = y;
        double r238496 = r238494 * r238495;
        double r238497 = z;
        double r238498 = r238496 + r238497;
        double r238499 = r238498 * r238495;
        double r238500 = t;
        double r238501 = r238499 + r238500;
        return r238501;
}
↓
double f(double x, double y, double z, double t) {
        double r238502 = x;
        double r238503 = y;
        double r238504 = z;
        double r238505 = fma(r238502, r238503, r238504);
        double r238506 = t;
        double r238507 = fma(r238505, r238503, r238506);
        return r238507;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

    \[\left(x \cdot y + z\right) \cdot y + t\]

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z t)
  :name "Language.Haskell.HsColour.ColourHighlight:unbase from hscolour-1.23"
  :precision binary64
  (+ (* (+ (* x y) z) y) t))