Average Error: 0.1 → 0.1
Time: 12.6s
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 r118528 = x;
        double r118529 = y;
        double r118530 = r118528 * r118529;
        double r118531 = z;
        double r118532 = r118530 + r118531;
        double r118533 = r118532 * r118529;
        double r118534 = t;
        double r118535 = r118533 + r118534;
        return r118535;
}
↓
double f(double x, double y, double z, double t) {
        double r118536 = x;
        double r118537 = y;
        double r118538 = z;
        double r118539 = fma(r118536, r118537, r118538);
        double r118540 = t;
        double r118541 = fma(r118539, r118537, r118540);
        return r118541;
}

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 2019305 +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))