Average Error: 0.1 → 0.1
Time: 5.0s
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 r127366 = x;
        double r127367 = y;
        double r127368 = r127366 * r127367;
        double r127369 = z;
        double r127370 = r127368 + r127369;
        double r127371 = r127370 * r127367;
        double r127372 = t;
        double r127373 = r127371 + r127372;
        return r127373;
}
↓
double f(double x, double y, double z, double t) {
        double r127374 = x;
        double r127375 = y;
        double r127376 = z;
        double r127377 = fma(r127374, r127375, r127376);
        double r127378 = t;
        double r127379 = fma(r127377, r127375, r127378);
        return r127379;
}

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 2020089 +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))