Average Error: 0.1 → 0.1
Time: 4.5s
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 r115491 = x;
        double r115492 = y;
        double r115493 = r115491 * r115492;
        double r115494 = z;
        double r115495 = r115493 + r115494;
        double r115496 = r115495 * r115492;
        double r115497 = t;
        double r115498 = r115496 + r115497;
        return r115498;
}
↓
double f(double x, double y, double z, double t) {
        double r115499 = x;
        double r115500 = y;
        double r115501 = z;
        double r115502 = fma(r115499, r115500, r115501);
        double r115503 = t;
        double r115504 = fma(r115502, r115500, r115503);
        return r115504;
}

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 2020002 +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))