Average Error: 0.1 → 0.1
Time: 1.3s
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 r164330 = x;
        double r164331 = y;
        double r164332 = r164330 * r164331;
        double r164333 = z;
        double r164334 = r164332 + r164333;
        double r164335 = r164334 * r164331;
        double r164336 = t;
        double r164337 = r164335 + r164336;
        return r164337;
}
↓
double f(double x, double y, double z, double t) {
        double r164338 = x;
        double r164339 = y;
        double r164340 = z;
        double r164341 = fma(r164338, r164339, r164340);
        double r164342 = t;
        double r164343 = fma(r164341, r164339, r164342);
        return r164343;
}

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 2020036 +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))