Average Error: 0.1 → 0.1
Time: 12.1s
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 r165412 = x;
        double r165413 = y;
        double r165414 = r165412 * r165413;
        double r165415 = z;
        double r165416 = r165414 + r165415;
        double r165417 = r165416 * r165413;
        double r165418 = t;
        double r165419 = r165417 + r165418;
        return r165419;
}
↓
double f(double x, double y, double z, double t) {
        double r165420 = x;
        double r165421 = y;
        double r165422 = z;
        double r165423 = fma(r165420, r165421, r165422);
        double r165424 = t;
        double r165425 = fma(r165423, r165421, r165424);
        return r165425;
}

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 2019209 +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))