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 r165033 = x;
        double r165034 = y;
        double r165035 = r165033 * r165034;
        double r165036 = z;
        double r165037 = r165035 + r165036;
        double r165038 = r165037 * r165034;
        double r165039 = t;
        double r165040 = r165038 + r165039;
        return r165040;
}
↓
double f(double x, double y, double z, double t) {
        double r165041 = x;
        double r165042 = y;
        double r165043 = z;
        double r165044 = fma(r165041, r165042, r165043);
        double r165045 = t;
        double r165046 = fma(r165044, r165042, r165045);
        return r165046;
}

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 2019354 +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))