Average Error: 17.7 → 0.0
Time: 2.2s
Precision: 64
\[\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y\]
↓
\[\mathsf{fma}\left(y, x - z, 0\right)\]
\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y
↓
\mathsf{fma}\left(y, x - z, 0\right)
double f(double x, double y, double z) {
        double r546893 = x;
        double r546894 = y;
        double r546895 = r546893 * r546894;
        double r546896 = r546894 * r546894;
        double r546897 = r546895 + r546896;
        double r546898 = z;
        double r546899 = r546894 * r546898;
        double r546900 = r546897 - r546899;
        double r546901 = r546900 - r546896;
        return r546901;
}
↓
double f(double x, double y, double z) {
        double r546902 = y;
        double r546903 = x;
        double r546904 = z;
        double r546905 = r546903 - r546904;
        double r546906 = 0.0;
        double r546907 = fma(r546902, r546905, r546906);
        return r546907;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.7
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.7

    \[\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - z, 0\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y, x - z, 0\right)\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, C"
  :precision binary64

  :herbie-target
  (* (- x z) y)

  (- (- (+ (* x y) (* y y)) (* y z)) (* y y)))