Average Error: 17.7 → 0.0
Time: 2.7s
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 r486895 = x;
        double r486896 = y;
        double r486897 = r486895 * r486896;
        double r486898 = r486896 * r486896;
        double r486899 = r486897 + r486898;
        double r486900 = z;
        double r486901 = r486896 * r486900;
        double r486902 = r486899 - r486901;
        double r486903 = r486902 - r486898;
        return r486903;
}
↓
double f(double x, double y, double z) {
        double r486904 = y;
        double r486905 = x;
        double r486906 = z;
        double r486907 = r486905 - r486906;
        double r486908 = 0.0;
        double r486909 = fma(r486904, r486907, r486908);
        return r486909;
}

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 2020033 +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)))