Average Error: 17.6 → 0.0
Time: 2.6s
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 r470908 = x;
        double r470909 = y;
        double r470910 = r470908 * r470909;
        double r470911 = r470909 * r470909;
        double r470912 = r470910 + r470911;
        double r470913 = z;
        double r470914 = r470909 * r470913;
        double r470915 = r470912 - r470914;
        double r470916 = r470915 - r470911;
        return r470916;
}
↓
double f(double x, double y, double z) {
        double r470917 = y;
        double r470918 = x;
        double r470919 = z;
        double r470920 = r470918 - r470919;
        double r470921 = 0.0;
        double r470922 = fma(r470917, r470920, r470921);
        return r470922;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original17.6
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y\]

Derivation

  1. Initial program 17.6

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

  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 2020039 +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)))