Average Error: 0.1 → 0.1
Time: 3.8s
Precision: 64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
↓
\[\mathsf{fma}\left(3, z \cdot z, x \cdot y\right)\]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\mathsf{fma}\left(3, z \cdot z, x \cdot y\right)
double f(double x, double y, double z) {
        double r489911 = x;
        double r489912 = y;
        double r489913 = r489911 * r489912;
        double r489914 = z;
        double r489915 = r489914 * r489914;
        double r489916 = r489913 + r489915;
        double r489917 = r489916 + r489915;
        double r489918 = r489917 + r489915;
        return r489918;
}
↓
double f(double x, double y, double z) {
        double r489919 = 3.0;
        double r489920 = z;
        double r489921 = r489920 * r489920;
        double r489922 = x;
        double r489923 = y;
        double r489924 = r489922 * r489923;
        double r489925 = fma(r489919, r489921, r489924);
        return r489925;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.1
\[\left(3 \cdot z\right) \cdot z + y \cdot x\]

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, z \cdot z, x \cdot y\right)}\]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(3, z \cdot z, x \cdot y\right)\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3 z) z) (* y x))

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