Average Error: 0.1 → 0.1
Time: 16.9s
Precision: 64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
↓
\[\mathsf{fma}\left(3 \cdot z, z, x \cdot y\right)\]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\mathsf{fma}\left(3 \cdot z, z, x \cdot y\right)
double f(double x, double y, double z) {
        double r351980 = x;
        double r351981 = y;
        double r351982 = r351980 * r351981;
        double r351983 = z;
        double r351984 = r351983 * r351983;
        double r351985 = r351982 + r351984;
        double r351986 = r351985 + r351984;
        double r351987 = r351986 + r351984;
        return r351987;
}
↓
double f(double x, double y, double z) {
        double r351988 = 3.0;
        double r351989 = z;
        double r351990 = r351988 * r351989;
        double r351991 = x;
        double r351992 = y;
        double r351993 = r351991 * r351992;
        double r351994 = fma(r351990, r351989, r351993);
        return r351994;
}

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. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{3 \cdot {z}^{2} + x \cdot y}\]

  3. Simplified0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"

  :herbie-target
  (+ (* (* 3.0 z) z) (* y x))

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