Average Error: 0.1 → 0.1
Time: 8.4s
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 r9058025 = x;
        double r9058026 = y;
        double r9058027 = r9058025 * r9058026;
        double r9058028 = z;
        double r9058029 = r9058028 * r9058028;
        double r9058030 = r9058027 + r9058029;
        double r9058031 = r9058030 + r9058029;
        double r9058032 = r9058031 + r9058029;
        return r9058032;
}
↓
double f(double x, double y, double z) {
        double r9058033 = 3.0;
        double r9058034 = z;
        double r9058035 = r9058034 * r9058034;
        double r9058036 = x;
        double r9058037 = y;
        double r9058038 = r9058036 * r9058037;
        double r9058039 = fma(r9058033, r9058035, r9058038);
        return r9058039;
}

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 2019156 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"

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

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