Average Error: 0.1 → 0.4
Time: 3.9s
Precision: 64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
↓
\[\left(x \cdot y + z \cdot z\right) + \left(\sqrt[3]{\mathsf{fma}\left(z, z, z \cdot z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z \cdot z\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z \cdot z\right)}\]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\left(x \cdot y + z \cdot z\right) + \left(\sqrt[3]{\mathsf{fma}\left(z, z, z \cdot z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z \cdot z\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z \cdot z\right)}
double f(double x, double y, double z) {
        double r620172 = x;
        double r620173 = y;
        double r620174 = r620172 * r620173;
        double r620175 = z;
        double r620176 = r620175 * r620175;
        double r620177 = r620174 + r620176;
        double r620178 = r620177 + r620176;
        double r620179 = r620178 + r620176;
        return r620179;
}
↓
double f(double x, double y, double z) {
        double r620180 = x;
        double r620181 = y;
        double r620182 = r620180 * r620181;
        double r620183 = z;
        double r620184 = r620183 * r620183;
        double r620185 = r620182 + r620184;
        double r620186 = fma(r620183, r620183, r620184);
        double r620187 = cbrt(r620186);
        double r620188 = r620187 * r620187;
        double r620189 = r620188 * r620187;
        double r620190 = r620185 + r620189;
        return r620190;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.4
\[\left(3 \cdot z\right) \cdot z + y \cdot x\]

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
  2. Using strategy rm

  3. Applied associate-+l+0.1

    \[\leadsto \color{blue}{\left(x \cdot y + z \cdot z\right) + \left(z \cdot z + z \cdot z\right)}\]

  4. Simplified0.1

    \[\leadsto \left(x \cdot y + z \cdot z\right) + \color{blue}{\mathsf{fma}\left(z, z, z \cdot z\right)}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019362 +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)))