Average Error: 0.1 → 0.1
Time: 14.8s
Precision: 64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
↓
\[\mathsf{fma}\left(y, x, \left(3 \cdot z\right) \cdot z\right)\]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\mathsf{fma}\left(y, x, \left(3 \cdot z\right) \cdot z\right)
double f(double x, double y, double z) {
        double r149171748 = x;
        double r149171749 = y;
        double r149171750 = r149171748 * r149171749;
        double r149171751 = z;
        double r149171752 = r149171751 * r149171751;
        double r149171753 = r149171750 + r149171752;
        double r149171754 = r149171753 + r149171752;
        double r149171755 = r149171754 + r149171752;
        return r149171755;
}
↓
double f(double x, double y, double z) {
        double r149171756 = y;
        double r149171757 = x;
        double r149171758 = 3.0;
        double r149171759 = z;
        double r149171760 = r149171758 * r149171759;
        double r149171761 = r149171760 * r149171759;
        double r149171762 = fma(r149171756, r149171757, r149171761);
        return r149171762;
}

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

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

  4. Applied associate-*r*0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019173 +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)))