Average Error: 0.1 → 0.1
Time: 16.4s
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 r12815754 = x;
        double r12815755 = y;
        double r12815756 = r12815754 * r12815755;
        double r12815757 = z;
        double r12815758 = r12815757 * r12815757;
        double r12815759 = r12815756 + r12815758;
        double r12815760 = r12815759 + r12815758;
        double r12815761 = r12815760 + r12815758;
        return r12815761;
}
↓
double f(double x, double y, double z) {
        double r12815762 = 3.0;
        double r12815763 = z;
        double r12815764 = r12815763 * r12815763;
        double r12815765 = x;
        double r12815766 = y;
        double r12815767 = r12815765 * r12815766;
        double r12815768 = fma(r12815762, r12815764, r12815767);
        return r12815768;
}

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 2019163 +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)))