Average Error: 0.1 → 0.1
Time: 4.3s
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 r484711 = x;
        double r484712 = y;
        double r484713 = r484711 * r484712;
        double r484714 = z;
        double r484715 = r484714 * r484714;
        double r484716 = r484713 + r484715;
        double r484717 = r484716 + r484715;
        double r484718 = r484717 + r484715;
        return r484718;
}
↓
double f(double x, double y, double z) {
        double r484719 = 3.0;
        double r484720 = z;
        double r484721 = r484720 * r484720;
        double r484722 = x;
        double r484723 = y;
        double r484724 = r484722 * r484723;
        double r484725 = fma(r484719, r484721, r484724);
        return r484725;
}

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 2020089 +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)))