Average Error: 0.1 → 0.1
Time: 3.9s
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 r610098 = x;
        double r610099 = y;
        double r610100 = r610098 * r610099;
        double r610101 = z;
        double r610102 = r610101 * r610101;
        double r610103 = r610100 + r610102;
        double r610104 = r610103 + r610102;
        double r610105 = r610104 + r610102;
        return r610105;
}
↓
double f(double x, double y, double z) {
        double r610106 = 3.0;
        double r610107 = z;
        double r610108 = r610107 * r610107;
        double r610109 = x;
        double r610110 = y;
        double r610111 = r610109 * r610110;
        double r610112 = fma(r610106, r610108, r610111);
        return r610112;
}

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 2020057 +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)))