Average Error: 0.1 → 0.1
Time: 9.2s
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 r21109046 = x;
        double r21109047 = y;
        double r21109048 = r21109046 * r21109047;
        double r21109049 = z;
        double r21109050 = r21109049 * r21109049;
        double r21109051 = r21109048 + r21109050;
        double r21109052 = r21109051 + r21109050;
        double r21109053 = r21109052 + r21109050;
        return r21109053;
}
↓
double f(double x, double y, double z) {
        double r21109054 = 3.0;
        double r21109055 = z;
        double r21109056 = r21109055 * r21109055;
        double r21109057 = x;
        double r21109058 = y;
        double r21109059 = r21109057 * r21109058;
        double r21109060 = fma(r21109054, r21109056, r21109059);
        return r21109060;
}

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 2019174 +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)))