Average Error: 0.1 → 0.1
Time: 9.4s
Precision: 64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
↓
\[\mathsf{fma}\left(y, x, 3 \cdot \left(z \cdot z\right)\right)\]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\mathsf{fma}\left(y, x, 3 \cdot \left(z \cdot z\right)\right)
double f(double x, double y, double z) {
        double r359460 = x;
        double r359461 = y;
        double r359462 = r359460 * r359461;
        double r359463 = z;
        double r359464 = r359463 * r359463;
        double r359465 = r359462 + r359464;
        double r359466 = r359465 + r359464;
        double r359467 = r359466 + r359464;
        return r359467;
}
↓
double f(double x, double y, double z) {
        double r359468 = y;
        double r359469 = x;
        double r359470 = 3.0;
        double r359471 = z;
        double r359472 = r359471 * r359471;
        double r359473 = r359470 * r359472;
        double r359474 = fma(r359468, r359469, r359473);
        return r359474;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019305 +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)))