Average Error: 0.1 → 0.1
Time: 6.3s
Precision: 64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
↓
\[\mathsf{fma}\left(y, x, \left(3 \cdot z\right) \cdot z\right)\]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\mathsf{fma}\left(y, x, \left(3 \cdot z\right) \cdot z\right)
double f(double x, double y, double z) {
        double r310545 = x;
        double r310546 = y;
        double r310547 = r310545 * r310546;
        double r310548 = z;
        double r310549 = r310548 * r310548;
        double r310550 = r310547 + r310549;
        double r310551 = r310550 + r310549;
        double r310552 = r310551 + r310549;
        return r310552;
}
↓
double f(double x, double y, double z) {
        double r310553 = y;
        double r310554 = x;
        double r310555 = 3.0;
        double r310556 = z;
        double r310557 = r310555 * r310556;
        double r310558 = r310557 * r310556;
        double r310559 = fma(r310553, r310554, r310558);
        return r310559;
}

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. Using strategy rm

  4. Applied associate-*r*0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019212 +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)))