Average Error: 0.1 → 0.1
Time: 4.0s
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 r604398 = x;
        double r604399 = y;
        double r604400 = r604398 * r604399;
        double r604401 = z;
        double r604402 = r604401 * r604401;
        double r604403 = r604400 + r604402;
        double r604404 = r604403 + r604402;
        double r604405 = r604404 + r604402;
        return r604405;
}
↓
double f(double x, double y, double z) {
        double r604406 = 3.0;
        double r604407 = z;
        double r604408 = r604407 * r604407;
        double r604409 = x;
        double r604410 = y;
        double r604411 = r604409 * r604410;
        double r604412 = fma(r604406, r604408, r604411);
        return r604412;
}

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 2020060 +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)))