Average Error: 0.1 → 0.1
Time: 7.7s
Precision: 64
\[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, x, \left(3 \cdot y\right) \cdot y\right)\]
\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, x, \left(3 \cdot y\right) \cdot y\right)
double f(double x, double y) {
        double r401505 = x;
        double r401506 = r401505 * r401505;
        double r401507 = y;
        double r401508 = r401507 * r401507;
        double r401509 = r401506 + r401508;
        double r401510 = r401509 + r401508;
        double r401511 = r401510 + r401508;
        return r401511;
}
↓
double f(double x, double y) {
        double r401512 = x;
        double r401513 = 3.0;
        double r401514 = y;
        double r401515 = r401513 * r401514;
        double r401516 = r401515 * r401514;
        double r401517 = fma(r401512, r401512, r401516);
        return r401517;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.1
Target0.1
Herbie0.1
\[x \cdot x + y \cdot \left(y + \left(y + y\right)\right)\]

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 3 \cdot \left(y \cdot y\right)\right)}\]
  3. Using strategy rm

  4. Applied associate-*r*0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

  :herbie-target
  (+ (* x x) (* y (+ y (+ y y))))

  (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))