Average Error: 0.1 → 0.1
Time: 12.7s
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 r502435 = x;
        double r502436 = r502435 * r502435;
        double r502437 = y;
        double r502438 = r502437 * r502437;
        double r502439 = r502436 + r502438;
        double r502440 = r502439 + r502438;
        double r502441 = r502440 + r502438;
        return r502441;
}
↓
double f(double x, double y) {
        double r502442 = x;
        double r502443 = 3.0;
        double r502444 = y;
        double r502445 = r502443 * r502444;
        double r502446 = r502445 * r502444;
        double r502447 = fma(r502442, r502442, r502446);
        return r502447;
}

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 2020047 +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)))