Average Error: 0.1 → 0.1
Time: 9.4s
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 r696323 = x;
        double r696324 = r696323 * r696323;
        double r696325 = y;
        double r696326 = r696325 * r696325;
        double r696327 = r696324 + r696326;
        double r696328 = r696327 + r696326;
        double r696329 = r696328 + r696326;
        return r696329;
}
↓
double f(double x, double y) {
        double r696330 = x;
        double r696331 = 3.0;
        double r696332 = y;
        double r696333 = r696331 * r696332;
        double r696334 = r696333 * r696332;
        double r696335 = fma(r696330, r696330, r696334);
        return r696335;
}

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 pow10.1

    \[\leadsto \mathsf{fma}\left(x, x, 3 \cdot \left(y \cdot \color{blue}{{y}^{1}}\right)\right)\]

  5. Applied pow10.1

    \[\leadsto \mathsf{fma}\left(x, x, 3 \cdot \left(\color{blue}{{y}^{1}} \cdot {y}^{1}\right)\right)\]

  6. Applied pow-prod-down0.1

    \[\leadsto \mathsf{fma}\left(x, x, 3 \cdot \color{blue}{{\left(y \cdot y\right)}^{1}}\right)\]

  7. Applied pow10.1

    \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{{3}^{1}} \cdot {\left(y \cdot y\right)}^{1}\right)\]

  8. Applied pow-prod-down0.1

    \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{{\left(3 \cdot \left(y \cdot y\right)\right)}^{1}}\right)\]

  9. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, x, {\color{blue}{\left(y \cdot \left(3 \cdot y\right)\right)}}^{1}\right)\]

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 +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)))