Average Error: 0.1 → 0.2
Time: 3.8s
Precision: 64
\[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(\sqrt{\left(x \cdot x + y \cdot y\right) + y \cdot y}, \sqrt{\left(x \cdot x + y \cdot y\right) + y \cdot y}, y \cdot y\right)\]
\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(\sqrt{\left(x \cdot x + y \cdot y\right) + y \cdot y}, \sqrt{\left(x \cdot x + y \cdot y\right) + y \cdot y}, y \cdot y\right)
double f(double x, double y) {
        double r596475 = x;
        double r596476 = r596475 * r596475;
        double r596477 = y;
        double r596478 = r596477 * r596477;
        double r596479 = r596476 + r596478;
        double r596480 = r596479 + r596478;
        double r596481 = r596480 + r596478;
        return r596481;
}
↓
double f(double x, double y) {
        double r596482 = x;
        double r596483 = r596482 * r596482;
        double r596484 = y;
        double r596485 = r596484 * r596484;
        double r596486 = r596483 + r596485;
        double r596487 = r596486 + r596485;
        double r596488 = sqrt(r596487);
        double r596489 = fma(r596488, r596488, r596485);
        return r596489;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.1
Target0.1
Herbie0.2
\[x \cdot x + y \cdot \left(y + \left(y + y\right)\right)\]

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.2

    \[\leadsto \color{blue}{\sqrt{\left(x \cdot x + y \cdot y\right) + y \cdot y} \cdot \sqrt{\left(x \cdot x + y \cdot y\right) + y \cdot y}} + y \cdot y\]

  4. Applied fma-def0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020057 +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)))