Average Error: 0.1 → 0.1
Time: 21.2s
Precision: 64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
↓
\[\mathsf{fma}\left(3, {z}^{2}, y \cdot x\right)\]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\mathsf{fma}\left(3, {z}^{2}, y \cdot x\right)
double f(double x, double y, double z) {
        double r391018 = x;
        double r391019 = y;
        double r391020 = r391018 * r391019;
        double r391021 = z;
        double r391022 = r391021 * r391021;
        double r391023 = r391020 + r391022;
        double r391024 = r391023 + r391022;
        double r391025 = r391024 + r391022;
        return r391025;
}
↓
double f(double x, double y, double z) {
        double r391026 = 3.0;
        double r391027 = z;
        double r391028 = 2.0;
        double r391029 = pow(r391027, r391028);
        double r391030 = y;
        double r391031 = x;
        double r391032 = r391030 * r391031;
        double r391033 = fma(r391026, r391029, r391032);
        return r391033;
}

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

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

  3. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{3 \cdot {z}^{2} + x \cdot y}\]

  4. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, {z}^{2}, y \cdot x\right)}\]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(3, {z}^{2}, y \cdot x\right)\]

Reproduce

herbie shell --seed 2019325 +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)))