Average Error: 0.1 → 0.1
Time: 9.3s
Precision: 64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
↓
\[\mathsf{fma}\left(3, z \cdot z, x \cdot y\right)\]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\mathsf{fma}\left(3, z \cdot z, x \cdot y\right)
double f(double x, double y, double z) {
        double r476732 = x;
        double r476733 = y;
        double r476734 = r476732 * r476733;
        double r476735 = z;
        double r476736 = r476735 * r476735;
        double r476737 = r476734 + r476736;
        double r476738 = r476737 + r476736;
        double r476739 = r476738 + r476736;
        return r476739;
}
↓
double f(double x, double y, double z) {
        double r476740 = 3.0;
        double r476741 = z;
        double r476742 = r476741 * r476741;
        double r476743 = x;
        double r476744 = y;
        double r476745 = r476743 * r476744;
        double r476746 = fma(r476740, r476742, r476745);
        return r476746;
}

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. Taylor expanded around 0 0.1

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

  3. Simplified0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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)))