Average Error: 0.1 → 0.1
Time: 15.6s
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 r21260138 = x;
        double r21260139 = y;
        double r21260140 = r21260138 * r21260139;
        double r21260141 = z;
        double r21260142 = r21260141 * r21260141;
        double r21260143 = r21260140 + r21260142;
        double r21260144 = r21260143 + r21260142;
        double r21260145 = r21260144 + r21260142;
        return r21260145;
}
↓
double f(double x, double y, double z) {
        double r21260146 = 3.0;
        double r21260147 = z;
        double r21260148 = r21260147 * r21260147;
        double r21260149 = x;
        double r21260150 = y;
        double r21260151 = r21260149 * r21260150;
        double r21260152 = fma(r21260146, r21260148, r21260151);
        return r21260152;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"

  :herbie-target
  (+ (* (* 3.0 z) z) (* y x))

  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))