Average Error: 0.1 → 0.1
Time: 6.6s
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 r2237 = x;
        double r2238 = y;
        double r2239 = r2237 * r2238;
        double r2240 = z;
        double r2241 = r2240 * r2240;
        double r2242 = r2239 + r2241;
        double r2243 = r2242 + r2241;
        double r2244 = r2243 + r2241;
        return r2244;
}
↓
double f(double x, double y, double z) {
        double r2245 = 3.0;
        double r2246 = z;
        double r2247 = r2246 * r2246;
        double r2248 = x;
        double r2249 = y;
        double r2250 = r2248 * r2249;
        double r2251 = fma(r2245, r2247, r2250);
        return r2251;
}

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 2020025 +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)))