Average Error: 0.1 → 0.1
Time: 20.1s
Precision: 64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
↓
\[\mathsf{fma}\left(y, x, \left(3 \cdot z\right) \cdot z\right)\]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\mathsf{fma}\left(y, x, \left(3 \cdot z\right) \cdot z\right)
double f(double x, double y, double z) {
        double r467149 = x;
        double r467150 = y;
        double r467151 = r467149 * r467150;
        double r467152 = z;
        double r467153 = r467152 * r467152;
        double r467154 = r467151 + r467153;
        double r467155 = r467154 + r467153;
        double r467156 = r467155 + r467153;
        return r467156;
}
↓
double f(double x, double y, double z) {
        double r467157 = y;
        double r467158 = x;
        double r467159 = 3.0;
        double r467160 = z;
        double r467161 = r467159 * r467160;
        double r467162 = r467161 * r467160;
        double r467163 = fma(r467157, r467158, r467162);
        return r467163;
}

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. Using strategy rm

  4. Applied associate-*r*0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019323 +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)))