Average Error: 0.1 → 0.1
Time: 4.2s
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 r1233132 = x;
        double r1233133 = y;
        double r1233134 = r1233132 * r1233133;
        double r1233135 = z;
        double r1233136 = r1233135 * r1233135;
        double r1233137 = r1233134 + r1233136;
        double r1233138 = r1233137 + r1233136;
        double r1233139 = r1233138 + r1233136;
        return r1233139;
}
↓
double f(double x, double y, double z) {
        double r1233140 = 3.0;
        double r1233141 = z;
        double r1233142 = r1233141 * r1233141;
        double r1233143 = x;
        double r1233144 = y;
        double r1233145 = r1233143 * r1233144;
        double r1233146 = fma(r1233140, r1233142, r1233145);
        return r1233146;
}

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 2020034 +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)))