Average Error: 0.1 → 0.1
Time: 15.0s
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 r21916001 = x;
        double r21916002 = y;
        double r21916003 = r21916001 * r21916002;
        double r21916004 = z;
        double r21916005 = r21916004 * r21916004;
        double r21916006 = r21916003 + r21916005;
        double r21916007 = r21916006 + r21916005;
        double r21916008 = r21916007 + r21916005;
        return r21916008;
}
↓
double f(double x, double y, double z) {
        double r21916009 = 3.0;
        double r21916010 = z;
        double r21916011 = r21916010 * r21916010;
        double r21916012 = x;
        double r21916013 = y;
        double r21916014 = r21916012 * r21916013;
        double r21916015 = fma(r21916009, r21916011, r21916014);
        return r21916015;
}

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 2019163 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"

  :herbie-target
  (+ (* (* 3 z) z) (* y x))

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