Average Error: 0.1 → 0.1
Time: 8.6s
Precision: 64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
↓
\[3 \cdot \left(z \cdot z\right) + x \cdot y\]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
3 \cdot \left(z \cdot z\right) + x \cdot y
double f(double x, double y, double z) {
        double r468721 = x;
        double r468722 = y;
        double r468723 = r468721 * r468722;
        double r468724 = z;
        double r468725 = r468724 * r468724;
        double r468726 = r468723 + r468725;
        double r468727 = r468726 + r468725;
        double r468728 = r468727 + r468725;
        return r468728;
}
↓
double f(double x, double y, double z) {
        double r468729 = 3.0;
        double r468730 = z;
        double r468731 = r468730 * r468730;
        double r468732 = r468729 * r468731;
        double r468733 = x;
        double r468734 = y;
        double r468735 = r468733 * r468734;
        double r468736 = r468732 + r468735;
        return r468736;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

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Results

Enter valid numbers for all inputs

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. Using strategy rm

  4. Applied fma-udef0.1

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

  5. Final simplification0.1

    \[\leadsto 3 \cdot \left(z \cdot z\right) + x \cdot y\]

Reproduce

herbie shell --seed 2019194 +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)))