Average Error: 0.1 → 0.1
Time: 11.9s
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 r22564162 = x;
        double r22564163 = y;
        double r22564164 = r22564162 * r22564163;
        double r22564165 = z;
        double r22564166 = r22564165 * r22564165;
        double r22564167 = r22564164 + r22564166;
        double r22564168 = r22564167 + r22564166;
        double r22564169 = r22564168 + r22564166;
        return r22564169;
}
↓
double f(double x, double y, double z) {
        double r22564170 = 3.0;
        double r22564171 = z;
        double r22564172 = r22564171 * r22564171;
        double r22564173 = r22564170 * r22564172;
        double r22564174 = x;
        double r22564175 = y;
        double r22564176 = r22564174 * r22564175;
        double r22564177 = r22564173 + r22564176;
        return r22564177;
}

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 2019169 +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)))