Average Error: 0.1 → 0.1
Time: 2.1s
Precision: 64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z\]
↓
\[\left(3 \cdot z\right) \cdot z + x \cdot y\]
\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\left(3 \cdot z\right) \cdot z + x \cdot y
double f(double x, double y, double z) {
        double r500903 = x;
        double r500904 = y;
        double r500905 = r500903 * r500904;
        double r500906 = z;
        double r500907 = r500906 * r500906;
        double r500908 = r500905 + r500907;
        double r500909 = r500908 + r500907;
        double r500910 = r500909 + r500907;
        return r500910;
}
↓
double f(double x, double y, double z) {
        double r500911 = 3.0;
        double r500912 = z;
        double r500913 = r500911 * r500912;
        double r500914 = r500913 * r500912;
        double r500915 = x;
        double r500916 = y;
        double r500917 = r500915 * r500916;
        double r500918 = r500914 + r500917;
        return r500918;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

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. Using strategy rm

  6. Applied associate-*r*0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020062 +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)))