Average Error: 0.1 → 0.1
Time: 2.0s
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 r440665 = x;
        double r440666 = y;
        double r440667 = r440665 * r440666;
        double r440668 = z;
        double r440669 = r440668 * r440668;
        double r440670 = r440667 + r440669;
        double r440671 = r440670 + r440669;
        double r440672 = r440671 + r440669;
        return r440672;
}
↓
double f(double x, double y, double z) {
        double r440673 = 3.0;
        double r440674 = z;
        double r440675 = r440673 * r440674;
        double r440676 = r440675 * r440674;
        double r440677 = x;
        double r440678 = y;
        double r440679 = r440677 * r440678;
        double r440680 = r440676 + r440679;
        return r440680;
}

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 2020039 +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)))