Average Error: 0.1 → 0.1
Time: 2.3s
Precision: 64
\[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y\]
↓
\[\left(3 \cdot y\right) \cdot y + x \cdot x\]
\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y
↓
\left(3 \cdot y\right) \cdot y + x \cdot x
double f(double x, double y) {
        double r405824 = x;
        double r405825 = r405824 * r405824;
        double r405826 = y;
        double r405827 = r405826 * r405826;
        double r405828 = r405825 + r405827;
        double r405829 = r405828 + r405827;
        double r405830 = r405829 + r405827;
        return r405830;
}
↓
double f(double x, double y) {
        double r405831 = 3.0;
        double r405832 = y;
        double r405833 = r405831 * r405832;
        double r405834 = r405833 * r405832;
        double r405835 = x;
        double r405836 = r405835 * r405835;
        double r405837 = r405834 + r405836;
        return r405837;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.1
Target0.1
Herbie0.1
\[x \cdot x + y \cdot \left(y + \left(y + y\right)\right)\]

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, y \cdot y, x \cdot x\right)}\]
  3. Using strategy rm

  4. Applied fma-udef0.1

    \[\leadsto \color{blue}{3 \cdot \left(y \cdot y\right) + x \cdot x}\]
  5. Using strategy rm

  6. Applied associate-*r*0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

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

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