Average Error: 0.0 → 0.0
Time: 1.2s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[x \cdot \left(2 \cdot y + x\right) + y \cdot y\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
x \cdot \left(2 \cdot y + x\right) + y \cdot y
double f(double x, double y) {
        double r849982 = x;
        double r849983 = r849982 * r849982;
        double r849984 = 2.0;
        double r849985 = r849982 * r849984;
        double r849986 = y;
        double r849987 = r849985 * r849986;
        double r849988 = r849983 + r849987;
        double r849989 = r849986 * r849986;
        double r849990 = r849988 + r849989;
        return r849990;
}
↓
double f(double x, double y) {
        double r849991 = x;
        double r849992 = 2.0;
        double r849993 = y;
        double r849994 = r849992 * r849993;
        double r849995 = r849994 + r849991;
        double r849996 = r849991 * r849995;
        double r849997 = r849993 * r849993;
        double r849998 = r849996 + r849997;
        return r849998;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.0
Target0.0
Herbie0.0
\[x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right)\]

Derivation

  1. Initial program 0.0

    \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left({x}^{2} + 2 \cdot \left(x \cdot y\right)\right)} + y \cdot y\]

  3. Simplified0.0

    \[\leadsto \color{blue}{x \cdot \left(2 \cdot y + x\right)} + y \cdot y\]

  4. Final simplification0.0

    \[\leadsto x \cdot \left(2 \cdot y + x\right) + y \cdot y\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2)))

  (+ (+ (* x x) (* (* x 2) y)) (* y y)))