Average Error: 0.0 → 0.0
Time: 1.4s
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 r635010 = x;
        double r635011 = r635010 * r635010;
        double r635012 = 2.0;
        double r635013 = r635010 * r635012;
        double r635014 = y;
        double r635015 = r635013 * r635014;
        double r635016 = r635011 + r635015;
        double r635017 = r635014 * r635014;
        double r635018 = r635016 + r635017;
        return r635018;
}
↓
double f(double x, double y) {
        double r635019 = x;
        double r635020 = 2.0;
        double r635021 = y;
        double r635022 = r635020 * r635021;
        double r635023 = r635022 + r635019;
        double r635024 = r635019 * r635023;
        double r635025 = r635021 * r635021;
        double r635026 = r635024 + r635025;
        return r635026;
}

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 2020001 
(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)))