Average Error: 39.0 → 0.0
Time: 12.4s
Precision: 64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
↓
\[x \cdot x + x \cdot 2\]
\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
x \cdot x + x \cdot 2
double f(double x) {
        double r570893 = x;
        double r570894 = 1.0;
        double r570895 = r570893 + r570894;
        double r570896 = r570895 * r570895;
        double r570897 = r570896 - r570894;
        return r570897;
}
↓
double f(double x) {
        double r570898 = x;
        double r570899 = r570898 * r570898;
        double r570900 = 2.0;
        double r570901 = r570898 * r570900;
        double r570902 = r570899 + r570901;
        return r570902;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 39.0

    \[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\left(x + 2\right) \cdot x}\]
  3. Using strategy rm

  4. Applied flip-+0.0

    \[\leadsto \color{blue}{\frac{x \cdot x - 2 \cdot 2}{x - 2}} \cdot x\]

  5. Applied associate-*l/6.7

    \[\leadsto \color{blue}{\frac{\left(x \cdot x - 2 \cdot 2\right) \cdot x}{x - 2}}\]

  6. Simplified6.7

    \[\leadsto \frac{\color{blue}{x \cdot \left(-4 + x \cdot x\right)}}{x - 2}\]

  7. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{2 \cdot x + {x}^{2}}\]

  8. Simplified0.0

    \[\leadsto \color{blue}{x \cdot x + 2 \cdot x}\]

  9. Final simplification0.0

    \[\leadsto x \cdot x + x \cdot 2\]

Reproduce

herbie shell --seed 2019141 
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1) (+ x 1)) 1))