Average Error: 38.6 → 0.0
Time: 28.7s
Precision: 64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
↓
\[x \cdot \left(x + 2\right)\]
\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
x \cdot \left(x + 2\right)
double f(double x) {
        double r499284 = x;
        double r499285 = 1.0;
        double r499286 = r499284 + r499285;
        double r499287 = r499286 * r499286;
        double r499288 = r499287 - r499285;
        return r499288;
}
↓
double f(double x) {
        double r499289 = x;
        double r499290 = 2.0;
        double r499291 = r499289 + r499290;
        double r499292 = r499289 * r499291;
        return r499292;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 38.6

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

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2, x \cdot x\right)}\]

  4. Taylor expanded around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))