Expanding a square

Average Error: 39.0 → 0

Time: 6.5s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r307305 = x;
        double r307306 = 1.0;
        double r307307 = r307305 + r307306;
        double r307308 = r307307 * r307307;
        double r307309 = r307308 - r307306;
        return r307309;
}

↓

double f(double x) {
        double r307310 = x;
        double r307311 = 2.0;
        double r307312 = r307310 * r307311;
        double r307313 = fma(r307310, r307310, r307312);
        return r307313;
}

Error

Bits error versus x

Derivation

1. Initial program 39.0

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

2. Simplified 0.0

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

3. Taylor expanded around 0 0.0

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

4. Simplified 0

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

5. Final simplification 0

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1) (+ x 1)) 1))