Expanding a square

Average Error: 39.4 → 0

Time: 6.1s

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 r307314 = x;
        double r307315 = 1.0;
        double r307316 = r307314 + r307315;
        double r307317 = r307316 * r307316;
        double r307318 = r307317 - r307315;
        return r307318;
}

↓

double f(double x) {
        double r307319 = x;
        double r307320 = 2.0;
        double r307321 = r307319 * r307320;
        double r307322 = fma(r307319, r307319, r307321);
        return r307322;
}

Error

Bits error versus x

Derivation

1. Initial program 39.4

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

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 2019164 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1) (+ x 1)) 1))