Expanding a square

Average Error: 38.9 → 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 r307301 = x;
        double r307302 = 1.0;
        double r307303 = r307301 + r307302;
        double r307304 = r307303 * r307303;
        double r307305 = r307304 - r307302;
        return r307305;
}

↓

double f(double x) {
        double r307306 = x;
        double r307307 = 2.0;
        double r307308 = r307306 * r307307;
        double r307309 = fma(r307306, r307306, r307308);
        return r307309;
}

Error

Bits error versus x

Derivation

1. Initial program 38.9

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

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 2019162 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1) (+ x 1)) 1))