Expanding a square

Average Error: 39.0 → 0

Time: 9.7s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r341442 = x;
        double r341443 = 1.0;
        double r341444 = r341442 + r341443;
        double r341445 = r341444 * r341444;
        double r341446 = r341445 - r341443;
        return r341446;
}

↓

double f(double x) {
        double r341447 = x;
        double r341448 = 2.0;
        double r341449 = r341447 * r341448;
        double r341450 = fma(r341447, r341447, r341449);
        return r341450;
}

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

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

4. Applied distribute-lft-in 0.0

   \[\leadsto \color{blue}{x \cdot x + x \cdot 2}\]
5. Using strategy rm

6. Applied fma-def 0

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

7. Final simplification 0

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1) (+ x 1)) 1))