Average Error: 39.6 → 0
Time: 3.8s
Precision: 64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
↓
\[\mathsf{fma}\left(x, x, x + x\right)\]
\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
\mathsf{fma}\left(x, x, x + x\right)
double f(double x) {
        double r99277 = x;
        double r99278 = 1.0;
        double r99279 = r99277 + r99278;
        double r99280 = r99279 * r99279;
        double r99281 = r99280 - r99278;
        return r99281;
}
↓
double f(double x) {
        double r99282 = x;
        double r99283 = r99282 + r99282;
        double r99284 = fma(r99282, r99282, r99283);
        return r99284;
}

Error

Bits error versus x

Derivation

  1. Initial program 39.6

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

  2. Simplified0

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

  3. Final simplification0

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1) (+ x 1)) 1))