Average Error: 38.3 → 0.0
Time: 1.6s
Precision: 64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
↓
\[\mathsf{fma}\left(x, 2, {x}^{2}\right)\]
\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
\mathsf{fma}\left(x, 2, {x}^{2}\right)
double f(double x) {
        double r2617 = x;
        double r2618 = 1.0;
        double r2619 = r2617 + r2618;
        double r2620 = r2619 * r2619;
        double r2621 = r2620 - r2618;
        return r2621;
}
↓
double f(double x) {
        double r2622 = x;
        double r2623 = 2.0;
        double r2624 = 2.0;
        double r2625 = pow(r2622, r2624);
        double r2626 = fma(r2622, r2623, r2625);
        return r2626;
}

Error

Bits error versus x

Derivation

  1. Initial program 38.3

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

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))