Average Error: 0.0 → 0.0
Time: 11.1s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\frac{x}{x + 1} + \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\frac{x}{x + 1} + \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)
double f(double x) {
        double r4293423 = 1.0;
        double r4293424 = x;
        double r4293425 = r4293424 - r4293423;
        double r4293426 = r4293423 / r4293425;
        double r4293427 = r4293424 + r4293423;
        double r4293428 = r4293424 / r4293427;
        double r4293429 = r4293426 + r4293428;
        return r4293429;
}
↓
double f(double x) {
        double r4293430 = x;
        double r4293431 = 1.0;
        double r4293432 = r4293430 + r4293431;
        double r4293433 = r4293430 / r4293432;
        double r4293434 = -1.0;
        double r4293435 = fma(r4293430, r4293430, r4293434);
        double r4293436 = r4293431 / r4293435;
        double r4293437 = r4293436 * r4293432;
        double r4293438 = r4293433 + r4293437;
        return r4293438;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{1}{x - 1} + \frac{x}{x + 1}\]
  2. Using strategy rm

  3. Applied flip--0.0

    \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}} + \frac{x}{x + 1}\]

  4. Applied associate-/r/0.0

    \[\leadsto \color{blue}{\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)} + \frac{x}{x + 1}\]

  5. Simplified0.0

    \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x, x, -1\right)}} \cdot \left(x + 1\right) + \frac{x}{x + 1}\]

  6. Final simplification0.0

    \[\leadsto \frac{x}{x + 1} + \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))