Average Error: 0.0 → 0.0
Time: 4.0s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\frac{{\left(\frac{1}{x - 1}\right)}^{3} + {\left(\frac{x}{x + 1}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{x + 1}, \frac{x}{x + 1} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\frac{{\left(\frac{1}{x - 1}\right)}^{3} + {\left(\frac{x}{x + 1}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{x + 1}, \frac{x}{x + 1} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}
double f(double x) {
        double r135424 = 1.0;
        double r135425 = x;
        double r135426 = r135425 - r135424;
        double r135427 = r135424 / r135426;
        double r135428 = r135425 + r135424;
        double r135429 = r135425 / r135428;
        double r135430 = r135427 + r135429;
        return r135430;
}
↓
double f(double x) {
        double r135431 = 1.0;
        double r135432 = x;
        double r135433 = r135432 - r135431;
        double r135434 = r135431 / r135433;
        double r135435 = 3.0;
        double r135436 = pow(r135434, r135435);
        double r135437 = r135432 + r135431;
        double r135438 = r135432 / r135437;
        double r135439 = pow(r135438, r135435);
        double r135440 = r135436 + r135439;
        double r135441 = r135438 - r135434;
        double r135442 = r135434 * r135434;
        double r135443 = fma(r135438, r135441, r135442);
        double r135444 = r135440 / r135443;
        return r135444;
}

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 flip3-+0.0

    \[\leadsto \color{blue}{\frac{{\left(\frac{1}{x - 1}\right)}^{3} + {\left(\frac{x}{x + 1}\right)}^{3}}{\frac{1}{x - 1} \cdot \frac{1}{x - 1} + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{1}{x - 1} \cdot \frac{x}{x + 1}\right)}}\]

  4. Simplified0.0

    \[\leadsto \frac{{\left(\frac{1}{x - 1}\right)}^{3} + {\left(\frac{x}{x + 1}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\frac{x}{x + 1}, \frac{x}{x + 1} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}}\]

  5. Final simplification0.0

    \[\leadsto \frac{{\left(\frac{1}{x - 1}\right)}^{3} + {\left(\frac{x}{x + 1}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{x + 1}, \frac{x}{x + 1} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))