Average Error: 29.4 → 0.2
Time: 8.2s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11098.444471894443 \lor \neg \left(x \le 0.998861424289309352\right):\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{{x}^{3}}, \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{x \cdot x - 1 \cdot 1}, x - 1, -\left(x + 1\right) \cdot \frac{x + 1}{x \cdot x - 1 \cdot 1}\right) + \frac{x + 1}{x \cdot x - 1 \cdot 1} \cdot \left(\left(-\left(x + 1\right)\right) + \left(x + 1\right)\right)} \cdot \sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -11098.444471894443 \lor \neg \left(x \le 0.998861424289309352\right):\\
\;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{{x}^{3}}, \frac{3}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{x \cdot x - 1 \cdot 1}, x - 1, -\left(x + 1\right) \cdot \frac{x + 1}{x \cdot x - 1 \cdot 1}\right) + \frac{x + 1}{x \cdot x - 1 \cdot 1} \cdot \left(\left(-\left(x + 1\right)\right) + \left(x + 1\right)\right)} \cdot \sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\\

\end{array}
double f(double x) {
        double r149346 = x;
        double r149347 = 1.0;
        double r149348 = r149346 + r149347;
        double r149349 = r149346 / r149348;
        double r149350 = r149346 - r149347;
        double r149351 = r149348 / r149350;
        double r149352 = r149349 - r149351;
        return r149352;
}
↓
double f(double x) {
        double r149353 = x;
        double r149354 = -11098.444471894443;
        bool r149355 = r149353 <= r149354;
        double r149356 = 0.9988614242893094;
        bool r149357 = r149353 <= r149356;
        double r149358 = !r149357;
        bool r149359 = r149355 || r149358;
        double r149360 = 1.0;
        double r149361 = -r149360;
        double r149362 = 2.0;
        double r149363 = pow(r149353, r149362);
        double r149364 = r149361 / r149363;
        double r149365 = 3.0;
        double r149366 = 1.0;
        double r149367 = 3.0;
        double r149368 = pow(r149353, r149367);
        double r149369 = r149366 / r149368;
        double r149370 = r149365 / r149353;
        double r149371 = fma(r149365, r149369, r149370);
        double r149372 = r149364 - r149371;
        double r149373 = r149353 * r149353;
        double r149374 = r149360 * r149360;
        double r149375 = r149373 - r149374;
        double r149376 = r149353 / r149375;
        double r149377 = r149353 - r149360;
        double r149378 = r149353 + r149360;
        double r149379 = r149378 / r149375;
        double r149380 = r149378 * r149379;
        double r149381 = -r149380;
        double r149382 = fma(r149376, r149377, r149381);
        double r149383 = -r149378;
        double r149384 = r149383 + r149378;
        double r149385 = r149379 * r149384;
        double r149386 = r149382 + r149385;
        double r149387 = sqrt(r149386);
        double r149388 = r149353 / r149378;
        double r149389 = r149378 / r149377;
        double r149390 = r149388 - r149389;
        double r149391 = sqrt(r149390);
        double r149392 = r149387 * r149391;
        double r149393 = r149359 ? r149372 : r149392;
        return r149393;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -11098.444471894443 or 0.9988614242893094 < x

    1. Initial program 58.9

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

    2. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)}\]

    3. Simplified0.6

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)}\]

    4. Taylor expanded around 0 0.6

      \[\leadsto \frac{-1}{{x}^{2}} - \color{blue}{\left(3 \cdot \frac{1}{{x}^{3}} + 3 \cdot \frac{1}{x}\right)}\]

    5. Simplified0.3

      \[\leadsto \frac{-1}{{x}^{2}} - \color{blue}{\mathsf{fma}\left(3, \frac{1}{{x}^{3}}, \frac{3}{x}\right)}\]

    if -11098.444471894443 < x < 0.9988614242893094

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

      \[\leadsto \color{blue}{\sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}}\]
    4. Using strategy rm

    5. Applied flip--0.1

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

    6. Applied associate-/r/0.1

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

    7. Applied flip-+0.1

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

    8. Applied associate-/r/0.1

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

    9. Applied prod-diff0.1

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

    10. Simplified0.1

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{x \cdot x - 1 \cdot 1}, x - 1, -\left(x + 1\right) \cdot \frac{x + 1}{x \cdot x - 1 \cdot 1}\right) + \color{blue}{\frac{x + 1}{x \cdot x - 1 \cdot 1} \cdot \left(\left(-\left(x + 1\right)\right) + \left(x + 1\right)\right)}} \cdot \sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11098.444471894443 \lor \neg \left(x \le 0.998861424289309352\right):\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{{x}^{3}}, \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{x \cdot x - 1 \cdot 1}, x - 1, -\left(x + 1\right) \cdot \frac{x + 1}{x \cdot x - 1 \cdot 1}\right) + \frac{x + 1}{x \cdot x - 1 \cdot 1} \cdot \left(\left(-\left(x + 1\right)\right) + \left(x + 1\right)\right)} \cdot \sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))