Average Error: 29.6 → 0.2
Time: 6.0s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10426.9285871911725 \lor \neg \left(x \le 9614.2427998710336\right):\\ \;\;\;\;-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + 1}\right)\right) - \log \left(e^{\frac{x + 1}{x - 1}}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10426.9285871911725 \lor \neg \left(x \le 9614.2427998710336\right):\\
\;\;\;\;-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + 1}\right)\right) - \log \left(e^{\frac{x + 1}{x - 1}}\right)\\

\end{array}
double f(double x) {
        double r136358 = x;
        double r136359 = 1.0;
        double r136360 = r136358 + r136359;
        double r136361 = r136358 / r136360;
        double r136362 = r136358 - r136359;
        double r136363 = r136360 / r136362;
        double r136364 = r136361 - r136363;
        return r136364;
}
↓
double f(double x) {
        double r136365 = x;
        double r136366 = -10426.928587191172;
        bool r136367 = r136365 <= r136366;
        double r136368 = 9614.242799871034;
        bool r136369 = r136365 <= r136368;
        double r136370 = !r136369;
        bool r136371 = r136367 || r136370;
        double r136372 = 1.0;
        double r136373 = 1.0;
        double r136374 = 2.0;
        double r136375 = pow(r136365, r136374);
        double r136376 = r136373 / r136375;
        double r136377 = r136372 * r136376;
        double r136378 = 3.0;
        double r136379 = r136373 / r136365;
        double r136380 = r136378 * r136379;
        double r136381 = 3.0;
        double r136382 = pow(r136365, r136381);
        double r136383 = r136373 / r136382;
        double r136384 = r136378 * r136383;
        double r136385 = r136380 + r136384;
        double r136386 = r136377 + r136385;
        double r136387 = -r136386;
        double r136388 = r136365 + r136372;
        double r136389 = r136365 / r136388;
        double r136390 = log1p(r136389);
        double r136391 = expm1(r136390);
        double r136392 = r136365 - r136372;
        double r136393 = r136388 / r136392;
        double r136394 = exp(r136393);
        double r136395 = log(r136394);
        double r136396 = r136391 - r136395;
        double r136397 = r136371 ? r136387 : r136396;
        return r136397;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -10426.928587191172 or 9614.242799871034 < x

    1. Initial program 59.3

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

    3. Applied flip--59.3

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

    4. Taylor expanded around inf 0.3

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

    if -10426.928587191172 < x < 9614.242799871034

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\log \left(e^{\frac{x + 1}{x - 1}}\right)}\]
    4. Using strategy rm

    5. Applied expm1-log1p-u0.1

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + 1}\right)\right)} - \log \left(e^{\frac{x + 1}{x - 1}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10426.9285871911725 \lor \neg \left(x \le 9614.2427998710336\right):\\ \;\;\;\;-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + 1}\right)\right) - \log \left(e^{\frac{x + 1}{x - 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))