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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -13946.2737556041702 \lor \neg \left(x \le 10454.066790945864\right):\\ \;\;\;\;\left(-\frac{3}{{x}^{3}}\right) - \left(\frac{3}{x} + \frac{1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, \frac{-\left(x + 1\right)}{x - 1}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -13946.2737556041702 \lor \neg \left(x \le 10454.066790945864\right):\\
\;\;\;\;\left(-\frac{3}{{x}^{3}}\right) - \left(\frac{3}{x} + \frac{1}{x \cdot x}\right)\\

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

\end{array}

double f(double x) {
        double r126338 = x;
        double r126339 = 1.0;
        double r126340 = r126338 + r126339;
        double r126341 = r126338 / r126340;
        double r126342 = r126338 - r126339;
        double r126343 = r126340 / r126342;
        double r126344 = r126341 - r126343;
        return r126344;
}

↓

double f(double x) {
        double r126345 = x;
        double r126346 = -13946.27375560417;
        bool r126347 = r126345 <= r126346;
        double r126348 = 10454.066790945864;
        bool r126349 = r126345 <= r126348;
        double r126350 = !r126349;
        bool r126351 = r126347 || r126350;
        double r126352 = 3.0;
        double r126353 = 3.0;
        double r126354 = pow(r126345, r126353);
        double r126355 = r126352 / r126354;
        double r126356 = -r126355;
        double r126357 = r126352 / r126345;
        double r126358 = 1.0;
        double r126359 = r126345 * r126345;
        double r126360 = r126358 / r126359;
        double r126361 = r126357 + r126360;
        double r126362 = r126356 - r126361;
        double r126363 = 1.0;
        double r126364 = r126345 + r126358;
        double r126365 = r126363 / r126364;
        double r126366 = -r126364;
        double r126367 = r126345 - r126358;
        double r126368 = r126366 / r126367;
        double r126369 = fma(r126345, r126365, r126368);
        double r126370 = r126351 ? r126362 : r126369;
        return r126370;
}

Derivation

Split input into 2 regimes
if x < -13946.27375560417 or 10454.066790945864 < x
1. Initial program 59.4
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Using strategy rm
3. Applied div-inv59.6
  \[\leadsto \color{blue}{x \cdot \frac{1}{x + 1}} - \frac{x + 1}{x - 1}\]
4. Applied fma-neg60.3
  \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{x + 1}{x - 1}\right)}\]
5. Simplified60.3
  \[\leadsto \mathsf{fma}\left(x, \frac{1}{x + 1}, \color{blue}{\frac{-\left(x + 1\right)}{x - 1}}\right)\]
6. Taylor expanded around inf 0.3
  \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(1 \cdot \frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]
7. Simplified0.0
  \[\leadsto \color{blue}{\left(-\frac{3}{{x}^{3}}\right) - \left(\frac{3}{x} + \frac{1}{x \cdot x}\right)}\]
if -13946.27375560417 < x < 10454.066790945864
1. Initial program 0.1
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Using strategy rm
3. Applied div-inv0.1
  \[\leadsto \color{blue}{x \cdot \frac{1}{x + 1}} - \frac{x + 1}{x - 1}\]
4. Applied fma-neg0.1
  \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{x + 1}, -\frac{x + 1}{x - 1}\right)}\]
5. Simplified0.1
  \[\leadsto \mathsf{fma}\left(x, \frac{1}{x + 1}, \color{blue}{\frac{-\left(x + 1\right)}{x - 1}}\right)\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13946.2737556041702 \lor \neg \left(x \le 10454.066790945864\right):\\ \;\;\;\;\left(-\frac{3}{{x}^{3}}\right) - \left(\frac{3}{x} + \frac{1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{x + 1}, \frac{-\left(x + 1\right)}{x - 1}\right)\\ \end{array}\]

Error

Derivation

`if x < -13946.27375560417 or 10454.066790945864 < x`

`if -13946.27375560417 < x < 10454.066790945864`

Reproduce