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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.9912853724188615:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 12935.179381933913:\\ \;\;\;\;\frac{x}{1 + x} - \left(\sqrt{1 + x} \cdot \frac{1}{x - 1}\right) \cdot \sqrt{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \end{array}\]

double f(double x) {
        double r13751295 = x;
        double r13751296 = 1.0;
        double r13751297 = r13751295 + r13751296;
        double r13751298 = r13751295 / r13751297;
        double r13751299 = r13751295 - r13751296;
        double r13751300 = r13751297 / r13751299;
        double r13751301 = r13751298 - r13751300;
        return r13751301;
}

↓

double f(double x) {
        double r13751302 = x;
        double r13751303 = -0.9912853724188615;
        bool r13751304 = r13751302 <= r13751303;
        double r13751305 = -1.0;
        double r13751306 = r13751302 * r13751302;
        double r13751307 = r13751305 / r13751306;
        double r13751308 = -3.0;
        double r13751309 = r13751306 * r13751302;
        double r13751310 = r13751308 / r13751309;
        double r13751311 = r13751308 / r13751302;
        double r13751312 = r13751310 + r13751311;
        double r13751313 = r13751307 + r13751312;
        double r13751314 = 12935.179381933913;
        bool r13751315 = r13751302 <= r13751314;
        double r13751316 = 1.0;
        double r13751317 = r13751316 + r13751302;
        double r13751318 = r13751302 / r13751317;
        double r13751319 = sqrt(r13751317);
        double r13751320 = r13751302 - r13751316;
        double r13751321 = r13751316 / r13751320;
        double r13751322 = r13751319 * r13751321;
        double r13751323 = r13751322 * r13751319;
        double r13751324 = r13751318 - r13751323;
        double r13751325 = r13751315 ? r13751324 : r13751313;
        double r13751326 = r13751304 ? r13751313 : r13751325;
        return r13751326;
}

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -0.9912853724188615:\\
\;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\

\mathbf{elif}\;x \le 12935.179381933913:\\
\;\;\;\;\frac{x}{1 + x} - \left(\sqrt{1 + x} \cdot \frac{1}{x - 1}\right) \cdot \sqrt{1 + x}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\

\end{array}

Derivation

Split input into 2 regimes
if x < -0.9912853724188615 or 12935.179381933913 < x
1. Initial program 59.0
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Taylor expanded around -inf 0.5
  \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]
3. Simplified0.5
  \[\leadsto \color{blue}{3 \cdot \left(\frac{\frac{-1}{x}}{x \cdot x} + \frac{-1}{x}\right) + \frac{-1}{x \cdot x}}\]
4. Taylor expanded around -inf 0.5
  \[\leadsto \color{blue}{\left(-\left(3 \cdot \frac{1}{{x}^{3}} + 3 \cdot \frac{1}{x}\right)\right)} + \frac{-1}{x \cdot x}\]
5. Simplified0.2
  \[\leadsto \color{blue}{\left(\frac{-3}{x} + \frac{-3}{x \cdot \left(x \cdot x\right)}\right)} + \frac{-1}{x \cdot x}\]
if -0.9912853724188615 < x < 12935.179381933913
1. Initial program 0.0
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Using strategy rm
3. Applied div-inv0.0
  \[\leadsto \frac{x}{x + 1} - \color{blue}{\left(x + 1\right) \cdot \frac{1}{x - 1}}\]
4. Using strategy rm
5. Applied add-sqr-sqrt0.1
  \[\leadsto \frac{x}{x + 1} - \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)} \cdot \frac{1}{x - 1}\]
6. Applied associate-*l*0.1
  \[\leadsto \frac{x}{x + 1} - \color{blue}{\sqrt{x + 1} \cdot \left(\sqrt{x + 1} \cdot \frac{1}{x - 1}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9912853724188615:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 12935.179381933913:\\ \;\;\;\;\frac{x}{1 + x} - \left(\sqrt{1 + x} \cdot \frac{1}{x - 1}\right) \cdot \sqrt{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \end{array}\]

Error

Derivation

`if x < -0.9912853724188615 or 12935.179381933913 < x`

`if -0.9912853724188615 < x < 12935.179381933913`

Reproduce