Average Error: 29.6 → 0.1
Time: 11.5s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10513.46193108966144791338592767715454102 \lor \neg \left(x \le 11268.5444087090854736743494868278503418\right):\\ \;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x + 1} - \left(x + 1\right) \cdot \frac{1}{x - 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10513.46193108966144791338592767715454102 \lor \neg \left(x \le 11268.5444087090854736743494868278503418\right):\\
\;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

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

\end{array}
double f(double x) {
        double r106322 = x;
        double r106323 = 1.0;
        double r106324 = r106322 + r106323;
        double r106325 = r106322 / r106324;
        double r106326 = r106322 - r106323;
        double r106327 = r106324 / r106326;
        double r106328 = r106325 - r106327;
        return r106328;
}
↓
double f(double x) {
        double r106329 = x;
        double r106330 = -10513.461931089661;
        bool r106331 = r106329 <= r106330;
        double r106332 = 11268.544408709085;
        bool r106333 = r106329 <= r106332;
        double r106334 = !r106333;
        bool r106335 = r106331 || r106334;
        double r106336 = 1.0;
        double r106337 = r106329 * r106329;
        double r106338 = r106336 / r106337;
        double r106339 = -r106338;
        double r106340 = 3.0;
        double r106341 = r106340 / r106329;
        double r106342 = 3.0;
        double r106343 = pow(r106329, r106342);
        double r106344 = r106340 / r106343;
        double r106345 = r106341 + r106344;
        double r106346 = r106339 - r106345;
        double r106347 = 1.0;
        double r106348 = r106329 + r106336;
        double r106349 = r106347 / r106348;
        double r106350 = r106329 * r106349;
        double r106351 = r106329 - r106336;
        double r106352 = r106347 / r106351;
        double r106353 = r106348 * r106352;
        double r106354 = r106350 - r106353;
        double r106355 = r106335 ? r106346 : r106354;
        return r106355;
}

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 < -10513.461931089661 or 11268.544408709085 < x

    1. Initial program 59.2

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

    3. Applied div-inv59.4

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

    4. Taylor expanded around inf 0.3

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

    5. Simplified0.0

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

    if -10513.461931089661 < x < 11268.544408709085

    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. Using strategy rm

    5. Applied div-inv0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10513.46193108966144791338592767715454102 \lor \neg \left(x \le 11268.5444087090854736743494868278503418\right):\\ \;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x + 1} - \left(x + 1\right) \cdot \frac{1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))