Average Error: 29.1 → 0.1
Time: 4.0s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11755.062723959316 \lor \neg \left(x \le 14618.9471916209168\right):\\ \;\;\;\;\frac{0}{{x}^{2}} - \left(\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -11755.062723959316 \lor \neg \left(x \le 14618.9471916209168\right):\\
\;\;\;\;\frac{0}{{x}^{2}} - \left(\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\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}}\\

\end{array}
double f(double x) {
        double r159337 = x;
        double r159338 = 1.0;
        double r159339 = r159337 + r159338;
        double r159340 = r159337 / r159339;
        double r159341 = r159337 - r159338;
        double r159342 = r159339 / r159341;
        double r159343 = r159340 - r159342;
        return r159343;
}
↓
double f(double x) {
        double r159344 = x;
        double r159345 = -11755.062723959316;
        bool r159346 = r159344 <= r159345;
        double r159347 = 14618.947191620917;
        bool r159348 = r159344 <= r159347;
        double r159349 = !r159348;
        bool r159350 = r159346 || r159349;
        double r159351 = 0.0;
        double r159352 = 2.0;
        double r159353 = pow(r159344, r159352);
        double r159354 = r159351 / r159353;
        double r159355 = 1.0;
        double r159356 = r159355 / r159353;
        double r159357 = 3.0;
        double r159358 = r159357 / r159344;
        double r159359 = r159356 + r159358;
        double r159360 = 3.0;
        double r159361 = pow(r159344, r159360);
        double r159362 = r159357 / r159361;
        double r159363 = r159359 + r159362;
        double r159364 = r159354 - r159363;
        double r159365 = r159344 + r159355;
        double r159366 = r159344 / r159365;
        double r159367 = r159366 * r159366;
        double r159368 = r159344 - r159355;
        double r159369 = r159365 / r159368;
        double r159370 = r159369 * r159369;
        double r159371 = r159367 - r159370;
        double r159372 = r159366 + r159369;
        double r159373 = r159371 / r159372;
        double r159374 = r159350 ? r159364 : r159373;
        return r159374;
}

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 < -11755.062723959316 or 14618.947191620917 < x

    1. Initial program 59.5

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

    5. Applied neg-sub00.3

      \[\leadsto \frac{\color{blue}{0 - 1}}{{x}^{2}} - 3 \cdot \left(\frac{1}{x} + \frac{1}{{x}^{3}}\right)\]

    6. Applied div-sub0.3

      \[\leadsto \color{blue}{\left(\frac{0}{{x}^{2}} - \frac{1}{{x}^{2}}\right)} - 3 \cdot \left(\frac{1}{x} + \frac{1}{{x}^{3}}\right)\]

    7. Applied associate--l-0.3

      \[\leadsto \color{blue}{\frac{0}{{x}^{2}} - \left(\frac{1}{{x}^{2}} + 3 \cdot \left(\frac{1}{x} + \frac{1}{{x}^{3}}\right)\right)}\]

    8. Simplified0.0

      \[\leadsto \frac{0}{{x}^{2}} - \color{blue}{\left(\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)}\]

    if -11755.062723959316 < x < 14618.947191620917

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11755.062723959316 \lor \neg \left(x \le 14618.9471916209168\right):\\ \;\;\;\;\frac{0}{{x}^{2}} - \left(\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))