Average Error: 29.1 → 0.1
Time: 8.7s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11755.062723959316 \lor \neg \left(x \le 14618.9471916209168\right):\\ \;\;\;\;-\left(\left(\frac{3}{x} + \frac{1}{x \cdot x}\right) + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]
\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):\\
\;\;\;\;-\left(\left(\frac{3}{x} + \frac{1}{x \cdot 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 r103425 = x;
        double r103426 = 1.0;
        double r103427 = r103425 + r103426;
        double r103428 = r103425 / r103427;
        double r103429 = r103425 - r103426;
        double r103430 = r103427 / r103429;
        double r103431 = r103428 - r103430;
        return r103431;
}
↓
double f(double x) {
        double r103432 = x;
        double r103433 = -11755.062723959316;
        bool r103434 = r103432 <= r103433;
        double r103435 = 14618.947191620917;
        bool r103436 = r103432 <= r103435;
        double r103437 = !r103436;
        bool r103438 = r103434 || r103437;
        double r103439 = 3.0;
        double r103440 = r103439 / r103432;
        double r103441 = 1.0;
        double r103442 = r103432 * r103432;
        double r103443 = r103441 / r103442;
        double r103444 = r103440 + r103443;
        double r103445 = 3.0;
        double r103446 = pow(r103432, r103445);
        double r103447 = r103439 / r103446;
        double r103448 = r103444 + r103447;
        double r103449 = -r103448;
        double r103450 = r103432 + r103441;
        double r103451 = r103432 / r103450;
        double r103452 = r103451 * r103451;
        double r103453 = r103432 - r103441;
        double r103454 = r103450 / r103453;
        double r103455 = r103454 * r103454;
        double r103456 = r103452 - r103455;
        double r103457 = r103451 + r103454;
        double r103458 = r103456 / r103457;
        double r103459 = r103438 ? r103449 : r103458;
        return r103459;
}

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. Simplified0.0

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

    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

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

Reproduce

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