Average Error: 29.0 → 0.2
Time: 16.7s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12435.32938867169286822900176048278808594 \lor \neg \left(x \le 11573.99856257406463555525988340377807617\right):\\ \;\;\;\;\frac{-1}{x} \cdot \left(3 + \frac{1}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -12435.32938867169286822900176048278808594 \lor \neg \left(x \le 11573.99856257406463555525988340377807617\right):\\
\;\;\;\;\frac{-1}{x} \cdot \left(3 + \frac{1}{x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r122536 = x;
        double r122537 = 1.0;
        double r122538 = r122536 + r122537;
        double r122539 = r122536 / r122538;
        double r122540 = r122536 - r122537;
        double r122541 = r122538 / r122540;
        double r122542 = r122539 - r122541;
        return r122542;
}
↓
double f(double x) {
        double r122543 = x;
        double r122544 = -12435.329388671693;
        bool r122545 = r122543 <= r122544;
        double r122546 = 11573.998562574065;
        bool r122547 = r122543 <= r122546;
        double r122548 = !r122547;
        bool r122549 = r122545 || r122548;
        double r122550 = -1.0;
        double r122551 = r122550 / r122543;
        double r122552 = 3.0;
        double r122553 = 1.0;
        double r122554 = r122553 / r122543;
        double r122555 = r122552 + r122554;
        double r122556 = r122551 * r122555;
        double r122557 = 3.0;
        double r122558 = pow(r122543, r122557);
        double r122559 = r122552 / r122558;
        double r122560 = r122556 - r122559;
        double r122561 = r122543 + r122553;
        double r122562 = r122543 / r122561;
        double r122563 = r122543 - r122553;
        double r122564 = r122561 / r122563;
        double r122565 = r122562 - r122564;
        double r122566 = exp(r122565);
        double r122567 = log(r122566);
        double r122568 = r122549 ? r122560 : r122567;
        return r122568;
}

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 < -12435.329388671693 or 11573.998562574065 < x

    1. Initial program 59.3

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

    3. Applied add-log-exp59.3

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

    4. Applied add-log-exp59.3

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

    5. Applied diff-log59.3

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

    6. Simplified59.3

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

    7. Taylor expanded around inf 0.3

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

    8. Simplified0.3

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

    if -12435.329388671693 < x < 11573.998562574065

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied add-log-exp0.1

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

    5. Applied diff-log0.1

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

    6. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12435.32938867169286822900176048278808594 \lor \neg \left(x \le 11573.99856257406463555525988340377807617\right):\\ \;\;\;\;\frac{-1}{x} \cdot \left(3 + \frac{1}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))