Average Error: 29.7 → 0.1
Time: 21.7s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -15309.12152708683424862101674079895019531 \lor \neg \left(x \le 12455.49900284703107899986207485198974609\right):\\ \;\;\;\;-\left(\left(\frac{1}{x \cdot x} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -15309.12152708683424862101674079895019531 \lor \neg \left(x \le 12455.49900284703107899986207485198974609\right):\\
\;\;\;\;-\left(\left(\frac{1}{x \cdot x} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\

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

\end{array}
double f(double x) {
        double r119394 = x;
        double r119395 = 1.0;
        double r119396 = r119394 + r119395;
        double r119397 = r119394 / r119396;
        double r119398 = r119394 - r119395;
        double r119399 = r119396 / r119398;
        double r119400 = r119397 - r119399;
        return r119400;
}
↓
double f(double x) {
        double r119401 = x;
        double r119402 = -15309.121527086834;
        bool r119403 = r119401 <= r119402;
        double r119404 = 12455.499002847031;
        bool r119405 = r119401 <= r119404;
        double r119406 = !r119405;
        bool r119407 = r119403 || r119406;
        double r119408 = 1.0;
        double r119409 = r119401 * r119401;
        double r119410 = r119408 / r119409;
        double r119411 = 3.0;
        double r119412 = r119411 / r119401;
        double r119413 = r119410 + r119412;
        double r119414 = 3.0;
        double r119415 = pow(r119401, r119414);
        double r119416 = r119411 / r119415;
        double r119417 = r119413 + r119416;
        double r119418 = -r119417;
        double r119419 = r119401 + r119408;
        double r119420 = r119401 / r119419;
        double r119421 = r119401 - r119408;
        double r119422 = r119419 / r119421;
        double r119423 = -r119422;
        double r119424 = r119420 + r119423;
        double r119425 = r119407 ? r119418 : r119424;
        return r119425;
}

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 < -15309.121527086834 or 12455.499002847031 < x

    1. Initial program 59.4

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

    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

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

    if -15309.121527086834 < x < 12455.499002847031

    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

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

    8. Applied sub-neg0.1

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

    9. Applied exp-sum0.1

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

    10. Applied log-prod0.1

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

    11. Simplified0.1

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

    12. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -15309.12152708683424862101674079895019531 \lor \neg \left(x \le 12455.49900284703107899986207485198974609\right):\\ \;\;\;\;-\left(\left(\frac{1}{x \cdot x} + \frac{3}{x}\right) + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))