Average Error: 29.5 → 0.1
Time: 5.0s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10127.72860477168615034315735101699829102 \lor \neg \left(x \le 7255.079150232641040929593145847320556641\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{x + 1} - \frac{x + 1}{x - 1}}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10127.72860477168615034315735101699829102 \lor \neg \left(x \le 7255.079150232641040929593145847320556641\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{x + 1} - \frac{x + 1}{x - 1}}\right)\\

\end{array}
double f(double x) {
        double r171409 = x;
        double r171410 = 1.0;
        double r171411 = r171409 + r171410;
        double r171412 = r171409 / r171411;
        double r171413 = r171409 - r171410;
        double r171414 = r171411 / r171413;
        double r171415 = r171412 - r171414;
        return r171415;
}
↓
double f(double x) {
        double r171416 = x;
        double r171417 = -10127.728604771686;
        bool r171418 = r171416 <= r171417;
        double r171419 = 7255.079150232641;
        bool r171420 = r171416 <= r171419;
        double r171421 = !r171420;
        bool r171422 = r171418 || r171421;
        double r171423 = 1.0;
        double r171424 = -r171423;
        double r171425 = 2.0;
        double r171426 = pow(r171416, r171425);
        double r171427 = r171424 / r171426;
        double r171428 = 3.0;
        double r171429 = r171428 / r171416;
        double r171430 = r171427 - r171429;
        double r171431 = 3.0;
        double r171432 = pow(r171416, r171431);
        double r171433 = r171428 / r171432;
        double r171434 = r171430 - r171433;
        double r171435 = cbrt(r171416);
        double r171436 = r171435 * r171435;
        double r171437 = r171416 + r171423;
        double r171438 = r171435 / r171437;
        double r171439 = r171436 * r171438;
        double r171440 = r171416 - r171423;
        double r171441 = r171437 / r171440;
        double r171442 = r171439 - r171441;
        double r171443 = exp(r171442);
        double r171444 = log(r171443);
        double r171445 = r171422 ? r171434 : r171444;
        return r171445;
}

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 < -10127.728604771686 or 7255.079150232641 < x

    1. Initial program 59.2

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

    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

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

    if -10127.728604771686 < x < 7255.079150232641

    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 *-un-lft-identity0.1

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

    9. Applied add-cube-cbrt0.1

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

    10. Applied times-frac0.1

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

    11. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10127.72860477168615034315735101699829102 \lor \neg \left(x \le 7255.079150232641040929593145847320556641\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{x + 1} - \frac{x + 1}{x - 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))