Average Error: 29.1 → 0.1
Time: 7.1s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10587.250289808897 \lor \neg \left(x \le 11866.5124804357783\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\left(x \cdot x\right) \cdot x}{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(x + 1\right)} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}\right)}{\frac{x + 1}{x - 1} \cdot \left(\frac{x + 1}{x - 1} + \frac{x}{x + 1}\right) + \frac{x}{x + 1} \cdot \frac{x}{x + 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10587.250289808897 \lor \neg \left(x \le 11866.5124804357783\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r121505 = x;
        double r121506 = 1.0;
        double r121507 = r121505 + r121506;
        double r121508 = r121505 / r121507;
        double r121509 = r121505 - r121506;
        double r121510 = r121507 / r121509;
        double r121511 = r121508 - r121510;
        return r121511;
}
↓
double f(double x) {
        double r121512 = x;
        double r121513 = -10587.250289808897;
        bool r121514 = r121512 <= r121513;
        double r121515 = 11866.512480435778;
        bool r121516 = r121512 <= r121515;
        double r121517 = !r121516;
        bool r121518 = r121514 || r121517;
        double r121519 = 1.0;
        double r121520 = -r121519;
        double r121521 = 2.0;
        double r121522 = pow(r121512, r121521);
        double r121523 = r121520 / r121522;
        double r121524 = 3.0;
        double r121525 = r121524 / r121512;
        double r121526 = r121523 - r121525;
        double r121527 = 3.0;
        double r121528 = pow(r121512, r121527);
        double r121529 = r121524 / r121528;
        double r121530 = r121526 - r121529;
        double r121531 = r121512 * r121512;
        double r121532 = r121531 * r121512;
        double r121533 = r121512 + r121519;
        double r121534 = r121533 * r121533;
        double r121535 = r121534 * r121533;
        double r121536 = r121532 / r121535;
        double r121537 = r121512 - r121519;
        double r121538 = r121533 / r121537;
        double r121539 = pow(r121538, r121527);
        double r121540 = r121536 - r121539;
        double r121541 = exp(r121540);
        double r121542 = log(r121541);
        double r121543 = r121512 / r121533;
        double r121544 = r121538 + r121543;
        double r121545 = r121538 * r121544;
        double r121546 = r121543 * r121543;
        double r121547 = r121545 + r121546;
        double r121548 = r121542 / r121547;
        double r121549 = r121518 ? r121530 : r121548;
        return r121549;
}

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 < -10587.250289808897 or 11866.512480435778 < 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 -10587.250289808897 < x < 11866.512480435778

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.1

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

    6. Applied add-log-exp0.1

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

    7. Applied add-log-exp0.1

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

    8. Applied diff-log0.1

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

    9. Simplified0.1

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

    11. Applied add-cbrt-cube0.1

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

    12. Applied add-cbrt-cube0.1

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

    13. Applied cbrt-undiv0.1

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

    14. Applied rem-cube-cbrt0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10587.250289808897 \lor \neg \left(x \le 11866.5124804357783\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\left(x \cdot x\right) \cdot x}{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(x + 1\right)} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}\right)}{\frac{x + 1}{x - 1} \cdot \left(\frac{x + 1}{x - 1} + \frac{x}{x + 1}\right) + \frac{x}{x + 1} \cdot \frac{x}{x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))