Average Error: 14.8 → 0.0
Time: 7.8s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -285.634106609297077739029191434383392334 \lor \neg \left(x \le 221.6495232693454227046458981931209564209\right):\\ \;\;\;\;\frac{-2}{{x}^{6}} - 2 \cdot \left({x}^{\left(-2\right)} + \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{x + 1}\right)}^{3} - {\left(\frac{1}{x - 1}\right)}^{3}}{\frac{1}{x - 1} \cdot \left(\frac{1}{x - 1} + \frac{1}{x + 1}\right) + \frac{1}{x + 1} \cdot \frac{1}{x + 1}}\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -285.634106609297077739029191434383392334 \lor \neg \left(x \le 221.6495232693454227046458981931209564209\right):\\
\;\;\;\;\frac{-2}{{x}^{6}} - 2 \cdot \left({x}^{\left(-2\right)} + \frac{1}{{x}^{4}}\right)\\

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

\end{array}
double f(double x) {
        double r151306 = 1.0;
        double r151307 = x;
        double r151308 = r151307 + r151306;
        double r151309 = r151306 / r151308;
        double r151310 = r151307 - r151306;
        double r151311 = r151306 / r151310;
        double r151312 = r151309 - r151311;
        return r151312;
}
↓
double f(double x) {
        double r151313 = x;
        double r151314 = -285.6341066092971;
        bool r151315 = r151313 <= r151314;
        double r151316 = 221.64952326934542;
        bool r151317 = r151313 <= r151316;
        double r151318 = !r151317;
        bool r151319 = r151315 || r151318;
        double r151320 = 2.0;
        double r151321 = -r151320;
        double r151322 = 6.0;
        double r151323 = pow(r151313, r151322);
        double r151324 = r151321 / r151323;
        double r151325 = 2.0;
        double r151326 = -r151325;
        double r151327 = pow(r151313, r151326);
        double r151328 = 1.0;
        double r151329 = 4.0;
        double r151330 = pow(r151313, r151329);
        double r151331 = r151328 / r151330;
        double r151332 = r151327 + r151331;
        double r151333 = r151320 * r151332;
        double r151334 = r151324 - r151333;
        double r151335 = 1.0;
        double r151336 = r151313 + r151335;
        double r151337 = r151335 / r151336;
        double r151338 = 3.0;
        double r151339 = pow(r151337, r151338);
        double r151340 = r151313 - r151335;
        double r151341 = r151335 / r151340;
        double r151342 = pow(r151341, r151338);
        double r151343 = r151339 - r151342;
        double r151344 = r151341 + r151337;
        double r151345 = r151341 * r151344;
        double r151346 = r151337 * r151337;
        double r151347 = r151345 + r151346;
        double r151348 = r151343 / r151347;
        double r151349 = r151319 ? r151334 : r151348;
        return r151349;
}

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 < -285.6341066092971 or 221.64952326934542 < x

    1. Initial program 29.4

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

    2. Taylor expanded around inf 0.8

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{6}} + \left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{{x}^{4}}\right)\right)}\]

    3. Simplified0.8

      \[\leadsto \color{blue}{\frac{-2}{{x}^{6}} - 2 \cdot \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)}\]
    4. Using strategy rm

    5. Applied pow-flip0.0

      \[\leadsto \frac{-2}{{x}^{6}} - 2 \cdot \left(\color{blue}{{x}^{\left(-2\right)}} + \frac{1}{{x}^{4}}\right)\]

    if -285.6341066092971 < x < 221.64952326934542

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -285.634106609297077739029191434383392334 \lor \neg \left(x \le 221.6495232693454227046458981931209564209\right):\\ \;\;\;\;\frac{-2}{{x}^{6}} - 2 \cdot \left({x}^{\left(-2\right)} + \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{x + 1}\right)}^{3} - {\left(\frac{1}{x - 1}\right)}^{3}}{\frac{1}{x - 1} \cdot \left(\frac{1}{x - 1} + \frac{1}{x + 1}\right) + \frac{1}{x + 1} \cdot \frac{1}{x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))