Average Error: 14.9 → 0.1
Time: 7.9s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9124183958286494 \lor \neg \left(x \le 14237.069114952126\right):\\ \;\;\;\;\left(-\frac{2}{{x}^{6}}\right) - \left(\frac{2}{{x}^{4}} + \frac{\frac{2}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right)\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9124183958286494 \lor \neg \left(x \le 14237.069114952126\right):\\
\;\;\;\;\left(-\frac{2}{{x}^{6}}\right) - \left(\frac{2}{{x}^{4}} + \frac{\frac{2}{x}}{x}\right)\\

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

\end{array}
double f(double x) {
        double r110207 = 1.0;
        double r110208 = x;
        double r110209 = r110208 + r110207;
        double r110210 = r110207 / r110209;
        double r110211 = r110208 - r110207;
        double r110212 = r110207 / r110211;
        double r110213 = r110210 - r110212;
        return r110213;
}
↓
double f(double x) {
        double r110214 = x;
        double r110215 = -9124183958286494.0;
        bool r110216 = r110214 <= r110215;
        double r110217 = 14237.069114952126;
        bool r110218 = r110214 <= r110217;
        double r110219 = !r110218;
        bool r110220 = r110216 || r110219;
        double r110221 = 2.0;
        double r110222 = 6.0;
        double r110223 = pow(r110214, r110222);
        double r110224 = r110221 / r110223;
        double r110225 = -r110224;
        double r110226 = 4.0;
        double r110227 = pow(r110214, r110226);
        double r110228 = r110221 / r110227;
        double r110229 = r110221 / r110214;
        double r110230 = r110229 / r110214;
        double r110231 = r110228 + r110230;
        double r110232 = r110225 - r110231;
        double r110233 = 1.0;
        double r110234 = r110214 * r110214;
        double r110235 = r110233 * r110233;
        double r110236 = r110234 - r110235;
        double r110237 = r110233 / r110236;
        double r110238 = r110214 - r110233;
        double r110239 = r110214 + r110233;
        double r110240 = r110238 - r110239;
        double r110241 = r110237 * r110240;
        double r110242 = r110220 ? r110232 : r110241;
        return r110242;
}

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 < -9124183958286494.0 or 14237.069114952126 < x

    1. Initial program 29.7

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

    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

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

    5. Applied unpow20.8

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

    6. Applied associate-/r*0.1

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

    if -9124183958286494.0 < x < 14237.069114952126

    1. Initial program 0.7

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

    3. Applied flip--0.7

      \[\leadsto \frac{1}{x + 1} - \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}}\]

    4. Applied associate-/r/0.7

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

    5. Applied flip-+0.7

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

    6. Applied associate-/r/0.7

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

    7. Applied distribute-lft-out--0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9124183958286494 \lor \neg \left(x \le 14237.069114952126\right):\\ \;\;\;\;\left(-\frac{2}{{x}^{6}}\right) - \left(\frac{2}{{x}^{4}} + \frac{\frac{2}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))