Average Error: 29.4 → 0.1
Time: 7.5s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12787.016553276117 \lor \neg \left(x \le 9971.293289135534\right):\\ \;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{1}{x \cdot x} + 1\right) \cdot \frac{3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{\left(x - 1\right) \cdot \left(x + 1\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -12787.016553276117 \lor \neg \left(x \le 9971.293289135534\right):\\
\;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{1}{x \cdot x} + 1\right) \cdot \frac{3}{x}\\

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

\end{array}
double f(double x) {
        double r132887 = x;
        double r132888 = 1.0;
        double r132889 = r132887 + r132888;
        double r132890 = r132887 / r132889;
        double r132891 = r132887 - r132888;
        double r132892 = r132889 / r132891;
        double r132893 = r132890 - r132892;
        return r132893;
}
↓
double f(double x) {
        double r132894 = x;
        double r132895 = -12787.016553276117;
        bool r132896 = r132894 <= r132895;
        double r132897 = 9971.293289135534;
        bool r132898 = r132894 <= r132897;
        double r132899 = !r132898;
        bool r132900 = r132896 || r132899;
        double r132901 = 1.0;
        double r132902 = r132894 * r132894;
        double r132903 = r132901 / r132902;
        double r132904 = -r132903;
        double r132905 = 1.0;
        double r132906 = r132905 / r132902;
        double r132907 = r132906 + r132905;
        double r132908 = 3.0;
        double r132909 = r132908 / r132894;
        double r132910 = r132907 * r132909;
        double r132911 = r132904 - r132910;
        double r132912 = r132894 - r132901;
        double r132913 = r132894 * r132912;
        double r132914 = r132894 + r132901;
        double r132915 = r132914 * r132914;
        double r132916 = r132913 - r132915;
        double r132917 = r132912 * r132914;
        double r132918 = r132916 / r132917;
        double r132919 = r132900 ? r132911 : r132918;
        return r132919;
}

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 < -12787.016553276117 or 9971.293289135534 < x

    1. Initial program 59.3

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

    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

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

    5. Applied unpow30.0

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

    6. Applied *-un-lft-identity0.0

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

    7. Applied times-frac0.0

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

    8. Applied distribute-lft1-in0.0

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

    if -12787.016553276117 < x < 9971.293289135534

    1. Initial program 0.1

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

    3. Applied frac-sub0.1

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12787.016553276117 \lor \neg \left(x \le 9971.293289135534\right):\\ \;\;\;\;\left(-\frac{1}{x \cdot x}\right) - \left(\frac{1}{x \cdot x} + 1\right) \cdot \frac{3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{\left(x - 1\right) \cdot \left(x + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))