Average Error: 29.4 → 0.1
Time: 5.3s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11931.4119690201605 \lor \neg \left(x \le 12679.85899306799\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right) - \left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -11931.4119690201605 \lor \neg \left(x \le 12679.85899306799\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r125885 = x;
        double r125886 = 1.0;
        double r125887 = r125885 + r125886;
        double r125888 = r125885 / r125887;
        double r125889 = r125885 - r125886;
        double r125890 = r125887 / r125889;
        double r125891 = r125888 - r125890;
        return r125891;
}
↓
double f(double x) {
        double r125892 = x;
        double r125893 = -11931.41196902016;
        bool r125894 = r125892 <= r125893;
        double r125895 = 12679.85899306799;
        bool r125896 = r125892 <= r125895;
        double r125897 = !r125896;
        bool r125898 = r125894 || r125897;
        double r125899 = 1.0;
        double r125900 = -r125899;
        double r125901 = 2.0;
        double r125902 = pow(r125892, r125901);
        double r125903 = r125900 / r125902;
        double r125904 = 3.0;
        double r125905 = r125904 / r125892;
        double r125906 = r125903 - r125905;
        double r125907 = 3.0;
        double r125908 = pow(r125892, r125907);
        double r125909 = r125904 / r125908;
        double r125910 = r125906 - r125909;
        double r125911 = r125892 * r125892;
        double r125912 = r125892 - r125899;
        double r125913 = r125912 * r125912;
        double r125914 = r125911 * r125913;
        double r125915 = r125892 + r125899;
        double r125916 = r125915 * r125915;
        double r125917 = r125916 * r125916;
        double r125918 = r125914 - r125917;
        double r125919 = r125892 / r125915;
        double r125920 = r125915 / r125912;
        double r125921 = r125919 + r125920;
        double r125922 = r125916 * r125913;
        double r125923 = r125921 * r125922;
        double r125924 = r125918 / r125923;
        double r125925 = r125898 ? r125910 : r125924;
        return r125925;
}

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 < -11931.41196902016 or 12679.85899306799 < 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

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

    if -11931.41196902016 < x < 12679.85899306799

    1. Initial program 0.1

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

    3. Applied flip--0.1

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}}\]
    4. Using strategy rm

    5. Applied frac-times0.1

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

    6. Applied frac-times0.1

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

    7. Applied frac-sub0.1

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

    8. Applied associate-/l/0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11931.4119690201605 \lor \neg \left(x \le 12679.85899306799\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right) - \left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))