Average Error: 29.5 → 0.1
Time: 12.3s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4683.545058486523:\\ \;\;\;\;\frac{\log \left(1 + N\right) \cdot \log \left(1 + N\right) - \log N \cdot \log N}{\log \left(1 + N\right) + \log N}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{2}}{N \cdot N} + \left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4683.545058486523:\\
\;\;\;\;\frac{\log \left(1 + N\right) \cdot \log \left(1 + N\right) - \log N \cdot \log N}{\log \left(1 + N\right) + \log N}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{-1}{2}}{N \cdot N} + \left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \frac{1}{N}\right)\\

\end{array}
double f(double N) {
        double r1164753 = N;
        double r1164754 = 1.0;
        double r1164755 = r1164753 + r1164754;
        double r1164756 = log(r1164755);
        double r1164757 = log(r1164753);
        double r1164758 = r1164756 - r1164757;
        return r1164758;
}
↓
double f(double N) {
        double r1164759 = N;
        double r1164760 = 4683.545058486523;
        bool r1164761 = r1164759 <= r1164760;
        double r1164762 = 1.0;
        double r1164763 = r1164762 + r1164759;
        double r1164764 = log(r1164763);
        double r1164765 = r1164764 * r1164764;
        double r1164766 = log(r1164759);
        double r1164767 = r1164766 * r1164766;
        double r1164768 = r1164765 - r1164767;
        double r1164769 = r1164764 + r1164766;
        double r1164770 = r1164768 / r1164769;
        double r1164771 = -0.5;
        double r1164772 = r1164759 * r1164759;
        double r1164773 = r1164771 / r1164772;
        double r1164774 = 0.3333333333333333;
        double r1164775 = r1164774 / r1164759;
        double r1164776 = r1164775 / r1164772;
        double r1164777 = r1164762 / r1164759;
        double r1164778 = r1164776 + r1164777;
        double r1164779 = r1164773 + r1164778;
        double r1164780 = r1164761 ? r1164770 : r1164779;
        return r1164780;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if N < 4683.545058486523

    1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied flip--0.1

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

    if 4683.545058486523 < N

    1. Initial program 59.5

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied diff-log59.3

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]

    4. Taylor expanded around inf 0.1

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

    5. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4683.545058486523:\\ \;\;\;\;\frac{\log \left(1 + N\right) \cdot \log \left(1 + N\right) - \log N \cdot \log N}{\log \left(1 + N\right) + \log N}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{2}}{N \cdot N} + \left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))