Average Error: 29.4 → 0.1
Time: 13.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7627.9955927630945:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7627.9955927630945:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r5815881 = N;
        double r5815882 = 1.0;
        double r5815883 = r5815881 + r5815882;
        double r5815884 = log(r5815883);
        double r5815885 = log(r5815881);
        double r5815886 = r5815884 - r5815885;
        return r5815886;
}
↓
double f(double N) {
        double r5815887 = N;
        double r5815888 = 7627.9955927630945;
        bool r5815889 = r5815887 <= r5815888;
        double r5815890 = 1.0;
        double r5815891 = r5815890 + r5815887;
        double r5815892 = r5815891 / r5815887;
        double r5815893 = log(r5815892);
        double r5815894 = r5815890 / r5815887;
        double r5815895 = -0.5;
        double r5815896 = r5815887 * r5815887;
        double r5815897 = r5815895 / r5815896;
        double r5815898 = r5815894 + r5815897;
        double r5815899 = 0.3333333333333333;
        double r5815900 = r5815899 / r5815896;
        double r5815901 = r5815900 / r5815887;
        double r5815902 = r5815898 + r5815901;
        double r5815903 = r5815889 ? r5815893 : r5815902;
        return r5815903;
}

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 < 7627.9955927630945

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    if 7627.9955927630945 < 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 59.3

      \[\leadsto \log \color{blue}{\left(\frac{1}{N} + 1\right)}\]
    5. Using strategy rm

    6. Applied add-exp-log59.3

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

    7. Taylor expanded around -inf 0.0

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

    8. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7627.9955927630945:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))