Average Error: 29.2 → 0.1
Time: 11.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7883.767016394317579397466033697128295898:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1}{N}\right) - \frac{\frac{0.5}{N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7883.767016394317579397466033697128295898:\\
\;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1}{N}\right) - \frac{\frac{0.5}{N}}{N}\\

\end{array}
double f(double N) {
        double r45407 = N;
        double r45408 = 1.0;
        double r45409 = r45407 + r45408;
        double r45410 = log(r45409);
        double r45411 = log(r45407);
        double r45412 = r45410 - r45411;
        return r45412;
}
↓
double f(double N) {
        double r45413 = N;
        double r45414 = 7883.767016394318;
        bool r45415 = r45413 <= r45414;
        double r45416 = 1.0;
        double r45417 = r45416 + r45413;
        double r45418 = r45417 / r45413;
        double r45419 = sqrt(r45418);
        double r45420 = log(r45419);
        double r45421 = r45420 + r45420;
        double r45422 = 0.3333333333333333;
        double r45423 = 3.0;
        double r45424 = pow(r45413, r45423);
        double r45425 = r45422 / r45424;
        double r45426 = r45416 / r45413;
        double r45427 = r45425 + r45426;
        double r45428 = 0.5;
        double r45429 = r45428 / r45413;
        double r45430 = r45429 / r45413;
        double r45431 = r45427 - r45430;
        double r45432 = r45415 ? r45421 : r45431;
        return r45432;
}

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 < 7883.767016394318

    1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]

    2. Simplified0.1

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

    4. Applied diff-log0.1

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

    5. Simplified0.1

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

    7. Applied add-sqr-sqrt0.1

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

    8. Applied log-prod0.1

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

    9. Simplified0.1

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

    10. Simplified0.1

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

    if 7883.767016394318 < N

    1. Initial program 59.5

      \[\log \left(N + 1\right) - \log N\]

    2. Simplified59.5

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

    4. Applied diff-log59.3

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

    5. Simplified59.3

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

    6. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    7. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1}{N}\right) - \frac{\frac{0.5}{N}}{N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7883.767016394317579397466033697128295898:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1}{N}\right) - \frac{\frac{0.5}{N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))