Average Error: 29.4 → 0.1
Time: 15.7s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4593.354257030818189377896487712860107422:\\ \;\;\;\;\log \left(\left(\sqrt[3]{\frac{N + 1}{N}} \cdot \sqrt[3]{\frac{N + 1}{N}}\right) \cdot \sqrt[3]{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{N \cdot \left(N \cdot N\right)} + \left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4593.354257030818189377896487712860107422:\\
\;\;\;\;\log \left(\left(\sqrt[3]{\frac{N + 1}{N}} \cdot \sqrt[3]{\frac{N + 1}{N}}\right) \cdot \sqrt[3]{\frac{N + 1}{N}}\right)\\

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

\end{array}
double f(double N) {
        double r8194681 = N;
        double r8194682 = 1.0;
        double r8194683 = r8194681 + r8194682;
        double r8194684 = log(r8194683);
        double r8194685 = log(r8194681);
        double r8194686 = r8194684 - r8194685;
        return r8194686;
}
↓
double f(double N) {
        double r8194687 = N;
        double r8194688 = 4593.354257030818;
        bool r8194689 = r8194687 <= r8194688;
        double r8194690 = 1.0;
        double r8194691 = r8194687 + r8194690;
        double r8194692 = r8194691 / r8194687;
        double r8194693 = cbrt(r8194692);
        double r8194694 = r8194693 * r8194693;
        double r8194695 = r8194694 * r8194693;
        double r8194696 = log(r8194695);
        double r8194697 = 0.3333333333333333;
        double r8194698 = r8194687 * r8194687;
        double r8194699 = r8194687 * r8194698;
        double r8194700 = r8194697 / r8194699;
        double r8194701 = r8194690 / r8194687;
        double r8194702 = 0.5;
        double r8194703 = r8194702 / r8194698;
        double r8194704 = r8194701 - r8194703;
        double r8194705 = r8194700 + r8194704;
        double r8194706 = r8194689 ? r8194696 : r8194705;
        return r8194706;
}

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 < 4593.354257030818

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    5. Applied add-cube-cbrt0.1

      \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\frac{N + 1}{N}} \cdot \sqrt[3]{\frac{N + 1}{N}}\right) \cdot \sqrt[3]{\frac{N + 1}{N}}\right)}\]

    if 4593.354257030818 < N

    1. Initial program 59.5

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4593.354257030818189377896487712860107422:\\ \;\;\;\;\log \left(\left(\sqrt[3]{\frac{N + 1}{N}} \cdot \sqrt[3]{\frac{N + 1}{N}}\right) \cdot \sqrt[3]{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{N \cdot \left(N \cdot N\right)} + \left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))