Average Error: 29.4 → 0.1
Time: 18.9s
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 r6629089 = N;
        double r6629090 = 1.0;
        double r6629091 = r6629089 + r6629090;
        double r6629092 = log(r6629091);
        double r6629093 = log(r6629089);
        double r6629094 = r6629092 - r6629093;
        return r6629094;
}
↓
double f(double N) {
        double r6629095 = N;
        double r6629096 = 4593.354257030818;
        bool r6629097 = r6629095 <= r6629096;
        double r6629098 = 1.0;
        double r6629099 = r6629095 + r6629098;
        double r6629100 = r6629099 / r6629095;
        double r6629101 = cbrt(r6629100);
        double r6629102 = r6629101 * r6629101;
        double r6629103 = r6629102 * r6629101;
        double r6629104 = log(r6629103);
        double r6629105 = 0.3333333333333333;
        double r6629106 = r6629095 * r6629095;
        double r6629107 = r6629095 * r6629106;
        double r6629108 = r6629105 / r6629107;
        double r6629109 = r6629098 / r6629095;
        double r6629110 = 0.5;
        double r6629111 = r6629110 / r6629106;
        double r6629112 = r6629109 - r6629111;
        double r6629113 = r6629108 + r6629112;
        double r6629114 = r6629097 ? r6629104 : r6629113;
        return r6629114;
}

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. 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

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

    5. 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 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))