Average Error: 29.2 → 0.1
Time: 10.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 5567.039381079165:\\ \;\;\;\;\sqrt[3]{\log \left(1 + N\right) \cdot \left(\log \left(1 + N\right) \cdot \log \left(1 + N\right)\right)} - \log N\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right) + \frac{\frac{1}{3}}{\left(N \cdot N\right) \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 5567.039381079165:\\
\;\;\;\;\sqrt[3]{\log \left(1 + N\right) \cdot \left(\log \left(1 + N\right) \cdot \log \left(1 + N\right)\right)} - \log N\\

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

\end{array}
double f(double N) {
        double r1215786 = N;
        double r1215787 = 1.0;
        double r1215788 = r1215786 + r1215787;
        double r1215789 = log(r1215788);
        double r1215790 = log(r1215786);
        double r1215791 = r1215789 - r1215790;
        return r1215791;
}
↓
double f(double N) {
        double r1215792 = N;
        double r1215793 = 5567.039381079165;
        bool r1215794 = r1215792 <= r1215793;
        double r1215795 = 1.0;
        double r1215796 = r1215795 + r1215792;
        double r1215797 = log(r1215796);
        double r1215798 = r1215797 * r1215797;
        double r1215799 = r1215797 * r1215798;
        double r1215800 = cbrt(r1215799);
        double r1215801 = log(r1215792);
        double r1215802 = r1215800 - r1215801;
        double r1215803 = r1215795 / r1215792;
        double r1215804 = 0.5;
        double r1215805 = r1215792 * r1215792;
        double r1215806 = r1215804 / r1215805;
        double r1215807 = r1215803 - r1215806;
        double r1215808 = 0.3333333333333333;
        double r1215809 = r1215805 * r1215792;
        double r1215810 = r1215808 / r1215809;
        double r1215811 = r1215807 + r1215810;
        double r1215812 = r1215794 ? r1215802 : r1215811;
        return r1215812;
}

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 < 5567.039381079165

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    if 5567.039381079165 < N

    1. Initial program 59.4

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

    3. Applied add-cbrt-cube59.7

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

    4. Taylor expanded around inf 0.0

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

    5. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5567.039381079165:\\ \;\;\;\;\sqrt[3]{\log \left(1 + N\right) \cdot \left(\log \left(1 + N\right) \cdot \log \left(1 + N\right)\right)} - \log N\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right) + \frac{\frac{1}{3}}{\left(N \cdot N\right) \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))