Average Error: 28.9 → 0.0
Time: 14.3s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8906.11117804379:\\ \;\;\;\;-\left(\log \left(\sqrt{\frac{N}{N + 1}}\right) + \frac{1}{2} \cdot \log \left(\frac{N}{N + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} - \left(\frac{\frac{1}{2}}{N \cdot N} - \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8906.11117804379:\\
\;\;\;\;-\left(\log \left(\sqrt{\frac{N}{N + 1}}\right) + \frac{1}{2} \cdot \log \left(\frac{N}{N + 1}\right)\right)\\

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

\end{array}
double f(double N) {
        double r1610839 = N;
        double r1610840 = 1.0;
        double r1610841 = r1610839 + r1610840;
        double r1610842 = log(r1610841);
        double r1610843 = log(r1610839);
        double r1610844 = r1610842 - r1610843;
        return r1610844;
}
↓
double f(double N) {
        double r1610845 = N;
        double r1610846 = 8906.11117804379;
        bool r1610847 = r1610845 <= r1610846;
        double r1610848 = 1.0;
        double r1610849 = r1610845 + r1610848;
        double r1610850 = r1610845 / r1610849;
        double r1610851 = sqrt(r1610850);
        double r1610852 = log(r1610851);
        double r1610853 = 0.5;
        double r1610854 = log(r1610850);
        double r1610855 = r1610853 * r1610854;
        double r1610856 = r1610852 + r1610855;
        double r1610857 = -r1610856;
        double r1610858 = 0.3333333333333333;
        double r1610859 = r1610858 / r1610845;
        double r1610860 = r1610845 * r1610845;
        double r1610861 = r1610859 / r1610860;
        double r1610862 = r1610853 / r1610860;
        double r1610863 = r1610848 / r1610845;
        double r1610864 = r1610862 - r1610863;
        double r1610865 = r1610861 - r1610864;
        double r1610866 = r1610847 ? r1610857 : r1610865;
        return r1610866;
}

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 < 8906.11117804379

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied diff-log0.1

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

    5. Simplified0.1

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

    7. Applied clear-num0.1

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

    8. Applied log-rec0.1

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

    10. Applied add-sqr-sqrt0.1

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

    11. Applied log-prod0.1

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

    13. Applied pow1/20.1

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

    14. Applied log-pow0.1

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

    if 8906.11117804379 < N

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8906.11117804379:\\ \;\;\;\;-\left(\log \left(\sqrt{\frac{N}{N + 1}}\right) + \frac{1}{2} \cdot \log \left(\frac{N}{N + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} - \left(\frac{\frac{1}{2}}{N \cdot N} - \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))