Average Error: 29.7 → 0.1
Time: 14.9s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7072.329313818584:\\ \;\;\;\;\log \left(\sqrt{1 + \frac{1}{N}}\right) + \log \left(\sqrt{1 + \frac{1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N \cdot N}}{N} + \left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7072.329313818584:\\
\;\;\;\;\log \left(\sqrt{1 + \frac{1}{N}}\right) + \log \left(\sqrt{1 + \frac{1}{N}}\right)\\

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

\end{array}
double f(double N) {
        double r2085973 = N;
        double r2085974 = 1.0;
        double r2085975 = r2085973 + r2085974;
        double r2085976 = log(r2085975);
        double r2085977 = log(r2085973);
        double r2085978 = r2085976 - r2085977;
        return r2085978;
}
↓
double f(double N) {
        double r2085979 = N;
        double r2085980 = 7072.329313818584;
        bool r2085981 = r2085979 <= r2085980;
        double r2085982 = 1.0;
        double r2085983 = r2085982 / r2085979;
        double r2085984 = r2085982 + r2085983;
        double r2085985 = sqrt(r2085984);
        double r2085986 = log(r2085985);
        double r2085987 = r2085986 + r2085986;
        double r2085988 = 0.3333333333333333;
        double r2085989 = r2085979 * r2085979;
        double r2085990 = r2085988 / r2085989;
        double r2085991 = r2085990 / r2085979;
        double r2085992 = -0.5;
        double r2085993 = r2085992 / r2085989;
        double r2085994 = r2085983 + r2085993;
        double r2085995 = r2085991 + r2085994;
        double r2085996 = r2085981 ? r2085987 : r2085995;
        return r2085996;
}

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 < 7072.329313818584

    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. Taylor expanded around -inf 0.1

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

    6. Applied add-sqr-sqrt0.1

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

    7. Applied log-prod0.1

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

    if 7072.329313818584 < N

    1. Initial program 59.6

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

    3. Applied diff-log59.3

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

    4. Taylor expanded around inf 0.0

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

    5. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7072.329313818584:\\ \;\;\;\;\log \left(\sqrt{1 + \frac{1}{N}}\right) + \log \left(\sqrt{1 + \frac{1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N \cdot N}}{N} + \left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))