Average Error: 29.2 → 0.2
Time: 14.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4798.860931730865559075027704238891601562:\\ \;\;\;\;\left(\log \left(1 + N\right) - 2 \cdot \log \left(\sqrt[3]{N}\right)\right) - \log \left(\sqrt[3]{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} - \frac{\frac{0.5}{N}}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4798.860931730865559075027704238891601562:\\
\;\;\;\;\left(\log \left(1 + N\right) - 2 \cdot \log \left(\sqrt[3]{N}\right)\right) - \log \left(\sqrt[3]{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} - \frac{\frac{0.5}{N}}{N}\right)\\

\end{array}
double f(double N) {
        double r45948 = N;
        double r45949 = 1.0;
        double r45950 = r45948 + r45949;
        double r45951 = log(r45950);
        double r45952 = log(r45948);
        double r45953 = r45951 - r45952;
        return r45953;
}
↓
double f(double N) {
        double r45954 = N;
        double r45955 = 4798.860931730866;
        bool r45956 = r45954 <= r45955;
        double r45957 = 1.0;
        double r45958 = r45957 + r45954;
        double r45959 = log(r45958);
        double r45960 = 2.0;
        double r45961 = cbrt(r45954);
        double r45962 = log(r45961);
        double r45963 = r45960 * r45962;
        double r45964 = r45959 - r45963;
        double r45965 = r45964 - r45962;
        double r45966 = r45957 / r45954;
        double r45967 = 0.3333333333333333;
        double r45968 = 3.0;
        double r45969 = pow(r45954, r45968);
        double r45970 = r45967 / r45969;
        double r45971 = 0.5;
        double r45972 = r45971 / r45954;
        double r45973 = r45972 / r45954;
        double r45974 = r45970 - r45973;
        double r45975 = r45966 + r45974;
        double r45976 = r45956 ? r45965 : r45975;
        return r45976;
}

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 < 4798.860931730866

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied log-prod0.4

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

    6. Applied associate--r+0.4

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

    7. Simplified0.4

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

    if 4798.860931730866 < N

    1. Initial program 59.7

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

    2. Simplified59.7

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4798.860931730865559075027704238891601562:\\ \;\;\;\;\left(\log \left(1 + N\right) - 2 \cdot \log \left(\sqrt[3]{N}\right)\right) - \log \left(\sqrt[3]{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} - \frac{\frac{0.5}{N}}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))