Average Error: 29.5 → 0.1
Time: 9.2s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7622.50412124721061:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7622.50412124721061:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}\\

\end{array}
double f(double N) {
        double r37857 = N;
        double r37858 = 1.0;
        double r37859 = r37857 + r37858;
        double r37860 = log(r37859);
        double r37861 = log(r37857);
        double r37862 = r37860 - r37861;
        return r37862;
}
↓
double f(double N) {
        double r37863 = N;
        double r37864 = 7622.504121247211;
        bool r37865 = r37863 <= r37864;
        double r37866 = 1.0;
        double r37867 = r37863 + r37866;
        double r37868 = r37867 / r37863;
        double r37869 = log(r37868);
        double r37870 = r37866 / r37863;
        double r37871 = 0.3333333333333333;
        double r37872 = 3.0;
        double r37873 = pow(r37863, r37872);
        double r37874 = r37871 / r37873;
        double r37875 = r37870 + r37874;
        double r37876 = 0.5;
        double r37877 = r37863 * r37863;
        double r37878 = r37876 / r37877;
        double r37879 = r37875 - r37878;
        double r37880 = r37865 ? r37869 : r37879;
        return r37880;
}

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 < 7622.504121247211

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    if 7622.504121247211 < N

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7622.50412124721061:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))