Average Error: 30.2 → 0.1
Time: 13.2s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 11281.62749418612293084152042865753173828:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 11281.62749418612293084152042865753173828:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

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

\end{array}
double f(double N) {
        double r45868 = N;
        double r45869 = 1.0;
        double r45870 = r45868 + r45869;
        double r45871 = log(r45870);
        double r45872 = log(r45868);
        double r45873 = r45871 - r45872;
        return r45873;
}
↓
double f(double N) {
        double r45874 = N;
        double r45875 = 11281.627494186123;
        bool r45876 = r45874 <= r45875;
        double r45877 = 1.0;
        double r45878 = r45874 + r45877;
        double r45879 = r45878 / r45874;
        double r45880 = log(r45879);
        double r45881 = 0.3333333333333333;
        double r45882 = 3.0;
        double r45883 = pow(r45874, r45882);
        double r45884 = r45881 / r45883;
        double r45885 = r45877 / r45874;
        double r45886 = 0.5;
        double r45887 = r45874 * r45874;
        double r45888 = r45886 / r45887;
        double r45889 = r45885 - r45888;
        double r45890 = r45884 + r45889;
        double r45891 = r45876 ? r45880 : r45890;
        return r45891;
}

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 < 11281.627494186123

    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 11281.627494186123 < N

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 11281.62749418612293084152042865753173828:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))