Average Error: 29.8 → 0.1
Time: 4.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9757.882283737008037860505282878875732422:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9757.882283737008037860505282878875732422:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r47930 = N;
        double r47931 = 1.0;
        double r47932 = r47930 + r47931;
        double r47933 = log(r47932);
        double r47934 = log(r47930);
        double r47935 = r47933 - r47934;
        return r47935;
}
↓
double f(double N) {
        double r47936 = N;
        double r47937 = 9757.882283737008;
        bool r47938 = r47936 <= r47937;
        double r47939 = 1.0;
        double r47940 = r47936 + r47939;
        double r47941 = r47940 / r47936;
        double r47942 = log(r47941);
        double r47943 = 1.0;
        double r47944 = 2.0;
        double r47945 = pow(r47936, r47944);
        double r47946 = r47943 / r47945;
        double r47947 = 0.3333333333333333;
        double r47948 = r47947 / r47936;
        double r47949 = 0.5;
        double r47950 = r47948 - r47949;
        double r47951 = r47946 * r47950;
        double r47952 = r47939 / r47936;
        double r47953 = r47951 + r47952;
        double r47954 = r47938 ? r47942 : r47953;
        return r47954;
}

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 < 9757.882283737008

    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 9757.882283737008 < 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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9757.882283737008037860505282878875732422:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))