Average Error: 29.6 → 0.1
Time: 11.9s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7426.105103924180184549186378717422485352:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) \cdot \frac{\frac{1}{N}}{N} + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7426.105103924180184549186378717422485352:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r67910 = N;
        double r67911 = 1.0;
        double r67912 = r67910 + r67911;
        double r67913 = log(r67912);
        double r67914 = log(r67910);
        double r67915 = r67913 - r67914;
        return r67915;
}
↓
double f(double N) {
        double r67916 = N;
        double r67917 = 7426.10510392418;
        bool r67918 = r67916 <= r67917;
        double r67919 = 1.0;
        double r67920 = r67919 + r67916;
        double r67921 = r67920 / r67916;
        double r67922 = log(r67921);
        double r67923 = 0.3333333333333333;
        double r67924 = r67923 / r67916;
        double r67925 = 0.5;
        double r67926 = r67924 - r67925;
        double r67927 = 1.0;
        double r67928 = r67927 / r67916;
        double r67929 = r67928 / r67916;
        double r67930 = r67926 * r67929;
        double r67931 = r67919 / r67916;
        double r67932 = r67930 + r67931;
        double r67933 = r67918 ? r67922 : r67932;
        return r67933;
}

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 < 7426.10510392418

    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 diff-log0.1

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

    5. Simplified0.1

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

    if 7426.10510392418 < N

    1. Initial program 59.6

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

    2. Simplified59.6

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7426.105103924180184549186378717422485352:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) \cdot \frac{\frac{1}{N}}{N} + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))