Average Error: 29.7 → 0.0
Time: 17.2s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9843.297794559382964507676661014556884766:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9843.297794559382964507676661014556884766:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

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

\end{array}
double f(double N) {
        double r27948 = N;
        double r27949 = 1.0;
        double r27950 = r27948 + r27949;
        double r27951 = log(r27950);
        double r27952 = log(r27948);
        double r27953 = r27951 - r27952;
        return r27953;
}
↓
double f(double N) {
        double r27954 = N;
        double r27955 = 9843.297794559383;
        bool r27956 = r27954 <= r27955;
        double r27957 = 1.0;
        double r27958 = r27954 + r27957;
        double r27959 = r27958 / r27954;
        double r27960 = log(r27959);
        double r27961 = 0.3333333333333333;
        double r27962 = 3.0;
        double r27963 = pow(r27954, r27962);
        double r27964 = r27961 / r27963;
        double r27965 = 0.5;
        double r27966 = r27965 / r27954;
        double r27967 = r27957 - r27966;
        double r27968 = r27967 / r27954;
        double r27969 = r27964 + r27968;
        double r27970 = r27956 ? r27960 : r27969;
        return r27970;
}

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 < 9843.297794559383

    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 9843.297794559383 < N

    1. Initial program 59.5

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

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9843.297794559382964507676661014556884766:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))