Average Error: 30.0 → 0.1
Time: 3.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7004.8538811289454:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7004.8538811289454:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r37960 = N;
        double r37961 = 1.0;
        double r37962 = r37960 + r37961;
        double r37963 = log(r37962);
        double r37964 = log(r37960);
        double r37965 = r37963 - r37964;
        return r37965;
}
↓
double f(double N) {
        double r37966 = N;
        double r37967 = 7004.853881128945;
        bool r37968 = r37966 <= r37967;
        double r37969 = 1.0;
        double r37970 = r37966 + r37969;
        double r37971 = r37970 / r37966;
        double r37972 = log(r37971);
        double r37973 = 1.0;
        double r37974 = 2.0;
        double r37975 = pow(r37966, r37974);
        double r37976 = r37973 / r37975;
        double r37977 = 0.3333333333333333;
        double r37978 = r37977 / r37966;
        double r37979 = 0.5;
        double r37980 = r37978 - r37979;
        double r37981 = r37976 * r37980;
        double r37982 = r37969 / r37966;
        double r37983 = r37981 + r37982;
        double r37984 = r37968 ? r37972 : r37983;
        return r37984;
}

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 < 7004.853881128945

    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 7004.853881128945 < N

    1. Initial program 59.5

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7004.8538811289454:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))