Average Error: 29.6 → 0.1
Time: 12.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7804.869598936953:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{N}}{N} + \left(\frac{1}{N} + \frac{1}{3} \cdot \frac{\frac{\frac{1}{N}}{N}}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7804.869598936953:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{-1}{2}}{N}}{N} + \left(\frac{1}{N} + \frac{1}{3} \cdot \frac{\frac{\frac{1}{N}}{N}}{N}\right)\\

\end{array}
double f(double N) {
        double r1154969 = N;
        double r1154970 = 1.0;
        double r1154971 = r1154969 + r1154970;
        double r1154972 = log(r1154971);
        double r1154973 = log(r1154969);
        double r1154974 = r1154972 - r1154973;
        return r1154974;
}
↓
double f(double N) {
        double r1154975 = N;
        double r1154976 = 7804.869598936953;
        bool r1154977 = r1154975 <= r1154976;
        double r1154978 = 1.0;
        double r1154979 = r1154978 + r1154975;
        double r1154980 = r1154979 / r1154975;
        double r1154981 = log(r1154980);
        double r1154982 = -0.5;
        double r1154983 = r1154982 / r1154975;
        double r1154984 = r1154983 / r1154975;
        double r1154985 = r1154978 / r1154975;
        double r1154986 = 0.3333333333333333;
        double r1154987 = r1154985 / r1154975;
        double r1154988 = r1154987 / r1154975;
        double r1154989 = r1154986 * r1154988;
        double r1154990 = r1154985 + r1154989;
        double r1154991 = r1154984 + r1154990;
        double r1154992 = r1154977 ? r1154981 : r1154991;
        return r1154992;
}

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 < 7804.869598936953

    1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied add-log-exp0.1

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

    4. Simplified0.1

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

    if 7804.869598936953 < N

    1. Initial program 59.5

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied add-log-exp59.5

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

    4. Simplified59.2

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

    5. Taylor expanded around inf 0.0

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

    6. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7804.869598936953:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{N}}{N} + \left(\frac{1}{N} + \frac{1}{3} \cdot \frac{\frac{\frac{1}{N}}{N}}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))