Average Error: 28.9 → 0.1
Time: 14.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8585.191610072276:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right) \cdot \frac{\frac{1}{N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8585.191610072276:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r1655059 = N;
        double r1655060 = 1.0;
        double r1655061 = r1655059 + r1655060;
        double r1655062 = log(r1655061);
        double r1655063 = log(r1655059);
        double r1655064 = r1655062 - r1655063;
        return r1655064;
}
↓
double f(double N) {
        double r1655065 = N;
        double r1655066 = 8585.191610072276;
        bool r1655067 = r1655065 <= r1655066;
        double r1655068 = 1.0;
        double r1655069 = r1655068 + r1655065;
        double r1655070 = r1655069 / r1655065;
        double r1655071 = log(r1655070);
        double r1655072 = r1655068 / r1655065;
        double r1655073 = 0.5;
        double r1655074 = 0.3333333333333333;
        double r1655075 = r1655074 / r1655065;
        double r1655076 = r1655073 - r1655075;
        double r1655077 = r1655072 / r1655065;
        double r1655078 = r1655076 * r1655077;
        double r1655079 = r1655072 - r1655078;
        double r1655080 = r1655067 ? r1655071 : r1655079;
        return r1655080;
}

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 < 8585.191610072276

    1. Initial program 0.1

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

    3. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
    4. Using strategy rm

    5. Applied add-log-exp0.1

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

    6. Simplified0.1

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

    if 8585.191610072276 < N

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8585.191610072276:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right) \cdot \frac{\frac{1}{N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))