Average Error: 28.9 → 0.1
Time: 15.7s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8041.189519456958:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right) \cdot \frac{\frac{1}{N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8041.189519456958:\\
\;\;\;\;\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 r2668241 = N;
        double r2668242 = 1.0;
        double r2668243 = r2668241 + r2668242;
        double r2668244 = log(r2668243);
        double r2668245 = log(r2668241);
        double r2668246 = r2668244 - r2668245;
        return r2668246;
}
↓
double f(double N) {
        double r2668247 = N;
        double r2668248 = 8041.189519456958;
        bool r2668249 = r2668247 <= r2668248;
        double r2668250 = 1.0;
        double r2668251 = r2668250 + r2668247;
        double r2668252 = r2668251 / r2668247;
        double r2668253 = log(r2668252);
        double r2668254 = r2668250 / r2668247;
        double r2668255 = 0.5;
        double r2668256 = 0.3333333333333333;
        double r2668257 = r2668256 / r2668247;
        double r2668258 = r2668255 - r2668257;
        double r2668259 = r2668254 / r2668247;
        double r2668260 = r2668258 * r2668259;
        double r2668261 = r2668254 - r2668260;
        double r2668262 = r2668249 ? r2668253 : r2668261;
        return r2668262;
}

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 < 8041.189519456958

    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 8041.189519456958 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8041.189519456958:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right) \cdot \frac{\frac{1}{N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))