Average Error: 29.9 → 0.1
Time: 13.9s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 12631.98063969317081500776112079620361328:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{{N}^{2}} + \left(1 - \frac{0.5}{N}\right)\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 12631.98063969317081500776112079620361328:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{{N}^{2}} + \left(1 - \frac{0.5}{N}\right)\right)\\

\end{array}
double f(double N) {
        double r47259 = N;
        double r47260 = 1.0;
        double r47261 = r47259 + r47260;
        double r47262 = log(r47261);
        double r47263 = log(r47259);
        double r47264 = r47262 - r47263;
        return r47264;
}
↓
double f(double N) {
        double r47265 = N;
        double r47266 = 12631.98063969317;
        bool r47267 = r47265 <= r47266;
        double r47268 = 1.0;
        double r47269 = r47265 + r47268;
        double r47270 = r47269 / r47265;
        double r47271 = log(r47270);
        double r47272 = 1.0;
        double r47273 = r47272 / r47265;
        double r47274 = 0.3333333333333333;
        double r47275 = 2.0;
        double r47276 = pow(r47265, r47275);
        double r47277 = r47274 / r47276;
        double r47278 = 0.5;
        double r47279 = r47278 / r47265;
        double r47280 = r47268 - r47279;
        double r47281 = r47277 + r47280;
        double r47282 = r47273 * r47281;
        double r47283 = r47267 ? r47271 : r47282;
        return r47283;
}

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 < 12631.98063969317

    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 12631.98063969317 < 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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 12631.98063969317081500776112079620361328:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{{N}^{2}} + \left(1 - \frac{0.5}{N}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))