Average Error: 28.8 → 0.1
Time: 11.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 12267.95132095184089848771691322326660156:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 12267.95132095184089848771691322326660156:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\

\end{array}
double f(double N) {
        double r39319 = N;
        double r39320 = 1.0;
        double r39321 = r39319 + r39320;
        double r39322 = log(r39321);
        double r39323 = log(r39319);
        double r39324 = r39322 - r39323;
        return r39324;
}
↓
double f(double N) {
        double r39325 = N;
        double r39326 = 12267.95132095184;
        bool r39327 = r39325 <= r39326;
        double r39328 = 1.0;
        double r39329 = r39325 + r39328;
        double r39330 = r39329 / r39325;
        double r39331 = log(r39330);
        double r39332 = r39328 / r39325;
        double r39333 = 0.5;
        double r39334 = r39325 * r39325;
        double r39335 = r39333 / r39334;
        double r39336 = r39332 - r39335;
        double r39337 = 0.3333333333333333;
        double r39338 = 3.0;
        double r39339 = pow(r39325, r39338);
        double r39340 = r39337 / r39339;
        double r39341 = r39336 + r39340;
        double r39342 = r39327 ? r39331 : r39341;
        return r39342;
}

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 < 12267.95132095184

    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 12267.95132095184 < N

    1. Initial program 59.4

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

    3. Applied diff-log59.2

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

    4. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    5. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 12267.95132095184089848771691322326660156:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))