Average Error: 29.7 → 0.1
Time: 13.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 5705.402280155900371028110384941101074219:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \left(\log \left(\sqrt{1 + N}\right) - \log \left(\sqrt{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{0.5}{N \cdot N} - \frac{\frac{0.3333333333333333148296162562473909929395}{N}}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 5705.402280155900371028110384941101074219:\\
\;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \left(\log \left(\sqrt{1 + N}\right) - \log \left(\sqrt{N}\right)\right)\\

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

\end{array}
double f(double N) {
        double r3094312 = N;
        double r3094313 = 1.0;
        double r3094314 = r3094312 + r3094313;
        double r3094315 = log(r3094314);
        double r3094316 = log(r3094312);
        double r3094317 = r3094315 - r3094316;
        return r3094317;
}
↓
double f(double N) {
        double r3094318 = N;
        double r3094319 = 5705.4022801559;
        bool r3094320 = r3094318 <= r3094319;
        double r3094321 = 1.0;
        double r3094322 = r3094321 + r3094318;
        double r3094323 = r3094322 / r3094318;
        double r3094324 = sqrt(r3094323);
        double r3094325 = log(r3094324);
        double r3094326 = sqrt(r3094322);
        double r3094327 = log(r3094326);
        double r3094328 = sqrt(r3094318);
        double r3094329 = log(r3094328);
        double r3094330 = r3094327 - r3094329;
        double r3094331 = r3094325 + r3094330;
        double r3094332 = r3094321 / r3094318;
        double r3094333 = 0.5;
        double r3094334 = r3094318 * r3094318;
        double r3094335 = r3094333 / r3094334;
        double r3094336 = 0.3333333333333333;
        double r3094337 = r3094336 / r3094318;
        double r3094338 = r3094337 / r3094334;
        double r3094339 = r3094335 - r3094338;
        double r3094340 = r3094332 - r3094339;
        double r3094341 = r3094320 ? r3094331 : r3094340;
        return r3094341;
}

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 < 5705.4022801559

    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-sqr-sqrt0.1

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

    6. Applied log-prod0.1

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

    8. Applied sqrt-div0.1

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

    9. Applied log-div0.1

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

    if 5705.4022801559 < N

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5705.402280155900371028110384941101074219:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \left(\log \left(\sqrt{1 + N}\right) - \log \left(\sqrt{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{0.5}{N \cdot N} - \frac{\frac{0.3333333333333333148296162562473909929395}{N}}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))