Average Error: 29.6 → 0.1
Time: 9.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4688.1335077464264:\\ \;\;\;\;\left(-\log \left(\sqrt{N}\right)\right) + \log \left(\frac{N + 1}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4688.1335077464264:\\
\;\;\;\;\left(-\log \left(\sqrt{N}\right)\right) + \log \left(\frac{N + 1}{\sqrt{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}\\

\end{array}
double f(double N) {
        double r36446 = N;
        double r36447 = 1.0;
        double r36448 = r36446 + r36447;
        double r36449 = log(r36448);
        double r36450 = log(r36446);
        double r36451 = r36449 - r36450;
        return r36451;
}
↓
double f(double N) {
        double r36452 = N;
        double r36453 = 4688.133507746426;
        bool r36454 = r36452 <= r36453;
        double r36455 = sqrt(r36452);
        double r36456 = log(r36455);
        double r36457 = -r36456;
        double r36458 = 1.0;
        double r36459 = r36452 + r36458;
        double r36460 = r36459 / r36455;
        double r36461 = log(r36460);
        double r36462 = r36457 + r36461;
        double r36463 = r36458 / r36452;
        double r36464 = 0.3333333333333333;
        double r36465 = 3.0;
        double r36466 = pow(r36452, r36465);
        double r36467 = r36464 / r36466;
        double r36468 = r36463 + r36467;
        double r36469 = 0.5;
        double r36470 = r36452 * r36452;
        double r36471 = r36469 / r36470;
        double r36472 = r36468 - r36471;
        double r36473 = r36454 ? r36462 : r36472;
        return r36473;
}

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 < 4688.133507746426

    1. Initial program 0.1

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

    3. Applied diff-log0.0

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

    5. Applied add-sqr-sqrt0.0

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

    6. Applied *-un-lft-identity0.0

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

    7. Applied times-frac0.0

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

    8. Applied log-prod0.1

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

    9. Simplified0.1

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

    if 4688.133507746426 < N

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4688.1335077464264:\\ \;\;\;\;\left(-\log \left(\sqrt{N}\right)\right) + \log \left(\frac{N + 1}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))