Average Error: 29.6 → 0.1
Time: 6.9s
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 r29041 = N;
        double r29042 = 1.0;
        double r29043 = r29041 + r29042;
        double r29044 = log(r29043);
        double r29045 = log(r29041);
        double r29046 = r29044 - r29045;
        return r29046;
}
↓
double f(double N) {
        double r29047 = N;
        double r29048 = 4688.133507746426;
        bool r29049 = r29047 <= r29048;
        double r29050 = sqrt(r29047);
        double r29051 = log(r29050);
        double r29052 = -r29051;
        double r29053 = 1.0;
        double r29054 = r29047 + r29053;
        double r29055 = r29054 / r29050;
        double r29056 = log(r29055);
        double r29057 = r29052 + r29056;
        double r29058 = r29053 / r29047;
        double r29059 = 0.3333333333333333;
        double r29060 = 3.0;
        double r29061 = pow(r29047, r29060);
        double r29062 = r29059 / r29061;
        double r29063 = r29058 + r29062;
        double r29064 = 0.5;
        double r29065 = r29047 * r29047;
        double r29066 = r29064 / r29065;
        double r29067 = r29063 - r29066;
        double r29068 = r29049 ? r29057 : r29067;
        return r29068;
}

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)))