Average Error: 29.7 → 0.1
Time: 13.9s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 5705.402280155900371028110384941101074219:\\ \;\;\;\;\left(\log \left(\sqrt{N + 1}\right) - \log \left(\sqrt{N}\right)\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{N \cdot \left(N \cdot N\right)} + \left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 5705.402280155900371028110384941101074219:\\
\;\;\;\;\left(\log \left(\sqrt{N + 1}\right) - \log \left(\sqrt{N}\right)\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\

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

\end{array}
double f(double N) {
        double r4793123 = N;
        double r4793124 = 1.0;
        double r4793125 = r4793123 + r4793124;
        double r4793126 = log(r4793125);
        double r4793127 = log(r4793123);
        double r4793128 = r4793126 - r4793127;
        return r4793128;
}
↓
double f(double N) {
        double r4793129 = N;
        double r4793130 = 5705.4022801559;
        bool r4793131 = r4793129 <= r4793130;
        double r4793132 = 1.0;
        double r4793133 = r4793129 + r4793132;
        double r4793134 = sqrt(r4793133);
        double r4793135 = log(r4793134);
        double r4793136 = sqrt(r4793129);
        double r4793137 = log(r4793136);
        double r4793138 = r4793135 - r4793137;
        double r4793139 = r4793133 / r4793129;
        double r4793140 = sqrt(r4793139);
        double r4793141 = log(r4793140);
        double r4793142 = r4793138 + r4793141;
        double r4793143 = 0.3333333333333333;
        double r4793144 = r4793129 * r4793129;
        double r4793145 = r4793129 * r4793144;
        double r4793146 = r4793143 / r4793145;
        double r4793147 = r4793132 / r4793129;
        double r4793148 = 0.5;
        double r4793149 = r4793148 / r4793144;
        double r4793150 = r4793147 - r4793149;
        double r4793151 = r4793146 + r4793150;
        double r4793152 = r4793131 ? r4793142 : r4793151;
        return r4793152;
}

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

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

  4. Final simplification0.1

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

Reproduce

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