Average Error: 29.6 → 0.1
Time: 8.7s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4688.1335077464264:\\ \;\;\;\;\left(-\log \left(\sqrt{N}\right)\right) + \log \left(\frac{N + 1}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.333333333333333315}{{N}^{3}}\\ \end{array}\]
\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{0.5}{N \cdot N}\right) + \frac{0.333333333333333315}{{N}^{3}}\\

\end{array}
double f(double N) {
        double r48762 = N;
        double r48763 = 1.0;
        double r48764 = r48762 + r48763;
        double r48765 = log(r48764);
        double r48766 = log(r48762);
        double r48767 = r48765 - r48766;
        return r48767;
}
↓
double f(double N) {
        double r48768 = N;
        double r48769 = 4688.133507746426;
        bool r48770 = r48768 <= r48769;
        double r48771 = sqrt(r48768);
        double r48772 = log(r48771);
        double r48773 = -r48772;
        double r48774 = 1.0;
        double r48775 = r48768 + r48774;
        double r48776 = r48775 / r48771;
        double r48777 = log(r48776);
        double r48778 = r48773 + r48777;
        double r48779 = r48774 / r48768;
        double r48780 = 0.5;
        double r48781 = r48768 * r48768;
        double r48782 = r48780 / r48781;
        double r48783 = r48779 - r48782;
        double r48784 = 0.3333333333333333;
        double r48785 = 3.0;
        double r48786 = pow(r48768, r48785);
        double r48787 = r48784 / r48786;
        double r48788 = r48783 + r48787;
        double r48789 = r48770 ? r48778 : r48788;
        return r48789;
}

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. Using strategy rm

    3. Applied diff-log59.0

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

    4. Taylor expanded around inf 0.0

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

    5. Simplified0.0

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

  4. Final simplification0.1

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

Reproduce

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