Average Error: 29.8 → 0.1
Time: 15.7s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 5899.4445172695305:\\ \;\;\;\;\left(\log \left(\sqrt{\frac{1}{\sqrt{N}}}\right) + \log \left(\sqrt{\frac{1 + N}{\sqrt{N}}}\right)\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right) + \frac{\frac{1}{3}}{N \cdot \left(N \cdot N\right)}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 5899.4445172695305:\\
\;\;\;\;\left(\log \left(\sqrt{\frac{1}{\sqrt{N}}}\right) + \log \left(\sqrt{\frac{1 + N}{\sqrt{N}}}\right)\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right) + \frac{\frac{1}{3}}{N \cdot \left(N \cdot N\right)}\\

\end{array}
double f(double N) {
        double r1656842 = N;
        double r1656843 = 1.0;
        double r1656844 = r1656842 + r1656843;
        double r1656845 = log(r1656844);
        double r1656846 = log(r1656842);
        double r1656847 = r1656845 - r1656846;
        return r1656847;
}
↓
double f(double N) {
        double r1656848 = N;
        double r1656849 = 5899.4445172695305;
        bool r1656850 = r1656848 <= r1656849;
        double r1656851 = 1.0;
        double r1656852 = sqrt(r1656848);
        double r1656853 = r1656851 / r1656852;
        double r1656854 = sqrt(r1656853);
        double r1656855 = log(r1656854);
        double r1656856 = r1656851 + r1656848;
        double r1656857 = r1656856 / r1656852;
        double r1656858 = sqrt(r1656857);
        double r1656859 = log(r1656858);
        double r1656860 = r1656855 + r1656859;
        double r1656861 = r1656856 / r1656848;
        double r1656862 = sqrt(r1656861);
        double r1656863 = log(r1656862);
        double r1656864 = r1656860 + r1656863;
        double r1656865 = r1656851 / r1656848;
        double r1656866 = 0.5;
        double r1656867 = r1656848 * r1656848;
        double r1656868 = r1656866 / r1656867;
        double r1656869 = r1656865 - r1656868;
        double r1656870 = 0.3333333333333333;
        double r1656871 = r1656848 * r1656867;
        double r1656872 = r1656870 / r1656871;
        double r1656873 = r1656869 + r1656872;
        double r1656874 = r1656850 ? r1656864 : r1656873;
        return r1656874;
}

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 < 5899.4445172695305

    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 add-sqr-sqrt0.1

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

    9. Applied *-un-lft-identity0.1

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

    10. Applied *-un-lft-identity0.1

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

    11. Applied distribute-lft-out0.1

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

    12. Applied times-frac0.1

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

    13. Applied sqrt-prod0.1

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

    14. Applied log-prod0.1

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

    if 5899.4445172695305 < N

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5899.4445172695305:\\ \;\;\;\;\left(\log \left(\sqrt{\frac{1}{\sqrt{N}}}\right) + \log \left(\sqrt{\frac{1 + N}{\sqrt{N}}}\right)\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right) + \frac{\frac{1}{3}}{N \cdot \left(N \cdot N\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))