Average Error: 29.2 → 0.1
Time: 12.1s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7140.532646773332999146077781915664672852:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt{\frac{N + 1}{N}}}\right) + \log \left(\sqrt{\sqrt{\frac{N + 1}{N}}}\right)\right) + \log \left(\frac{N + 1}{N}\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7140.532646773332999146077781915664672852:\\
\;\;\;\;\left(\log \left(\sqrt{\sqrt{\frac{N + 1}{N}}}\right) + \log \left(\sqrt{\sqrt{\frac{N + 1}{N}}}\right)\right) + \log \left(\frac{N + 1}{N}\right) \cdot \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\

\end{array}
double f(double N) {
        double r58723 = N;
        double r58724 = 1.0;
        double r58725 = r58723 + r58724;
        double r58726 = log(r58725);
        double r58727 = log(r58723);
        double r58728 = r58726 - r58727;
        return r58728;
}
↓
double f(double N) {
        double r58729 = N;
        double r58730 = 7140.532646773333;
        bool r58731 = r58729 <= r58730;
        double r58732 = 1.0;
        double r58733 = r58729 + r58732;
        double r58734 = r58733 / r58729;
        double r58735 = sqrt(r58734);
        double r58736 = sqrt(r58735);
        double r58737 = log(r58736);
        double r58738 = r58737 + r58737;
        double r58739 = log(r58734);
        double r58740 = 0.5;
        double r58741 = r58739 * r58740;
        double r58742 = r58738 + r58741;
        double r58743 = r58732 / r58729;
        double r58744 = 0.5;
        double r58745 = r58729 * r58729;
        double r58746 = r58744 / r58745;
        double r58747 = r58743 - r58746;
        double r58748 = 0.3333333333333333;
        double r58749 = 3.0;
        double r58750 = pow(r58729, r58749);
        double r58751 = r58748 / r58750;
        double r58752 = r58747 + r58751;
        double r58753 = r58731 ? r58742 : r58752;
        return r58753;
}

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 < 7140.532646773333

    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

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

    9. Applied sqrt-prod0.1

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

    10. Applied log-prod0.1

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

    12. Applied pow10.1

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

    13. Applied sqrt-pow10.1

      \[\leadsto \left(\log \left(\sqrt{\sqrt{\frac{N + 1}{N}}}\right) + \log \left(\sqrt{\sqrt{\frac{N + 1}{N}}}\right)\right) + \log \color{blue}{\left({\left(\frac{N + 1}{N}\right)}^{\left(\frac{1}{2}\right)}\right)}\]

    14. Applied log-pow0.1

      \[\leadsto \left(\log \left(\sqrt{\sqrt{\frac{N + 1}{N}}}\right) + \log \left(\sqrt{\sqrt{\frac{N + 1}{N}}}\right)\right) + \color{blue}{\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right)}\]

    if 7140.532646773333 < N

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7140.532646773332999146077781915664672852:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt{\frac{N + 1}{N}}}\right) + \log \left(\sqrt{\sqrt{\frac{N + 1}{N}}}\right)\right) + \log \left(\frac{N + 1}{N}\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))