Average Error: 29.1 → 0.1
Time: 4.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8430.538053514888815698213875293731689453:\\ \;\;\;\;\log \left(\sqrt{N + 1} \cdot \sqrt{\frac{1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8430.538053514888815698213875293731689453:\\
\;\;\;\;\log \left(\sqrt{N + 1} \cdot \sqrt{\frac{1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r50900 = N;
        double r50901 = 1.0;
        double r50902 = r50900 + r50901;
        double r50903 = log(r50902);
        double r50904 = log(r50900);
        double r50905 = r50903 - r50904;
        return r50905;
}
↓
double f(double N) {
        double r50906 = N;
        double r50907 = 8430.538053514889;
        bool r50908 = r50906 <= r50907;
        double r50909 = 1.0;
        double r50910 = r50906 + r50909;
        double r50911 = sqrt(r50910);
        double r50912 = 1.0;
        double r50913 = r50912 / r50906;
        double r50914 = sqrt(r50913);
        double r50915 = r50911 * r50914;
        double r50916 = log(r50915);
        double r50917 = r50910 / r50906;
        double r50918 = sqrt(r50917);
        double r50919 = log(r50918);
        double r50920 = r50916 + r50919;
        double r50921 = 2.0;
        double r50922 = pow(r50906, r50921);
        double r50923 = r50912 / r50922;
        double r50924 = 0.3333333333333333;
        double r50925 = r50924 / r50906;
        double r50926 = 0.5;
        double r50927 = r50925 - r50926;
        double r50928 = r50923 * r50927;
        double r50929 = r50909 / r50906;
        double r50930 = r50928 + r50929;
        double r50931 = r50908 ? r50920 : r50930;
        return r50931;
}

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 < 8430.538053514889

    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 div-inv0.1

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

    9. Applied sqrt-prod0.1

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

    if 8430.538053514889 < N

    1. Initial program 59.4

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8430.538053514888815698213875293731689453:\\ \;\;\;\;\log \left(\sqrt{N + 1} \cdot \sqrt{\frac{1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))