Average Error: 29.8 → 0.1
Time: 4.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8116.362270388288379763253033161163330078:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8116.362270388288379763253033161163330078:\\
\;\;\;\;\frac{1}{2} \cdot \log \left(\frac{N + 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 r41818 = N;
        double r41819 = 1.0;
        double r41820 = r41818 + r41819;
        double r41821 = log(r41820);
        double r41822 = log(r41818);
        double r41823 = r41821 - r41822;
        return r41823;
}
↓
double f(double N) {
        double r41824 = N;
        double r41825 = 8116.362270388288;
        bool r41826 = r41824 <= r41825;
        double r41827 = 0.5;
        double r41828 = 1.0;
        double r41829 = r41824 + r41828;
        double r41830 = r41829 / r41824;
        double r41831 = log(r41830);
        double r41832 = r41827 * r41831;
        double r41833 = sqrt(r41830);
        double r41834 = log(r41833);
        double r41835 = r41832 + r41834;
        double r41836 = 1.0;
        double r41837 = 2.0;
        double r41838 = pow(r41824, r41837);
        double r41839 = r41836 / r41838;
        double r41840 = 0.3333333333333333;
        double r41841 = r41840 / r41824;
        double r41842 = 0.5;
        double r41843 = r41841 - r41842;
        double r41844 = r41839 * r41843;
        double r41845 = r41828 / r41824;
        double r41846 = r41844 + r41845;
        double r41847 = r41826 ? r41835 : r41846;
        return r41847;
}

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 < 8116.362270388288

    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 pow10.1

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

    9. Applied sqrt-pow10.1

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

    10. Applied log-pow0.1

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

    if 8116.362270388288 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8116.362270388288379763253033161163330078:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))