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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 6762.824049100211:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \frac{1}{2} \cdot \left(\log \left(\frac{\sqrt{N + 1}}{N}\right) + \log \left(\sqrt{N + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) - \frac{\frac{-1}{3}}{N \cdot \left(N \cdot N\right)}\\ \end{array}\]

double f(double N) {
        double r1340745 = N;
        double r1340746 = 1.0;
        double r1340747 = r1340745 + r1340746;
        double r1340748 = log(r1340747);
        double r1340749 = log(r1340745);
        double r1340750 = r1340748 - r1340749;
        return r1340750;
}

↓

double f(double N) {
        double r1340751 = N;
        double r1340752 = 6762.824049100211;
        bool r1340753 = r1340751 <= r1340752;
        double r1340754 = 1.0;
        double r1340755 = r1340751 + r1340754;
        double r1340756 = r1340755 / r1340751;
        double r1340757 = sqrt(r1340756);
        double r1340758 = log(r1340757);
        double r1340759 = 0.5;
        double r1340760 = sqrt(r1340755);
        double r1340761 = r1340760 / r1340751;
        double r1340762 = log(r1340761);
        double r1340763 = log(r1340760);
        double r1340764 = r1340762 + r1340763;
        double r1340765 = r1340759 * r1340764;
        double r1340766 = r1340758 + r1340765;
        double r1340767 = r1340754 / r1340751;
        double r1340768 = -0.5;
        double r1340769 = r1340751 * r1340751;
        double r1340770 = r1340768 / r1340769;
        double r1340771 = r1340767 + r1340770;
        double r1340772 = -0.3333333333333333;
        double r1340773 = r1340751 * r1340769;
        double r1340774 = r1340772 / r1340773;
        double r1340775 = r1340771 - r1340774;
        double r1340776 = r1340753 ? r1340766 : r1340775;
        return r1340776;
}

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

↓

\begin{array}{l}
\mathbf{if}\;N \le 6762.824049100211:\\
\;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \frac{1}{2} \cdot \left(\log \left(\frac{\sqrt{N + 1}}{N}\right) + \log \left(\sqrt{N + 1}\right)\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}

Derivation

Split input into 2 regimes
if N < 6762.824049100211
1. Initial program 0.1
  \[\log \left(N + 1\right) - \log N\]
2. Simplified0.1
  \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]
3. Using strategy rm
4. Applied log1p-udef0.1
  \[\leadsto \color{blue}{\log \left(1 + N\right)} - \log N\]
5. Applied diff-log0.1
  \[\leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
6. Using strategy rm
7. Applied add-sqr-sqrt0.1
  \[\leadsto \log \color{blue}{\left(\sqrt{\frac{1 + N}{N}} \cdot \sqrt{\frac{1 + N}{N}}\right)}\]
8. Applied log-prod0.1
  \[\leadsto \color{blue}{\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)}\]
9. Using strategy rm
10. Applied pow1/20.1
  \[\leadsto \log \color{blue}{\left({\left(\frac{1 + N}{N}\right)}^{\frac{1}{2}}\right)} + \log \left(\sqrt{\frac{1 + N}{N}}\right)\]
11. Applied log-pow0.1
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \log \left(\frac{1 + N}{N}\right)} + \log \left(\sqrt{\frac{1 + N}{N}}\right)\]
12. Using strategy rm
13. Applied *-un-lft-identity0.1
  \[\leadsto \frac{1}{2} \cdot \log \left(\frac{1 + N}{\color{blue}{1 \cdot N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\]
14. Applied add-sqr-sqrt0.1
  \[\leadsto \frac{1}{2} \cdot \log \left(\frac{\color{blue}{\sqrt{1 + N} \cdot \sqrt{1 + N}}}{1 \cdot N}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\]
15. Applied times-frac0.1
  \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left(\frac{\sqrt{1 + N}}{1} \cdot \frac{\sqrt{1 + N}}{N}\right)} + \log \left(\sqrt{\frac{1 + N}{N}}\right)\]
16. Applied log-prod0.1
  \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\log \left(\frac{\sqrt{1 + N}}{1}\right) + \log \left(\frac{\sqrt{1 + N}}{N}\right)\right)} + \log \left(\sqrt{\frac{1 + N}{N}}\right)\]
if 6762.824049100211 < N
1. Initial program 59.5
  \[\log \left(N + 1\right) - \log N\]
2. Simplified59.5
  \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]
3. Using strategy rm
4. Applied log1p-udef59.5
  \[\leadsto \color{blue}{\log \left(1 + N\right)} - \log N\]
5. Applied diff-log59.3
  \[\leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
6. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^{2}}}\]
7. Simplified0.0
  \[\leadsto \color{blue}{\left(\frac{\frac{-1}{2}}{N \cdot N} + \frac{1}{N}\right) - \frac{\frac{-1}{3}}{\left(N \cdot N\right) \cdot N}}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;N \le 6762.824049100211:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \frac{1}{2} \cdot \left(\log \left(\frac{\sqrt{N + 1}}{N}\right) + \log \left(\sqrt{N + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) - \frac{\frac{-1}{3}}{N \cdot \left(N \cdot N\right)}\\ \end{array}\]

Error

Derivation

`if N < 6762.824049100211`

`if 6762.824049100211 < N`

Reproduce