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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 8873.677015963014:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}\\ \end{array}\]

double f(double N) {
        double r1890987 = N;
        double r1890988 = 1.0;
        double r1890989 = r1890987 + r1890988;
        double r1890990 = log(r1890989);
        double r1890991 = log(r1890987);
        double r1890992 = r1890990 - r1890991;
        return r1890992;
}

↓

double f(double N) {
        double r1890993 = N;
        double r1890994 = 8873.677015963014;
        bool r1890995 = r1890993 <= r1890994;
        double r1890996 = 1.0;
        double r1890997 = r1890996 + r1890993;
        double r1890998 = r1890997 / r1890993;
        double r1890999 = log(r1890998);
        double r1891000 = r1890996 / r1890993;
        double r1891001 = -0.5;
        double r1891002 = r1890993 * r1890993;
        double r1891003 = r1891001 / r1891002;
        double r1891004 = r1891000 + r1891003;
        double r1891005 = 0.3333333333333333;
        double r1891006 = r1891005 / r1891002;
        double r1891007 = r1891006 / r1890993;
        double r1891008 = r1891004 + r1891007;
        double r1891009 = r1890995 ? r1890999 : r1891008;
        return r1891009;
}

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

↓

\begin{array}{l}
\mathbf{if}\;N \le 8873.677015963014:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}

Derivation

Split input into 2 regimes
if N < 8873.677015963014
1. Initial program 0.1
  \[\log \left(N + 1\right) - \log N\]
2. Using strategy rm
3. Applied diff-log0.1
  \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
if 8873.677015963014 < N
1. Initial program 59.5
  \[\log \left(N + 1\right) - \log N\]
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
  \[\leadsto \color{blue}{\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8873.677015963014:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}\\ \end{array}\]

Error

Derivation

`if N < 8873.677015963014`

`if 8873.677015963014 < N`

Reproduce