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

↓

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

double f(double N) {
        double r2840073 = N;
        double r2840074 = 1.0;
        double r2840075 = r2840073 + r2840074;
        double r2840076 = log(r2840075);
        double r2840077 = log(r2840073);
        double r2840078 = r2840076 - r2840077;
        return r2840078;
}

↓

double f(double N) {
        double r2840079 = N;
        double r2840080 = 8873.677015963014;
        bool r2840081 = r2840079 <= r2840080;
        double r2840082 = 1.0;
        double r2840083 = r2840082 + r2840079;
        double r2840084 = r2840083 / r2840079;
        double r2840085 = log(r2840084);
        double r2840086 = r2840082 / r2840079;
        double r2840087 = -0.5;
        double r2840088 = r2840079 * r2840079;
        double r2840089 = r2840087 / r2840088;
        double r2840090 = r2840086 + r2840089;
        double r2840091 = -0.3333333333333333;
        double r2840092 = r2840079 * r2840088;
        double r2840093 = r2840091 / r2840092;
        double r2840094 = r2840090 - r2840093;
        double r2840095 = r2840081 ? r2840085 : r2840094;
        return r2840095;
}

\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{-1}{3}}{N \cdot \left(N \cdot N\right)}\\

\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. 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
  \[\leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
if 8873.677015963014 < 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-log59.2
  \[\leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
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
\[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8873.677015963014:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) - \frac{\frac{-1}{3}}{N \cdot \left(N \cdot N\right)}\\ \end{array}\]

Error

Derivation

`if N < 8873.677015963014`

`if 8873.677015963014 < N`

Reproduce