Average Error: 29.0 → 0.1
Time: 14.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7726.51655081009:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N}}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7726.51655081009:\\
\;\;\;\;\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}}{N \cdot N}\\

\end{array}
double f(double N) {
        double r1023059 = N;
        double r1023060 = 1.0;
        double r1023061 = r1023059 + r1023060;
        double r1023062 = log(r1023061);
        double r1023063 = log(r1023059);
        double r1023064 = r1023062 - r1023063;
        return r1023064;
}
↓
double f(double N) {
        double r1023065 = N;
        double r1023066 = 7726.51655081009;
        bool r1023067 = r1023065 <= r1023066;
        double r1023068 = 1.0;
        double r1023069 = r1023068 + r1023065;
        double r1023070 = r1023069 / r1023065;
        double r1023071 = log(r1023070);
        double r1023072 = r1023068 / r1023065;
        double r1023073 = -0.5;
        double r1023074 = r1023065 * r1023065;
        double r1023075 = r1023073 / r1023074;
        double r1023076 = r1023072 + r1023075;
        double r1023077 = 0.3333333333333333;
        double r1023078 = r1023077 / r1023065;
        double r1023079 = r1023078 / r1023074;
        double r1023080 = r1023076 + r1023079;
        double r1023081 = r1023067 ? r1023071 : r1023080;
        return r1023081;
}

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 < 7726.51655081009

    1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied add-log-exp0.1

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

    4. Simplified0.1

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

    if 7726.51655081009 < 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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7726.51655081009:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N}}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))