Average Error: 29.0 → 0.1
Time: 11.8s
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 r1198152 = N;
        double r1198153 = 1.0;
        double r1198154 = r1198152 + r1198153;
        double r1198155 = log(r1198154);
        double r1198156 = log(r1198152);
        double r1198157 = r1198155 - r1198156;
        return r1198157;
}
↓
double f(double N) {
        double r1198158 = N;
        double r1198159 = 7726.51655081009;
        bool r1198160 = r1198158 <= r1198159;
        double r1198161 = 1.0;
        double r1198162 = r1198161 + r1198158;
        double r1198163 = r1198162 / r1198158;
        double r1198164 = log(r1198163);
        double r1198165 = r1198161 / r1198158;
        double r1198166 = -0.5;
        double r1198167 = r1198158 * r1198158;
        double r1198168 = r1198166 / r1198167;
        double r1198169 = r1198165 + r1198168;
        double r1198170 = 0.3333333333333333;
        double r1198171 = r1198170 / r1198158;
        double r1198172 = r1198171 / r1198167;
        double r1198173 = r1198169 + r1198172;
        double r1198174 = r1198160 ? r1198164 : r1198173;
        return r1198174;
}

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 diff-log0.1

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

    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)))