Average Error: 29.3 → 0.1
Time: 27.1s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7250.408229147694:\\ \;\;\;\;\log \left(\frac{\frac{1 + N}{\sqrt{N}}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{\frac{-1}{2}}{N \cdot N} + \frac{\frac{1}{N}}{N \cdot N} \cdot \frac{1}{3}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7250.408229147694:\\
\;\;\;\;\log \left(\frac{\frac{1 + N}{\sqrt{N}}}{\sqrt{N}}\right)\\

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

\end{array}
double f(double N) {
        double r1574220 = N;
        double r1574221 = 1.0;
        double r1574222 = r1574220 + r1574221;
        double r1574223 = log(r1574222);
        double r1574224 = log(r1574220);
        double r1574225 = r1574223 - r1574224;
        return r1574225;
}
↓
double f(double N) {
        double r1574226 = N;
        double r1574227 = 7250.408229147694;
        bool r1574228 = r1574226 <= r1574227;
        double r1574229 = 1.0;
        double r1574230 = r1574229 + r1574226;
        double r1574231 = sqrt(r1574226);
        double r1574232 = r1574230 / r1574231;
        double r1574233 = r1574232 / r1574231;
        double r1574234 = log(r1574233);
        double r1574235 = r1574229 / r1574226;
        double r1574236 = -0.5;
        double r1574237 = r1574226 * r1574226;
        double r1574238 = r1574236 / r1574237;
        double r1574239 = r1574235 / r1574237;
        double r1574240 = 0.3333333333333333;
        double r1574241 = r1574239 * r1574240;
        double r1574242 = r1574238 + r1574241;
        double r1574243 = r1574235 + r1574242;
        double r1574244 = r1574228 ? r1574234 : r1574243;
        return r1574244;
}

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 < 7250.408229147694

    1. Initial program 0.1

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

    3. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt0.1

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

    6. Applied associate-/r*0.1

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

    if 7250.408229147694 < N

    1. Initial program 59.7

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

    3. Applied diff-log59.5

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt60.0

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

    6. Applied associate-/r*60.0

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

    7. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^{2}}}\]

    8. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7250.408229147694:\\ \;\;\;\;\log \left(\frac{\frac{1 + N}{\sqrt{N}}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{\frac{-1}{2}}{N \cdot N} + \frac{\frac{1}{N}}{N \cdot N} \cdot \frac{1}{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))