Average Error: 29.6 → 0.1
Time: 28.2s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8072.727785295212:\\ \;\;\;\;\log \left(\frac{1 + \left(N \cdot N\right) \cdot N}{N \cdot \left(N \cdot N + \left(1 - N\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{N \cdot N} + \frac{1}{N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8072.727785295212:\\
\;\;\;\;\log \left(\frac{1 + \left(N \cdot N\right) \cdot N}{N \cdot \left(N \cdot N + \left(1 - N\right)\right)}\right)\\

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

\end{array}
double f(double N) {
        double r2505226 = N;
        double r2505227 = 1.0;
        double r2505228 = r2505226 + r2505227;
        double r2505229 = log(r2505228);
        double r2505230 = log(r2505226);
        double r2505231 = r2505229 - r2505230;
        return r2505231;
}
↓
double f(double N) {
        double r2505232 = N;
        double r2505233 = 8072.727785295212;
        bool r2505234 = r2505232 <= r2505233;
        double r2505235 = 1.0;
        double r2505236 = r2505232 * r2505232;
        double r2505237 = r2505236 * r2505232;
        double r2505238 = r2505235 + r2505237;
        double r2505239 = r2505235 - r2505232;
        double r2505240 = r2505236 + r2505239;
        double r2505241 = r2505232 * r2505240;
        double r2505242 = r2505238 / r2505241;
        double r2505243 = log(r2505242);
        double r2505244 = -0.5;
        double r2505245 = r2505244 / r2505236;
        double r2505246 = r2505235 / r2505232;
        double r2505247 = r2505245 + r2505246;
        double r2505248 = 0.3333333333333333;
        double r2505249 = r2505248 / r2505236;
        double r2505250 = r2505249 / r2505232;
        double r2505251 = r2505247 + r2505250;
        double r2505252 = r2505234 ? r2505243 : r2505251;
        return r2505252;
}

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 < 8072.727785295212

    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 flip3-+0.1

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

    6. Applied associate-/l/0.1

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

    7. Simplified0.1

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

    if 8072.727785295212 < N

    1. Initial program 59.6

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

    3. Applied diff-log59.4

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

    5. Applied flip3-+61.2

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

    6. Applied associate-/l/61.2

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

    7. Simplified61.2

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

    8. Taylor expanded around inf 0.0

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

    9. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8072.727785295212:\\ \;\;\;\;\log \left(\frac{1 + \left(N \cdot N\right) \cdot N}{N \cdot \left(N \cdot N + \left(1 - N\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{N \cdot N} + \frac{1}{N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))