Average Error: 29.9 → 0.1
Time: 15.7s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8020.572324824966926826164126396179199219:\\ \;\;\;\;\log \left(\frac{{1}^{3} + {N}^{3}}{\left(\left(1 - N\right) \cdot 1 + N \cdot N\right) \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8020.572324824966926826164126396179199219:\\
\;\;\;\;\log \left(\frac{{1}^{3} + {N}^{3}}{\left(\left(1 - N\right) \cdot 1 + N \cdot N\right) \cdot N}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\

\end{array}
double f(double N) {
        double r40340 = N;
        double r40341 = 1.0;
        double r40342 = r40340 + r40341;
        double r40343 = log(r40342);
        double r40344 = log(r40340);
        double r40345 = r40343 - r40344;
        return r40345;
}
↓
double f(double N) {
        double r40346 = N;
        double r40347 = 8020.572324824967;
        bool r40348 = r40346 <= r40347;
        double r40349 = 1.0;
        double r40350 = 3.0;
        double r40351 = pow(r40349, r40350);
        double r40352 = pow(r40346, r40350);
        double r40353 = r40351 + r40352;
        double r40354 = r40349 - r40346;
        double r40355 = r40354 * r40349;
        double r40356 = r40346 * r40346;
        double r40357 = r40355 + r40356;
        double r40358 = r40357 * r40346;
        double r40359 = r40353 / r40358;
        double r40360 = log(r40359);
        double r40361 = r40349 / r40346;
        double r40362 = 0.5;
        double r40363 = r40362 / r40346;
        double r40364 = r40363 / r40346;
        double r40365 = r40361 - r40364;
        double r40366 = 0.3333333333333333;
        double r40367 = r40366 / r40352;
        double r40368 = r40365 + r40367;
        double r40369 = r40348 ? r40360 : r40368;
        return r40369;
}

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 < 8020.572324824967

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied diff-log0.1

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

    5. Simplified0.1

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

    7. Applied flip3-+0.1

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

    8. Applied associate-/l/0.1

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

    9. Simplified0.1

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

    if 8020.572324824967 < N

    1. Initial program 59.5

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

    2. Simplified59.5

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

    4. Applied diff-log59.3

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

    5. Simplified59.3

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

    6. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    7. Simplified0.0

      \[\leadsto \color{blue}{\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8020.572324824966926826164126396179199219:\\ \;\;\;\;\log \left(\frac{{1}^{3} + {N}^{3}}{\left(\left(1 - N\right) \cdot 1 + N \cdot N\right) \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))