Average Error: 29.6 → 0.1
Time: 11.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7426.105103924180184549186378717422485352:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{0.5}{N \cdot N} - \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7426.105103924180184549186378717422485352:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r24484 = N;
        double r24485 = 1.0;
        double r24486 = r24484 + r24485;
        double r24487 = log(r24486);
        double r24488 = log(r24484);
        double r24489 = r24487 - r24488;
        return r24489;
}
↓
double f(double N) {
        double r24490 = N;
        double r24491 = 7426.10510392418;
        bool r24492 = r24490 <= r24491;
        double r24493 = 1.0;
        double r24494 = r24493 + r24490;
        double r24495 = r24494 / r24490;
        double r24496 = log(r24495);
        double r24497 = r24493 / r24490;
        double r24498 = 0.5;
        double r24499 = r24490 * r24490;
        double r24500 = r24498 / r24499;
        double r24501 = 0.3333333333333333;
        double r24502 = 3.0;
        double r24503 = pow(r24490, r24502);
        double r24504 = r24501 / r24503;
        double r24505 = r24500 - r24504;
        double r24506 = r24497 - r24505;
        double r24507 = r24492 ? r24496 : r24506;
        return r24507;
}

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 < 7426.10510392418

    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. Simplified0.1

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

    if 7426.10510392418 < N

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7426.105103924180184549186378717422485352:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{0.5}{N \cdot N} - \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))