Average Error: 29.3 → 0.1
Time: 3.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9398.8604648901419:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9398.8604648901419:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r36468 = N;
        double r36469 = 1.0;
        double r36470 = r36468 + r36469;
        double r36471 = log(r36470);
        double r36472 = log(r36468);
        double r36473 = r36471 - r36472;
        return r36473;
}
↓
double f(double N) {
        double r36474 = N;
        double r36475 = 9398.860464890142;
        bool r36476 = r36474 <= r36475;
        double r36477 = 1.0;
        double r36478 = r36474 + r36477;
        double r36479 = r36478 / r36474;
        double r36480 = log(r36479);
        double r36481 = 1.0;
        double r36482 = 2.0;
        double r36483 = pow(r36474, r36482);
        double r36484 = r36481 / r36483;
        double r36485 = 0.3333333333333333;
        double r36486 = r36485 / r36474;
        double r36487 = 0.5;
        double r36488 = r36486 - r36487;
        double r36489 = r36484 * r36488;
        double r36490 = r36477 / r36474;
        double r36491 = r36489 + r36490;
        double r36492 = r36476 ? r36480 : r36491;
        return r36492;
}

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 < 9398.860464890142

    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 9398.860464890142 < N

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9398.8604648901419:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))