Average Error: 29.1 → 0.1
Time: 4.9s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9158.53286535175539:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9158.53286535175539:\\
\;\;\;\;\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 r53395 = N;
        double r53396 = 1.0;
        double r53397 = r53395 + r53396;
        double r53398 = log(r53397);
        double r53399 = log(r53395);
        double r53400 = r53398 - r53399;
        return r53400;
}
↓
double f(double N) {
        double r53401 = N;
        double r53402 = 9158.532865351755;
        bool r53403 = r53401 <= r53402;
        double r53404 = 1.0;
        double r53405 = r53401 + r53404;
        double r53406 = r53405 / r53401;
        double r53407 = log(r53406);
        double r53408 = 1.0;
        double r53409 = 2.0;
        double r53410 = pow(r53401, r53409);
        double r53411 = r53408 / r53410;
        double r53412 = 0.3333333333333333;
        double r53413 = r53412 / r53401;
        double r53414 = 0.5;
        double r53415 = r53413 - r53414;
        double r53416 = r53411 * r53415;
        double r53417 = r53404 / r53401;
        double r53418 = r53416 + r53417;
        double r53419 = r53403 ? r53407 : r53418;
        return r53419;
}

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 < 9158.532865351755

    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 9158.532865351755 < N

    1. Initial program 59.3

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9158.53286535175539:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))