Average Error: 29.7 → 0.1
Time: 2.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7877.19975446712033:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7877.19975446712033:\\
\;\;\;\;\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 r35550 = N;
        double r35551 = 1.0;
        double r35552 = r35550 + r35551;
        double r35553 = log(r35552);
        double r35554 = log(r35550);
        double r35555 = r35553 - r35554;
        return r35555;
}
↓
double f(double N) {
        double r35556 = N;
        double r35557 = 7877.19975446712;
        bool r35558 = r35556 <= r35557;
        double r35559 = 1.0;
        double r35560 = r35556 + r35559;
        double r35561 = r35560 / r35556;
        double r35562 = log(r35561);
        double r35563 = 1.0;
        double r35564 = 2.0;
        double r35565 = pow(r35556, r35564);
        double r35566 = r35563 / r35565;
        double r35567 = 0.3333333333333333;
        double r35568 = r35567 / r35556;
        double r35569 = 0.5;
        double r35570 = r35568 - r35569;
        double r35571 = r35566 * r35570;
        double r35572 = r35559 / r35556;
        double r35573 = r35571 + r35572;
        double r35574 = r35558 ? r35562 : r35573;
        return r35574;
}

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 < 7877.19975446712

    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 7877.19975446712 < N

    1. Initial program 59.5

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7877.19975446712033:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))