Average Error: 29.5 → 0.1
Time: 8.7s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7622.50412124721061:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7622.50412124721061:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}\\

\end{array}
double f(double N) {
        double r41562 = N;
        double r41563 = 1.0;
        double r41564 = r41562 + r41563;
        double r41565 = log(r41564);
        double r41566 = log(r41562);
        double r41567 = r41565 - r41566;
        return r41567;
}
↓
double f(double N) {
        double r41568 = N;
        double r41569 = 7622.504121247211;
        bool r41570 = r41568 <= r41569;
        double r41571 = 1.0;
        double r41572 = r41568 + r41571;
        double r41573 = r41572 / r41568;
        double r41574 = log(r41573);
        double r41575 = r41571 / r41568;
        double r41576 = 0.3333333333333333;
        double r41577 = 3.0;
        double r41578 = pow(r41568, r41577);
        double r41579 = r41576 / r41578;
        double r41580 = r41575 + r41579;
        double r41581 = 0.5;
        double r41582 = r41568 * r41568;
        double r41583 = r41581 / r41582;
        double r41584 = r41580 - r41583;
        double r41585 = r41570 ? r41574 : r41584;
        return r41585;
}

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 < 7622.504121247211

    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 7622.504121247211 < N

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7622.50412124721061:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))