Average Error: 30.3 → 0.1
Time: 4.9s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 20452.1348824906744994223117828369140625:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 20452.1348824906744994223117828369140625:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r42629 = N;
        double r42630 = 1.0;
        double r42631 = r42629 + r42630;
        double r42632 = log(r42631);
        double r42633 = log(r42629);
        double r42634 = r42632 - r42633;
        return r42634;
}
↓
double f(double N) {
        double r42635 = N;
        double r42636 = 20452.134882490674;
        bool r42637 = r42635 <= r42636;
        double r42638 = 1.0;
        double r42639 = r42635 + r42638;
        double r42640 = r42639 / r42635;
        double r42641 = log(r42640);
        double r42642 = 1.0;
        double r42643 = 2.0;
        double r42644 = pow(r42635, r42643);
        double r42645 = r42642 / r42644;
        double r42646 = 0.3333333333333333;
        double r42647 = r42646 / r42635;
        double r42648 = 0.5;
        double r42649 = r42647 - r42648;
        double r42650 = r42645 * r42649;
        double r42651 = r42638 / r42635;
        double r42652 = r42650 + r42651;
        double r42653 = r42637 ? r42641 : r42652;
        return r42653;
}

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 < 20452.134882490674

    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 20452.134882490674 < N

    1. Initial program 59.5

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 20452.1348824906744994223117828369140625:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))