Average Error: 29.8 → 0.1
Time: 4.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9757.882283737008037860505282878875732422:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9757.882283737008037860505282878875732422:\\
\;\;\;\;\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 r45672 = N;
        double r45673 = 1.0;
        double r45674 = r45672 + r45673;
        double r45675 = log(r45674);
        double r45676 = log(r45672);
        double r45677 = r45675 - r45676;
        return r45677;
}
↓
double f(double N) {
        double r45678 = N;
        double r45679 = 9757.882283737008;
        bool r45680 = r45678 <= r45679;
        double r45681 = 1.0;
        double r45682 = r45678 + r45681;
        double r45683 = r45682 / r45678;
        double r45684 = log(r45683);
        double r45685 = 1.0;
        double r45686 = 2.0;
        double r45687 = pow(r45678, r45686);
        double r45688 = r45685 / r45687;
        double r45689 = 0.3333333333333333;
        double r45690 = r45689 / r45678;
        double r45691 = 0.5;
        double r45692 = r45690 - r45691;
        double r45693 = r45688 * r45692;
        double r45694 = r45681 / r45678;
        double r45695 = r45693 + r45694;
        double r45696 = r45680 ? r45684 : r45695;
        return r45696;
}

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 < 9757.882283737008

    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 9757.882283737008 < N

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.0

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9757.882283737008037860505282878875732422:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))