Average Error: 29.4 → 0.1
Time: 5.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8616.08642354138101:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8616.08642354138101:\\
\;\;\;\;\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 r56723 = N;
        double r56724 = 1.0;
        double r56725 = r56723 + r56724;
        double r56726 = log(r56725);
        double r56727 = log(r56723);
        double r56728 = r56726 - r56727;
        return r56728;
}
↓
double f(double N) {
        double r56729 = N;
        double r56730 = 8616.086423541381;
        bool r56731 = r56729 <= r56730;
        double r56732 = 1.0;
        double r56733 = r56729 + r56732;
        double r56734 = r56733 / r56729;
        double r56735 = log(r56734);
        double r56736 = 1.0;
        double r56737 = 2.0;
        double r56738 = pow(r56729, r56737);
        double r56739 = r56736 / r56738;
        double r56740 = 0.3333333333333333;
        double r56741 = r56740 / r56729;
        double r56742 = 0.5;
        double r56743 = r56741 - r56742;
        double r56744 = r56739 * r56743;
        double r56745 = r56732 / r56729;
        double r56746 = r56744 + r56745;
        double r56747 = r56731 ? r56735 : r56746;
        return r56747;
}

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 < 8616.086423541381

    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 8616.086423541381 < N

    1. Initial program 59.4

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8616.08642354138101:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))