Average Error: 29.3 → 0.0
Time: 11.2s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9187.863337025455621187575161457061767578:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9187.863337025455621187575161457061767578:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\

\end{array}
double f(double N) {
        double r52604 = N;
        double r52605 = 1.0;
        double r52606 = r52604 + r52605;
        double r52607 = log(r52606);
        double r52608 = log(r52604);
        double r52609 = r52607 - r52608;
        return r52609;
}
↓
double f(double N) {
        double r52610 = N;
        double r52611 = 9187.863337025456;
        bool r52612 = r52610 <= r52611;
        double r52613 = 1.0;
        double r52614 = r52610 + r52613;
        double r52615 = r52614 / r52610;
        double r52616 = log(r52615);
        double r52617 = r52613 / r52610;
        double r52618 = 0.5;
        double r52619 = r52610 * r52610;
        double r52620 = r52618 / r52619;
        double r52621 = r52617 - r52620;
        double r52622 = 0.3333333333333333;
        double r52623 = 3.0;
        double r52624 = pow(r52610, r52623);
        double r52625 = r52622 / r52624;
        double r52626 = r52621 + r52625;
        double r52627 = r52612 ? r52616 : r52626;
        return r52627;
}

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 < 9187.863337025456

    1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied diff-log0.0

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]

    if 9187.863337025456 < N

    1. Initial program 59.5

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied diff-log59.3

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]

    4. Taylor expanded around inf 0.0

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

    5. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9187.863337025455621187575161457061767578:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))