Average Error: 29.3 → 0.0
Time: 15.0s
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 r47039 = N;
        double r47040 = 1.0;
        double r47041 = r47039 + r47040;
        double r47042 = log(r47041);
        double r47043 = log(r47039);
        double r47044 = r47042 - r47043;
        return r47044;
}
↓
double f(double N) {
        double r47045 = N;
        double r47046 = 9187.863337025456;
        bool r47047 = r47045 <= r47046;
        double r47048 = 1.0;
        double r47049 = r47045 + r47048;
        double r47050 = r47049 / r47045;
        double r47051 = log(r47050);
        double r47052 = r47048 / r47045;
        double r47053 = 0.5;
        double r47054 = r47045 * r47045;
        double r47055 = r47053 / r47054;
        double r47056 = r47052 - r47055;
        double r47057 = 0.3333333333333333;
        double r47058 = 3.0;
        double r47059 = pow(r47045, r47058);
        double r47060 = r47057 / r47059;
        double r47061 = r47056 + r47060;
        double r47062 = r47047 ? r47051 : r47061;
        return r47062;
}

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