Average Error: 29.5 → 0.1
Time: 8.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7622.50412124721061:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7622.50412124721061:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}\\

\end{array}
double f(double N) {
        double r37012 = N;
        double r37013 = 1.0;
        double r37014 = r37012 + r37013;
        double r37015 = log(r37014);
        double r37016 = log(r37012);
        double r37017 = r37015 - r37016;
        return r37017;
}
↓
double f(double N) {
        double r37018 = N;
        double r37019 = 7622.504121247211;
        bool r37020 = r37018 <= r37019;
        double r37021 = 1.0;
        double r37022 = r37018 + r37021;
        double r37023 = r37022 / r37018;
        double r37024 = log(r37023);
        double r37025 = r37021 / r37018;
        double r37026 = 0.3333333333333333;
        double r37027 = 3.0;
        double r37028 = pow(r37018, r37027);
        double r37029 = r37026 / r37028;
        double r37030 = r37025 + r37029;
        double r37031 = 0.5;
        double r37032 = r37018 * r37018;
        double r37033 = r37031 / r37032;
        double r37034 = r37030 - r37033;
        double r37035 = r37020 ? r37024 : r37034;
        return r37035;
}

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 < 7622.504121247211

    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 7622.504121247211 < N

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7622.50412124721061:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))