Average Error: 29.4 → 0.1
Time: 5.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7664.616617245729685237165540456771850586:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7664.616617245729685237165540456771850586:\\
\;\;\;\;\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 r46133 = N;
        double r46134 = 1.0;
        double r46135 = r46133 + r46134;
        double r46136 = log(r46135);
        double r46137 = log(r46133);
        double r46138 = r46136 - r46137;
        return r46138;
}
↓
double f(double N) {
        double r46139 = N;
        double r46140 = 7664.61661724573;
        bool r46141 = r46139 <= r46140;
        double r46142 = 1.0;
        double r46143 = r46139 + r46142;
        double r46144 = r46143 / r46139;
        double r46145 = log(r46144);
        double r46146 = 1.0;
        double r46147 = 2.0;
        double r46148 = pow(r46139, r46147);
        double r46149 = r46146 / r46148;
        double r46150 = 0.3333333333333333;
        double r46151 = r46150 / r46139;
        double r46152 = 0.5;
        double r46153 = r46151 - r46152;
        double r46154 = r46149 * r46153;
        double r46155 = r46142 / r46139;
        double r46156 = r46154 + r46155;
        double r46157 = r46141 ? r46145 : r46156;
        return r46157;
}

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 < 7664.61661724573

    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 7664.61661724573 < N

    1. Initial program 59.5

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7664.616617245729685237165540456771850586:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))