Average Error: 29.3 → 0.1
Time: 4.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8816.358696954577681026421487331390380859:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8816.358696954577681026421487331390380859:\\
\;\;\;\;\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 r71175 = N;
        double r71176 = 1.0;
        double r71177 = r71175 + r71176;
        double r71178 = log(r71177);
        double r71179 = log(r71175);
        double r71180 = r71178 - r71179;
        return r71180;
}
↓
double f(double N) {
        double r71181 = N;
        double r71182 = 8816.358696954578;
        bool r71183 = r71181 <= r71182;
        double r71184 = 1.0;
        double r71185 = r71181 + r71184;
        double r71186 = r71185 / r71181;
        double r71187 = log(r71186);
        double r71188 = 1.0;
        double r71189 = 2.0;
        double r71190 = pow(r71181, r71189);
        double r71191 = r71188 / r71190;
        double r71192 = 0.3333333333333333;
        double r71193 = r71192 / r71181;
        double r71194 = 0.5;
        double r71195 = r71193 - r71194;
        double r71196 = r71191 * r71195;
        double r71197 = r71184 / r71181;
        double r71198 = r71196 + r71197;
        double r71199 = r71183 ? r71187 : r71198;
        return r71199;
}

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 < 8816.358696954578

    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 8816.358696954578 < N

    1. Initial program 59.7

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8816.358696954577681026421487331390380859:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))