Average Error: 29.6 → 0.1
Time: 28.3s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7771.461980130583469872362911701202392578:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{0.3333333333333333148296162562473909929395}{N \cdot N}}{N}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7771.461980130583469872362911701202392578:\\
\;\;\;\;\log \left(1 + \frac{1}{N}\right)\\

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

\end{array}
double f(double N) {
        double r3073600 = N;
        double r3073601 = 1.0;
        double r3073602 = r3073600 + r3073601;
        double r3073603 = log(r3073602);
        double r3073604 = log(r3073600);
        double r3073605 = r3073603 - r3073604;
        return r3073605;
}
↓
double f(double N) {
        double r3073606 = N;
        double r3073607 = 7771.4619801305835;
        bool r3073608 = r3073606 <= r3073607;
        double r3073609 = 1.0;
        double r3073610 = 1.0;
        double r3073611 = r3073610 / r3073606;
        double r3073612 = r3073609 + r3073611;
        double r3073613 = log(r3073612);
        double r3073614 = 0.3333333333333333;
        double r3073615 = r3073606 * r3073606;
        double r3073616 = r3073614 / r3073615;
        double r3073617 = r3073616 / r3073606;
        double r3073618 = r3073611 + r3073617;
        double r3073619 = 0.5;
        double r3073620 = r3073619 / r3073615;
        double r3073621 = r3073618 - r3073620;
        double r3073622 = r3073608 ? r3073613 : r3073621;
        return r3073622;
}

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 < 7771.4619801305835

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Simplified0.1

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

    5. Taylor expanded around 0 0.1

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

    6. Simplified0.1

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

    if 7771.4619801305835 < N

    1. Initial program 59.5

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

    3. Applied add-log-exp59.5

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

    4. Simplified59.3

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

    5. Taylor expanded around 0 59.3

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

    6. Simplified59.3

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

    7. Taylor expanded around inf 0.0

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

    8. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7771.461980130583469872362911701202392578:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{0.3333333333333333148296162562473909929395}{N \cdot N}}{N}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))