Average Error: 29.7 → 0.1
Time: 4.2s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9843.297794559382964507676661014556884766:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9843.297794559382964507676661014556884766:\\
\;\;\;\;\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 r50479 = N;
        double r50480 = 1.0;
        double r50481 = r50479 + r50480;
        double r50482 = log(r50481);
        double r50483 = log(r50479);
        double r50484 = r50482 - r50483;
        return r50484;
}
↓
double f(double N) {
        double r50485 = N;
        double r50486 = 9843.297794559383;
        bool r50487 = r50485 <= r50486;
        double r50488 = 1.0;
        double r50489 = r50485 + r50488;
        double r50490 = r50489 / r50485;
        double r50491 = log(r50490);
        double r50492 = 1.0;
        double r50493 = 2.0;
        double r50494 = pow(r50485, r50493);
        double r50495 = r50492 / r50494;
        double r50496 = 0.3333333333333333;
        double r50497 = r50496 / r50485;
        double r50498 = 0.5;
        double r50499 = r50497 - r50498;
        double r50500 = r50495 * r50499;
        double r50501 = r50488 / r50485;
        double r50502 = r50500 + r50501;
        double r50503 = r50487 ? r50491 : r50502;
        return r50503;
}

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 < 9843.297794559383

    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 9843.297794559383 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9843.297794559382964507676661014556884766:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))