Average Error: 29.5 → 0.1
Time: 8.6s
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 r30514 = N;
        double r30515 = 1.0;
        double r30516 = r30514 + r30515;
        double r30517 = log(r30516);
        double r30518 = log(r30514);
        double r30519 = r30517 - r30518;
        return r30519;
}
↓
double f(double N) {
        double r30520 = N;
        double r30521 = 7622.504121247211;
        bool r30522 = r30520 <= r30521;
        double r30523 = 1.0;
        double r30524 = r30520 + r30523;
        double r30525 = r30524 / r30520;
        double r30526 = log(r30525);
        double r30527 = r30523 / r30520;
        double r30528 = 0.3333333333333333;
        double r30529 = 3.0;
        double r30530 = pow(r30520, r30529);
        double r30531 = r30528 / r30530;
        double r30532 = r30527 + r30531;
        double r30533 = 0.5;
        double r30534 = r30520 * r30520;
        double r30535 = r30533 / r30534;
        double r30536 = r30532 - r30535;
        double r30537 = r30522 ? r30526 : r30536;
        return r30537;
}

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)))