Average Error: 29.6 → 0.1
Time: 14.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7998.135430633447867876384407281875610352:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7998.135430633447867876384407281875610352:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\

\end{array}
double f(double N) {
        double r45550 = N;
        double r45551 = 1.0;
        double r45552 = r45550 + r45551;
        double r45553 = log(r45552);
        double r45554 = log(r45550);
        double r45555 = r45553 - r45554;
        return r45555;
}
↓
double f(double N) {
        double r45556 = N;
        double r45557 = 7998.135430633448;
        bool r45558 = r45556 <= r45557;
        double r45559 = 1.0;
        double r45560 = r45556 + r45559;
        double r45561 = r45560 / r45556;
        double r45562 = log(r45561);
        double r45563 = r45559 / r45556;
        double r45564 = 0.5;
        double r45565 = r45556 * r45556;
        double r45566 = r45564 / r45565;
        double r45567 = r45563 - r45566;
        double r45568 = 0.3333333333333333;
        double r45569 = 3.0;
        double r45570 = pow(r45556, r45569);
        double r45571 = r45568 / r45570;
        double r45572 = r45567 + r45571;
        double r45573 = r45558 ? r45562 : r45572;
        return r45573;
}

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 < 7998.135430633448

    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 7998.135430633448 < N

    1. Initial program 59.4

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7998.135430633447867876384407281875610352:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))