Average Error: 29.1 → 0.1
Time: 9.7s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 6535.778532396666378190275281667709350586:\\ \;\;\;\;\log \left(\frac{\frac{N + 1}{\sqrt{N}}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 6535.778532396666378190275281667709350586:\\
\;\;\;\;\log \left(\frac{\frac{N + 1}{\sqrt{N}}}{\sqrt{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}\\

\end{array}
double f(double N) {
        double r44516 = N;
        double r44517 = 1.0;
        double r44518 = r44516 + r44517;
        double r44519 = log(r44518);
        double r44520 = log(r44516);
        double r44521 = r44519 - r44520;
        return r44521;
}
↓
double f(double N) {
        double r44522 = N;
        double r44523 = 6535.778532396666;
        bool r44524 = r44522 <= r44523;
        double r44525 = 1.0;
        double r44526 = r44522 + r44525;
        double r44527 = sqrt(r44522);
        double r44528 = r44526 / r44527;
        double r44529 = r44528 / r44527;
        double r44530 = log(r44529);
        double r44531 = 0.3333333333333333;
        double r44532 = 3.0;
        double r44533 = pow(r44522, r44532);
        double r44534 = r44531 / r44533;
        double r44535 = 0.5;
        double r44536 = r44535 / r44522;
        double r44537 = r44525 - r44536;
        double r44538 = r44537 / r44522;
        double r44539 = r44534 + r44538;
        double r44540 = r44524 ? r44530 : r44539;
        return r44540;
}

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 < 6535.778532396666

    1. Initial program 0.1

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

    3. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt0.1

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

    6. Applied associate-/r*0.1

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

    if 6535.778532396666 < 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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 6535.778532396666378190275281667709350586:\\ \;\;\;\;\log \left(\frac{\frac{N + 1}{\sqrt{N}}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))