Average Error: 29.6 → 0.1
Time: 9.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4688.1335077464264:\\ \;\;\;\;\left(-\log \left(\sqrt{N}\right)\right) + \log \left(\frac{N + 1}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.333333333333333315}{{N}^{3}}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4688.1335077464264:\\
\;\;\;\;\left(-\log \left(\sqrt{N}\right)\right) + \log \left(\frac{N + 1}{\sqrt{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.333333333333333315}{{N}^{3}}\\

\end{array}
double f(double N) {
        double r37643 = N;
        double r37644 = 1.0;
        double r37645 = r37643 + r37644;
        double r37646 = log(r37645);
        double r37647 = log(r37643);
        double r37648 = r37646 - r37647;
        return r37648;
}
↓
double f(double N) {
        double r37649 = N;
        double r37650 = 4688.133507746426;
        bool r37651 = r37649 <= r37650;
        double r37652 = sqrt(r37649);
        double r37653 = log(r37652);
        double r37654 = -r37653;
        double r37655 = 1.0;
        double r37656 = r37649 + r37655;
        double r37657 = r37656 / r37652;
        double r37658 = log(r37657);
        double r37659 = r37654 + r37658;
        double r37660 = r37655 / r37649;
        double r37661 = 0.5;
        double r37662 = r37649 * r37649;
        double r37663 = r37661 / r37662;
        double r37664 = r37660 - r37663;
        double r37665 = 0.3333333333333333;
        double r37666 = 3.0;
        double r37667 = pow(r37649, r37666);
        double r37668 = r37665 / r37667;
        double r37669 = r37664 + r37668;
        double r37670 = r37651 ? r37659 : r37669;
        return r37670;
}

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 < 4688.133507746426

    1. Initial program 0.1

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

    3. Applied diff-log0.0

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

    5. Applied add-sqr-sqrt0.0

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

    6. Applied *-un-lft-identity0.0

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

    7. Applied times-frac0.0

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

    8. Applied log-prod0.1

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

    9. Simplified0.1

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

    if 4688.133507746426 < N

    1. Initial program 59.3

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

    3. Applied diff-log59.0

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

    4. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    5. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.333333333333333315}{{N}^{3}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4688.1335077464264:\\ \;\;\;\;\left(-\log \left(\sqrt{N}\right)\right) + \log \left(\frac{N + 1}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.333333333333333315}{{N}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))