Average Error: 29.0 → 0.2
Time: 4.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4600.68399624307585:\\ \;\;\;\;\left(\log \left(\sqrt{N + 1}\right) - \left(\frac{2}{3} \cdot \log N + \log \left(\sqrt[3]{1}\right)\right)\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4600.68399624307585:\\
\;\;\;\;\left(\log \left(\sqrt{N + 1}\right) - \left(\frac{2}{3} \cdot \log N + \log \left(\sqrt[3]{1}\right)\right)\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r54643 = N;
        double r54644 = 1.0;
        double r54645 = r54643 + r54644;
        double r54646 = log(r54645);
        double r54647 = log(r54643);
        double r54648 = r54646 - r54647;
        return r54648;
}
↓
double f(double N) {
        double r54649 = N;
        double r54650 = 4600.683996243076;
        bool r54651 = r54649 <= r54650;
        double r54652 = 1.0;
        double r54653 = r54649 + r54652;
        double r54654 = sqrt(r54653);
        double r54655 = log(r54654);
        double r54656 = 0.6666666666666666;
        double r54657 = log(r54649);
        double r54658 = r54656 * r54657;
        double r54659 = 1.0;
        double r54660 = cbrt(r54659);
        double r54661 = log(r54660);
        double r54662 = r54658 + r54661;
        double r54663 = r54655 - r54662;
        double r54664 = cbrt(r54649);
        double r54665 = r54654 / r54664;
        double r54666 = log(r54665);
        double r54667 = r54663 + r54666;
        double r54668 = 2.0;
        double r54669 = pow(r54649, r54668);
        double r54670 = r54659 / r54669;
        double r54671 = 0.3333333333333333;
        double r54672 = r54671 / r54649;
        double r54673 = 0.5;
        double r54674 = r54672 - r54673;
        double r54675 = r54670 * r54674;
        double r54676 = r54652 / r54649;
        double r54677 = r54675 + r54676;
        double r54678 = r54651 ? r54667 : r54677;
        return r54678;
}

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 < 4600.683996243076

    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-cube-cbrt0.2

      \[\leadsto \log \left(\frac{N + 1}{\color{blue}{\left(\sqrt[3]{N} \cdot \sqrt[3]{N}\right) \cdot \sqrt[3]{N}}}\right)\]

    6. Applied add-sqr-sqrt0.2

      \[\leadsto \log \left(\frac{\color{blue}{\sqrt{N + 1} \cdot \sqrt{N + 1}}}{\left(\sqrt[3]{N} \cdot \sqrt[3]{N}\right) \cdot \sqrt[3]{N}}\right)\]

    7. Applied times-frac0.2

      \[\leadsto \log \color{blue}{\left(\frac{\sqrt{N + 1}}{\sqrt[3]{N} \cdot \sqrt[3]{N}} \cdot \frac{\sqrt{N + 1}}{\sqrt[3]{N}}\right)}\]

    8. Applied log-prod0.4

      \[\leadsto \color{blue}{\log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N} \cdot \sqrt[3]{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N}}\right)}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity0.4

      \[\leadsto \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N} \cdot \sqrt[3]{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{\color{blue}{1 \cdot N}}}\right)\]

    11. Applied cbrt-prod0.4

      \[\leadsto \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N} \cdot \sqrt[3]{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{N}}}\right)\]

    12. Applied *-un-lft-identity0.4

      \[\leadsto \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N} \cdot \sqrt[3]{N}}\right) + \log \left(\frac{\sqrt{\color{blue}{1 \cdot \left(N + 1\right)}}}{\sqrt[3]{1} \cdot \sqrt[3]{N}}\right)\]

    13. Applied sqrt-prod0.4

      \[\leadsto \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N} \cdot \sqrt[3]{N}}\right) + \log \left(\frac{\color{blue}{\sqrt{1} \cdot \sqrt{N + 1}}}{\sqrt[3]{1} \cdot \sqrt[3]{N}}\right)\]

    14. Applied times-frac0.4

      \[\leadsto \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N} \cdot \sqrt[3]{N}}\right) + \log \color{blue}{\left(\frac{\sqrt{1}}{\sqrt[3]{1}} \cdot \frac{\sqrt{N + 1}}{\sqrt[3]{N}}\right)}\]

    15. Applied log-prod0.4

      \[\leadsto \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N} \cdot \sqrt[3]{N}}\right) + \color{blue}{\left(\log \left(\frac{\sqrt{1}}{\sqrt[3]{1}}\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N}}\right)\right)}\]

    16. Applied associate-+r+0.4

      \[\leadsto \color{blue}{\left(\log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N} \cdot \sqrt[3]{N}}\right) + \log \left(\frac{\sqrt{1}}{\sqrt[3]{1}}\right)\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N}}\right)}\]

    17. Simplified0.3

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

    if 4600.683996243076 < N

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4600.68399624307585:\\ \;\;\;\;\left(\log \left(\sqrt{N + 1}\right) - \left(\frac{2}{3} \cdot \log N + \log \left(\sqrt[3]{1}\right)\right)\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt[3]{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))