Average Error: 28.7 → 0.1
Time: 47.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9098.449974321133:\\ \;\;\;\;\log \left(\frac{1}{\sqrt{N}}\right) + \log \left(\frac{1 + N}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} - \frac{\frac{1}{2}}{N \cdot N}\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9098.449974321133:\\
\;\;\;\;\log \left(\frac{1}{\sqrt{N}}\right) + \log \left(\frac{1 + N}{\sqrt{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} - \frac{\frac{1}{2}}{N \cdot N}\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r4580997 = N;
        double r4580998 = 1.0;
        double r4580999 = r4580997 + r4580998;
        double r4581000 = log(r4580999);
        double r4581001 = log(r4580997);
        double r4581002 = r4581000 - r4581001;
        return r4581002;
}
↓
double f(double N) {
        double r4581003 = N;
        double r4581004 = 9098.449974321133;
        bool r4581005 = r4581003 <= r4581004;
        double r4581006 = 1.0;
        double r4581007 = sqrt(r4581003);
        double r4581008 = r4581006 / r4581007;
        double r4581009 = log(r4581008);
        double r4581010 = r4581006 + r4581003;
        double r4581011 = r4581010 / r4581007;
        double r4581012 = log(r4581011);
        double r4581013 = r4581009 + r4581012;
        double r4581014 = 0.3333333333333333;
        double r4581015 = r4581014 / r4581003;
        double r4581016 = r4581003 * r4581003;
        double r4581017 = r4581015 / r4581016;
        double r4581018 = 0.5;
        double r4581019 = r4581018 / r4581016;
        double r4581020 = r4581017 - r4581019;
        double r4581021 = r4581006 / r4581003;
        double r4581022 = r4581020 + r4581021;
        double r4581023 = r4581005 ? r4581013 : r4581022;
        return r4581023;
}

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 < 9098.449974321133

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]
    3. Using strategy rm

    4. Applied log1p-udef0.1

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

    5. Applied diff-log0.1

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

    7. Applied add-sqr-sqrt0.1

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

    8. Applied *-un-lft-identity0.1

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

    9. Applied *-un-lft-identity0.1

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

    10. Applied distribute-lft-out0.1

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

    11. Applied times-frac0.1

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

    12. Applied log-prod0.1

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

    if 9098.449974321133 < N

    1. Initial program 59.5

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

    2. Simplified59.5

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]
    3. Using strategy rm

    4. Applied log1p-udef59.5

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

    5. Applied diff-log59.3

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

    6. Taylor expanded around -inf 0.0

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

    7. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9098.449974321133:\\ \;\;\;\;\log \left(\frac{1}{\sqrt{N}}\right) + \log \left(\frac{1 + N}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} - \frac{\frac{1}{2}}{N \cdot N}\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))