Average Error: 29.2 → 0.1
Time: 10.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8226.710483527565884287469089031219482422:\\ \;\;\;\;\left(\log \left(\sqrt{N + 1}\right) + \log \left(\frac{\sqrt{1}}{\sqrt{N}}\right)\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{{N}^{2}} + \left(1 - \frac{0.5}{N}\right)\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8226.710483527565884287469089031219482422:\\
\;\;\;\;\left(\log \left(\sqrt{N + 1}\right) + \log \left(\frac{\sqrt{1}}{\sqrt{N}}\right)\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{{N}^{2}} + \left(1 - \frac{0.5}{N}\right)\right)\\

\end{array}
double f(double N) {
        double r59121 = N;
        double r59122 = 1.0;
        double r59123 = r59121 + r59122;
        double r59124 = log(r59123);
        double r59125 = log(r59121);
        double r59126 = r59124 - r59125;
        return r59126;
}
↓
double f(double N) {
        double r59127 = N;
        double r59128 = 8226.710483527566;
        bool r59129 = r59127 <= r59128;
        double r59130 = 1.0;
        double r59131 = r59127 + r59130;
        double r59132 = sqrt(r59131);
        double r59133 = log(r59132);
        double r59134 = 1.0;
        double r59135 = sqrt(r59134);
        double r59136 = sqrt(r59127);
        double r59137 = r59135 / r59136;
        double r59138 = log(r59137);
        double r59139 = r59133 + r59138;
        double r59140 = r59132 / r59136;
        double r59141 = log(r59140);
        double r59142 = r59139 + r59141;
        double r59143 = r59134 / r59127;
        double r59144 = 0.3333333333333333;
        double r59145 = 2.0;
        double r59146 = pow(r59127, r59145);
        double r59147 = r59144 / r59146;
        double r59148 = 0.5;
        double r59149 = r59148 / r59127;
        double r59150 = r59130 - r59149;
        double r59151 = r59147 + r59150;
        double r59152 = r59143 * r59151;
        double r59153 = r59129 ? r59142 : r59152;
        return r59153;
}

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 < 8226.710483527566

    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 *-un-lft-identity0.1

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

    6. Applied add-sqr-sqrt0.1

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

    7. Applied times-frac0.1

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

    8. Applied log-prod0.1

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

    9. Simplified0.1

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

    11. Applied add-sqr-sqrt0.1

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

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

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

    13. Applied sqrt-prod0.1

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

    14. Applied times-frac0.1

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

    15. Applied log-prod0.1

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

    16. Applied associate-+r+0.1

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

    if 8226.710483527566 < 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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8226.710483527565884287469089031219482422:\\ \;\;\;\;\left(\log \left(\sqrt{N + 1}\right) + \log \left(\frac{\sqrt{1}}{\sqrt{N}}\right)\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{{N}^{2}} + \left(1 - \frac{0.5}{N}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))