Average Error: 29.1 → 0.1
Time: 5.3s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8430.538053514888815698213875293731689453:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{N + 1} \cdot \sqrt{\frac{1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8430.538053514888815698213875293731689453:\\
\;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{N + 1} \cdot \sqrt{\frac{1}{N}}\right)\\

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

\end{array}
double f(double N) {
        double r57041 = N;
        double r57042 = 1.0;
        double r57043 = r57041 + r57042;
        double r57044 = log(r57043);
        double r57045 = log(r57041);
        double r57046 = r57044 - r57045;
        return r57046;
}
↓
double f(double N) {
        double r57047 = N;
        double r57048 = 8430.538053514889;
        bool r57049 = r57047 <= r57048;
        double r57050 = 1.0;
        double r57051 = r57047 + r57050;
        double r57052 = r57051 / r57047;
        double r57053 = sqrt(r57052);
        double r57054 = log(r57053);
        double r57055 = sqrt(r57051);
        double r57056 = 1.0;
        double r57057 = r57056 / r57047;
        double r57058 = sqrt(r57057);
        double r57059 = r57055 * r57058;
        double r57060 = log(r57059);
        double r57061 = r57054 + r57060;
        double r57062 = 2.0;
        double r57063 = pow(r57047, r57062);
        double r57064 = r57056 / r57063;
        double r57065 = 0.3333333333333333;
        double r57066 = r57065 / r57047;
        double r57067 = 0.5;
        double r57068 = r57066 - r57067;
        double r57069 = r57064 * r57068;
        double r57070 = r57050 / r57047;
        double r57071 = r57069 + r57070;
        double r57072 = r57049 ? r57061 : r57071;
        return r57072;
}

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 < 8430.538053514889

    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

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

    6. Applied log-prod0.1

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

    8. Applied div-inv0.1

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

    9. Applied sqrt-prod0.1

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

    if 8430.538053514889 < 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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8430.538053514888815698213875293731689453:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{N + 1} \cdot \sqrt{\frac{1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))