Average Error: 29.6 → 0.1
Time: 25.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 6613.323166361232:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{N \cdot N} + \frac{1}{N}\right) + \frac{\frac{1}{3}}{\left(N \cdot N\right) \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 6613.323166361232:\\
\;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\

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

\end{array}
double f(double N) {
        double r1644493 = N;
        double r1644494 = 1.0;
        double r1644495 = r1644493 + r1644494;
        double r1644496 = log(r1644495);
        double r1644497 = log(r1644493);
        double r1644498 = r1644496 - r1644497;
        return r1644498;
}
↓
double f(double N) {
        double r1644499 = N;
        double r1644500 = 6613.323166361232;
        bool r1644501 = r1644499 <= r1644500;
        double r1644502 = 1.0;
        double r1644503 = r1644502 + r1644499;
        double r1644504 = r1644503 / r1644499;
        double r1644505 = sqrt(r1644504);
        double r1644506 = log(r1644505);
        double r1644507 = r1644506 + r1644506;
        double r1644508 = -0.5;
        double r1644509 = r1644499 * r1644499;
        double r1644510 = r1644508 / r1644509;
        double r1644511 = r1644502 / r1644499;
        double r1644512 = r1644510 + r1644511;
        double r1644513 = 0.3333333333333333;
        double r1644514 = r1644509 * r1644499;
        double r1644515 = r1644513 / r1644514;
        double r1644516 = r1644512 + r1644515;
        double r1644517 = r1644501 ? r1644507 : r1644516;
        return r1644517;
}

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 < 6613.323166361232

    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

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

    if 6613.323166361232 < N

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 6613.323166361232:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-1}{2}}{N \cdot N} + \frac{1}{N}\right) + \frac{\frac{1}{3}}{\left(N \cdot N\right) \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))