Average Error: 29.4 → 0.1
Time: 14.1s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7627.9955927630945:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) - \frac{\frac{-1}{3}}{N \cdot \left(N \cdot N\right)}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7627.9955927630945:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r1804109 = N;
        double r1804110 = 1.0;
        double r1804111 = r1804109 + r1804110;
        double r1804112 = log(r1804111);
        double r1804113 = log(r1804109);
        double r1804114 = r1804112 - r1804113;
        return r1804114;
}
↓
double f(double N) {
        double r1804115 = N;
        double r1804116 = 7627.9955927630945;
        bool r1804117 = r1804115 <= r1804116;
        double r1804118 = 1.0;
        double r1804119 = r1804118 + r1804115;
        double r1804120 = r1804119 / r1804115;
        double r1804121 = log(r1804120);
        double r1804122 = r1804118 / r1804115;
        double r1804123 = -0.5;
        double r1804124 = r1804115 * r1804115;
        double r1804125 = r1804123 / r1804124;
        double r1804126 = r1804122 + r1804125;
        double r1804127 = -0.3333333333333333;
        double r1804128 = r1804115 * r1804124;
        double r1804129 = r1804127 / r1804128;
        double r1804130 = r1804126 - r1804129;
        double r1804131 = r1804117 ? r1804121 : r1804130;
        return r1804131;
}

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 < 7627.9955927630945

    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

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

    if 7627.9955927630945 < 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 59.3

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

    8. Applied add-exp-log59.3

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

    9. Taylor expanded around inf 0.0

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

    10. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7627.9955927630945:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) - \frac{\frac{-1}{3}}{N \cdot \left(N \cdot N\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))