Average Error: 29.7 → 0.1
Time: 16.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7072.329313818584:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) - \frac{\frac{-1}{3}}{N \cdot \left(N \cdot N\right)}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7072.329313818584:\\
\;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\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 r2836586 = N;
        double r2836587 = 1.0;
        double r2836588 = r2836586 + r2836587;
        double r2836589 = log(r2836588);
        double r2836590 = log(r2836586);
        double r2836591 = r2836589 - r2836590;
        return r2836591;
}
↓
double f(double N) {
        double r2836592 = N;
        double r2836593 = 7072.329313818584;
        bool r2836594 = r2836592 <= r2836593;
        double r2836595 = 1.0;
        double r2836596 = r2836595 + r2836592;
        double r2836597 = r2836596 / r2836592;
        double r2836598 = sqrt(r2836597);
        double r2836599 = log(r2836598);
        double r2836600 = r2836599 + r2836599;
        double r2836601 = r2836595 / r2836592;
        double r2836602 = -0.5;
        double r2836603 = r2836592 * r2836592;
        double r2836604 = r2836602 / r2836603;
        double r2836605 = r2836601 + r2836604;
        double r2836606 = -0.3333333333333333;
        double r2836607 = r2836592 * r2836603;
        double r2836608 = r2836606 / r2836607;
        double r2836609 = r2836605 - r2836608;
        double r2836610 = r2836594 ? r2836600 : r2836609;
        return r2836610;
}

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 < 7072.329313818584

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]
    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

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

    8. Applied log-prod0.1

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

    if 7072.329313818584 < N

    1. Initial program 59.6

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

    2. Simplified59.6

      \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]
    3. Using strategy rm

    4. Applied log1p-udef59.6

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

    5. Applied diff-log59.3

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

    7. Applied add-exp-log60.5

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

    8. Applied add-exp-log59.6

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

    9. Applied div-exp59.6

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

    10. Applied rem-log-exp59.6

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

    11. Simplified59.6

      \[\leadsto \color{blue}{\log_* (1 + N)} - \log N\]

    12. Taylor expanded around inf 0.0

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

    13. 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7072.329313818584:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) - \frac{\frac{-1}{3}}{N \cdot \left(N \cdot N\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))