Average Error: 29.3 → 0.1
Time: 12.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8810.738870329447:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{3}, \frac{\frac{1}{N}}{N \cdot N}, \frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8810.738870329447:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r996675 = N;
        double r996676 = 1.0;
        double r996677 = r996675 + r996676;
        double r996678 = log(r996677);
        double r996679 = log(r996675);
        double r996680 = r996678 - r996679;
        return r996680;
}
↓
double f(double N) {
        double r996681 = N;
        double r996682 = 8810.738870329447;
        bool r996683 = r996681 <= r996682;
        double r996684 = 1.0;
        double r996685 = r996684 + r996681;
        double r996686 = r996685 / r996681;
        double r996687 = log(r996686);
        double r996688 = 0.3333333333333333;
        double r996689 = r996684 / r996681;
        double r996690 = r996681 * r996681;
        double r996691 = r996689 / r996690;
        double r996692 = 0.5;
        double r996693 = r996692 / r996690;
        double r996694 = r996689 - r996693;
        double r996695 = fma(r996688, r996691, r996694);
        double r996696 = r996683 ? r996687 : r996695;
        return r996696;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 8810.738870329447

    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 8810.738870329447 < N

    1. Initial program 59.6

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

    2. Simplified59.6

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]
    3. Using strategy rm

    4. Applied log1p-udef59.6

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

    5. Applied diff-log59.4

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8810.738870329447:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{3}, \frac{\frac{1}{N}}{N \cdot N}, \frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))