Average Error: 28.9 → 0.1
Time: 17.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8585.191610072276:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{\frac{1}{3}}{N} + \frac{-1}{2}, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8585.191610072276:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r1983168 = N;
        double r1983169 = 1.0;
        double r1983170 = r1983168 + r1983169;
        double r1983171 = log(r1983170);
        double r1983172 = log(r1983168);
        double r1983173 = r1983171 - r1983172;
        return r1983173;
}
↓
double f(double N) {
        double r1983174 = N;
        double r1983175 = 8585.191610072276;
        bool r1983176 = r1983174 <= r1983175;
        double r1983177 = 1.0;
        double r1983178 = r1983177 + r1983174;
        double r1983179 = r1983178 / r1983174;
        double r1983180 = log(r1983179);
        double r1983181 = r1983177 / r1983174;
        double r1983182 = r1983181 / r1983174;
        double r1983183 = 0.3333333333333333;
        double r1983184 = r1983183 / r1983174;
        double r1983185 = -0.5;
        double r1983186 = r1983184 + r1983185;
        double r1983187 = fma(r1983182, r1983186, r1983181);
        double r1983188 = r1983176 ? r1983180 : r1983187;
        return r1983188;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 8585.191610072276

    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 8585.191610072276 < 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. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8585.191610072276:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{\frac{1}{3}}{N} + \frac{-1}{2}, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))