Average Error: 29.6 → 0.1
Time: 3.3s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 13931.9619516847724:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 13931.9619516847724:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\

\end{array}
double f(double N) {
        double r45292 = N;
        double r45293 = 1.0;
        double r45294 = r45292 + r45293;
        double r45295 = log(r45294);
        double r45296 = log(r45292);
        double r45297 = r45295 - r45296;
        return r45297;
}
↓
double f(double N) {
        double r45298 = N;
        double r45299 = 13931.961951684772;
        bool r45300 = r45298 <= r45299;
        double r45301 = 1.0;
        double r45302 = r45298 + r45301;
        double r45303 = r45302 / r45298;
        double r45304 = log(r45303);
        double r45305 = 1.0;
        double r45306 = 2.0;
        double r45307 = pow(r45298, r45306);
        double r45308 = r45305 / r45307;
        double r45309 = 0.3333333333333333;
        double r45310 = r45309 / r45298;
        double r45311 = 0.5;
        double r45312 = r45310 - r45311;
        double r45313 = r45301 / r45298;
        double r45314 = fma(r45308, r45312, r45313);
        double r45315 = r45300 ? r45304 : r45314;
        return r45315;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 13931.961951684772

    1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied diff-log0.1

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

    if 13931.961951684772 < N

    1. Initial program 59.5

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 13931.9619516847724:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))