Average Error: 29.1 → 0.1
Time: 3.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 10132.834563235498:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 10132.834563235498:\\
\;\;\;\;\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 r44374 = N;
        double r44375 = 1.0;
        double r44376 = r44374 + r44375;
        double r44377 = log(r44376);
        double r44378 = log(r44374);
        double r44379 = r44377 - r44378;
        return r44379;
}
↓
double f(double N) {
        double r44380 = N;
        double r44381 = 10132.834563235498;
        bool r44382 = r44380 <= r44381;
        double r44383 = 1.0;
        double r44384 = r44380 + r44383;
        double r44385 = r44384 / r44380;
        double r44386 = log(r44385);
        double r44387 = 1.0;
        double r44388 = 2.0;
        double r44389 = pow(r44380, r44388);
        double r44390 = r44387 / r44389;
        double r44391 = 0.3333333333333333;
        double r44392 = r44391 / r44380;
        double r44393 = 0.5;
        double r44394 = r44392 - r44393;
        double r44395 = r44383 / r44380;
        double r44396 = fma(r44390, r44394, r44395);
        double r44397 = r44382 ? r44386 : r44396;
        return r44397;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 10132.834563235498

    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 10132.834563235498 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 10132.834563235498:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))