Average Error: 29.2 → 0.1
Time: 6.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 1.113298076 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;\log \left(N + 1\right) - \log N \le 1.113298076 \cdot 10^{-7}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\end{array}
double f(double N) {
        double r67337 = N;
        double r67338 = 1.0;
        double r67339 = r67337 + r67338;
        double r67340 = log(r67339);
        double r67341 = log(r67337);
        double r67342 = r67340 - r67341;
        return r67342;
}
↓
double f(double N) {
        double r67343 = N;
        double r67344 = 1.0;
        double r67345 = r67343 + r67344;
        double r67346 = log(r67345);
        double r67347 = log(r67343);
        double r67348 = r67346 - r67347;
        double r67349 = 1.1132980759498423e-07;
        bool r67350 = r67348 <= r67349;
        double r67351 = 1.0;
        double r67352 = r67351 / r67343;
        double r67353 = 0.5;
        double r67354 = r67353 / r67343;
        double r67355 = r67344 - r67354;
        double r67356 = 0.3333333333333333;
        double r67357 = 3.0;
        double r67358 = pow(r67343, r67357);
        double r67359 = r67356 / r67358;
        double r67360 = fma(r67352, r67355, r67359);
        double r67361 = r67345 / r67343;
        double r67362 = log(r67361);
        double r67363 = r67350 ? r67360 : r67362;
        return r67363;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if (- (log (+ N 1.0)) (log N)) < 1.1132980759498423e-07

    1. Initial program 59.9

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

    3. Applied diff-log59.8

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

    4. Taylor expanded around inf 0.0

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

    5. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)}\]

    if 1.1132980759498423e-07 < (- (log (+ N 1.0)) (log N))

    1. Initial program 0.3

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

    3. Applied diff-log0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 1.113298076 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))