Average Error: 29.0 → 0.0
Time: 14.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8323.84667700594764:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.333333333333333315}{{N}^{3}} - \frac{0.5}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8323.84667700594764:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{N} + \left(\frac{0.333333333333333315}{{N}^{3}} - \frac{0.5}{N \cdot N}\right)\\

\end{array}
double f(double N) {
        double r24340 = N;
        double r24341 = 1.0;
        double r24342 = r24340 + r24341;
        double r24343 = log(r24342);
        double r24344 = log(r24340);
        double r24345 = r24343 - r24344;
        return r24345;
}
↓
double f(double N) {
        double r24346 = N;
        double r24347 = 8323.846677005948;
        bool r24348 = r24346 <= r24347;
        double r24349 = 1.0;
        double r24350 = r24346 + r24349;
        double r24351 = r24350 / r24346;
        double r24352 = log(r24351);
        double r24353 = r24349 / r24346;
        double r24354 = 0.3333333333333333;
        double r24355 = 3.0;
        double r24356 = pow(r24346, r24355);
        double r24357 = r24354 / r24356;
        double r24358 = 0.5;
        double r24359 = r24346 * r24346;
        double r24360 = r24358 / r24359;
        double r24361 = r24357 - r24360;
        double r24362 = r24353 + r24361;
        double r24363 = r24348 ? r24352 : r24362;
        return r24363;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if N < 8323.846677005948

    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 8323.846677005948 < 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

      \[\leadsto \color{blue}{\left(\frac{1}{N} + \frac{0.333333333333333315}{{N}^{3}}\right) - \frac{0.5}{N \cdot N}}\]
    4. Using strategy rm

    5. Applied associate--l+0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8323.84667700594764:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.333333333333333315}{{N}^{3}} - \frac{0.5}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))