Average Error: 29.4 → 0.0
Time: 19.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\mathsf{log1p}\left(\frac{1}{N}\right)\]
\log \left(N + 1\right) - \log N
↓
\mathsf{log1p}\left(\frac{1}{N}\right)
double f(double N) {
        double r39327 = N;
        double r39328 = 1.0;
        double r39329 = r39327 + r39328;
        double r39330 = log(r39329);
        double r39331 = log(r39327);
        double r39332 = r39330 - r39331;
        return r39332;
}
↓
double f(double N) {
        double r39333 = 1.0;
        double r39334 = N;
        double r39335 = r39333 / r39334;
        double r39336 = log1p(r39335);
        return r39336;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.4

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

  3. Applied diff-log29.3

    \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity29.3

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

  6. Applied *-un-lft-identity29.3

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

  7. Applied times-frac29.3

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

  8. Applied log-prod29.3

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

  9. Simplified29.3

    \[\leadsto \color{blue}{0} + \log \left(\frac{N + 1}{N}\right)\]

  10. Simplified0.0

    \[\leadsto 0 + \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)}\]

  11. Final simplification0.0

    \[\leadsto \mathsf{log1p}\left(\frac{1}{N}\right)\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))