Average Error: 29.6 → 0.0
Time: 11.7s
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 r1658007 = N;
        double r1658008 = 1.0;
        double r1658009 = r1658007 + r1658008;
        double r1658010 = log(r1658009);
        double r1658011 = log(r1658007);
        double r1658012 = r1658010 - r1658011;
        return r1658012;
}
↓
double f(double N) {
        double r1658013 = 1.0;
        double r1658014 = N;
        double r1658015 = r1658013 / r1658014;
        double r1658016 = log1p(r1658015);
        return r1658016;
}

Error

Bits error versus N

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.6

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

  2. Simplified29.6

    \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]
  3. Using strategy rm

  4. Applied log1p-udef29.6

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

  5. Applied diff-log29.5

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

  6. Taylor expanded around 0 29.5

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

  8. Applied log1p-expm1-u29.5

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{N} + 1\right)\right)\right)}\]

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))