Average Error: 29.4 → 0.0
Time: 2.7s
Precision: binary64
\[\log \left(N + 1\right) - \log N \]
↓
\[\mathsf{log1p}\left(\frac{1}{N}\right) \]
\log \left(N + 1\right) - \log N
↓
\mathsf{log1p}\left(\frac{1}{N}\right)

(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N) :precision binary64 (log1p (/ 1.0 N)))
double code(double N) {
	return log(N + 1.0) - log(N);
}
↓
double code(double N) {
	return log1p(1.0 / N);
}

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. Simplified29.4

    \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]

  3. Applied log1p-udef_binary6429.4

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

  4. Applied diff-log_binary6429.3

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

  5. Applied expm1-log1p-u_binary6429.9

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

  6. Simplified0.6

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

  7. Applied log1p-expm1-u_binary640.6

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2022067 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))