Average Error: 29.4 → 0.0
Time: 4.7s
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 add-cbrt-cube_binary6429.5

    \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(\frac{1 + N}{N}\right) \cdot \log \left(\frac{1 + N}{N}\right)\right) \cdot \log \left(\frac{1 + N}{N}\right)}} \]

  6. Applied cbrt-prod_binary6429.6

    \[\leadsto \color{blue}{\sqrt[3]{\log \left(\frac{1 + N}{N}\right) \cdot \log \left(\frac{1 + N}{N}\right)} \cdot \sqrt[3]{\log \left(\frac{1 + N}{N}\right)}} \]

  7. Simplified29.6

    \[\leadsto \color{blue}{\sqrt[3]{\mathsf{log1p}\left(\frac{1}{N}\right) \cdot \mathsf{log1p}\left(\frac{1}{N}\right)}} \cdot \sqrt[3]{\log \left(\frac{1 + N}{N}\right)} \]

  8. Simplified14.9

    \[\leadsto \sqrt[3]{\mathsf{log1p}\left(\frac{1}{N}\right) \cdot \mathsf{log1p}\left(\frac{1}{N}\right)} \cdot \color{blue}{\sqrt[3]{\mathsf{log1p}\left(\frac{1}{N}\right)}} \]

  9. Applied log1p-expm1-u_binary6414.9

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\mathsf{log1p}\left(\frac{1}{N}\right) \cdot \mathsf{log1p}\left(\frac{1}{N}\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(\frac{1}{N}\right)}\right)\right)} \]

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2021215 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))