Average Error: 30.0 → 0.0
Time: 2.5s
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 30.0

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

  2. Simplified30.0

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

  4. Applied add-sqr-sqrt_binary6430.3

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

  5. Simplified30.3

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

  6. Simplified0.5

    \[\leadsto \sqrt{\mathsf{log1p}\left(\frac{1}{N}\right)} \cdot \color{blue}{\sqrt{\mathsf{log1p}\left(\frac{1}{N}\right)}} \]
  7. Using strategy rm

  8. Applied rem-square-sqrt_binary640.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2021211 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))