Average Error: 28.9 → 28.8
Time: 3.7s
Precision: binary64
\[\log \left(N + 1\right) - \log N\]
↓
\[\log \left(\sqrt{\frac{N + 1}{N}}\right) + 0.5 \cdot \log \left(\frac{N + 1}{N}\right)\]
\log \left(N + 1\right) - \log N
↓
\log \left(\sqrt{\frac{N + 1}{N}}\right) + 0.5 \cdot \log \left(\frac{N + 1}{N}\right)
(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N)
 :precision binary64
 (+ (log (sqrt (/ (+ N 1.0) N))) (* 0.5 (log (/ (+ N 1.0) N)))))
double code(double N) {
	return log(N + 1.0) - log(N);
}
↓
double code(double N) {
	return log(sqrt((N + 1.0) / N)) + (0.5 * log((N + 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 28.9

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

  3. Applied diff-log_binary64_51128.8

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

  5. Applied add-sqr-sqrt_binary64_44128.9

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

  6. Applied log-prod_binary64_50528.9

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

  8. Applied pow1/2_binary64_49928.9

    \[\leadsto \log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \color{blue}{\left({\left(\frac{N + 1}{N}\right)}^{0.5}\right)}\]

  9. Applied log-pow_binary64_50828.8

    \[\leadsto \log \left(\sqrt{\frac{N + 1}{N}}\right) + \color{blue}{0.5 \cdot \log \left(\frac{N + 1}{N}\right)}\]

  10. Final simplification28.8

    \[\leadsto \log \left(\sqrt{\frac{N + 1}{N}}\right) + 0.5 \cdot \log \left(\frac{N + 1}{N}\right)\]

Reproduce

herbie shell --seed 2020344 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))