Average Error: 29.3 → 0.1
Time: 2.5s
Precision: binary64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \leq 4754.174426391358:\\ \;\;\;\;0.5 \cdot \log \left(\frac{N + 1}{N}\right) + \left(\log \left(\sqrt{N + 1}\right) - \log \left(\sqrt{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{N}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \leq 4754.174426391358:\\
\;\;\;\;0.5 \cdot \log \left(\frac{N + 1}{N}\right) + \left(\log \left(\sqrt{N + 1}\right) - \log \left(\sqrt{N}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{N}\right) - \frac{0.5}{N \cdot N}\\

\end{array}
(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N)
 :precision binary64
 (if (<= N 4754.174426391358)
   (+ (* 0.5 (log (/ (+ N 1.0) N))) (- (log (sqrt (+ N 1.0))) (log (sqrt N))))
   (- (+ (/ 0.3333333333333333 (pow N 3.0)) (/ 1.0 N)) (/ 0.5 (* N N)))))
double code(double N) {
	return ((double) (((double) log(((double) (N + 1.0)))) - ((double) log(N))));
}
↓
double code(double N) {
	double tmp;
	if ((N <= 4754.174426391358)) {
		tmp = ((double) (((double) (0.5 * ((double) log((((double) (N + 1.0)) / N))))) + ((double) (((double) log(((double) sqrt(((double) (N + 1.0)))))) - ((double) log(((double) sqrt(N))))))));
	} else {
		tmp = ((double) (((double) ((0.3333333333333333 / ((double) pow(N, 3.0))) + (1.0 / N))) - (0.5 / ((double) (N * N)))));
	}
	return tmp;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if N < 4754.17442639135788

    1. Initial program 0.1

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

    3. Applied diff-log_binary640.1

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

    5. Applied add-sqr-sqrt_binary640.1

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

    6. Applied log-prod_binary640.1

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

    8. Applied sqrt-div_binary640.1

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

    9. Applied log-div_binary640.1

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

    11. Applied pow1/2_binary640.1

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

    12. Applied log-pow_binary640.1

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

    if 4754.17442639135788 < N

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.1

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{N}\right) - \frac{0.5}{N \cdot N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 4754.174426391358:\\ \;\;\;\;0.5 \cdot \log \left(\frac{N + 1}{N}\right) + \left(\log \left(\sqrt{N + 1}\right) - \log \left(\sqrt{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{N}\right) - \frac{0.5}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))