Average Error: 30.0 → 0.1
Time: 2.2s
Precision: binary64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 123639.612736954281:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 123639.612736954281:\\
\;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\

\end{array}
double code(double N) {
	return ((double) (((double) log(((double) (N + 1.0)))) - ((double) log(N))));
}
↓
double code(double N) {
	double VAR;
	if ((N <= 123639.61273695428)) {
		VAR = ((double) (((double) log(((double) sqrt(((double) (((double) (N + 1.0)) / N)))))) + ((double) log(((double) sqrt(((double) (((double) (N + 1.0)) / N))))))));
	} else {
		VAR = ((double) (((double) (1.0 - ((double) (0.5 / N)))) / N));
	}
	return VAR;
}

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 < 123639.612736954281

    1. Initial program 0.2

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

    3. Applied diff-log0.2

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

    5. Applied add-sqr-sqrt0.2

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

    6. Applied log-prod0.2

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

    if 123639.612736954281 < N

    1. Initial program 59.7

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

    2. Taylor expanded around -inf 64.0

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

    3. Simplified0.1

      \[\leadsto \color{blue}{\frac{1}{N} \cdot \left(1 - \frac{0.5}{N}\right)}\]
    4. Using strategy rm

    5. Applied associate-*l/0.1

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

    6. Simplified0.1

      \[\leadsto \frac{\color{blue}{1 - \frac{0.5}{N}}}{N}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 123639.612736954281:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))