Average Error: 29.5 → 0.1
Time: 2.5s
Precision: binary64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \leq 292431.2368045783:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \leq 292431.2368045783:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{N} - \frac{0.5}{N \cdot 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 <= 292431.2368045783)) {
		VAR = ((double) log((((double) (N + 1.0)) / N)));
	} else {
		VAR = ((double) ((1.0 / N) - (0.5 / ((double) (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 < 292431.236804578279

    1. Initial program Error: 0.2 bits

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

    3. Applied diff-logError: 0.2 bits

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

    if 292431.236804578279 < N

    1. Initial program Error: 59.7 bits

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

    2. Taylor expanded around -inf Error: 64.0 bits

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

    3. SimplifiedError: 0.1 bits

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

  4. Final simplificationError: 0.1 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 292431.2368045783:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))