Average Error: 29.3 → 0.0
Time: 3.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8891.4495036160115:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{\frac{0.5}{N}}{N} + \left(\frac{-0.333333333333333315}{{N}^{3}} - \frac{1}{N}\right)\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8891.4495036160115:\\
\;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\

\mathbf{else}:\\
\;\;\;\;-\left(\frac{\frac{0.5}{N}}{N} + \left(\frac{-0.333333333333333315}{{N}^{3}} - \frac{1}{N}\right)\right)\\

\end{array}
double code(double N) {
	return (log((N + 1.0)) - log(N));
}
↓
double code(double N) {
	double VAR;
	if ((N <= 8891.449503616011)) {
		VAR = -log((N / (N + 1.0)));
	} else {
		VAR = -(((0.5 / N) / N) + ((-0.3333333333333333 / pow(N, 3.0)) - (1.0 / 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 < 8891.449503616011

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    5. Applied clear-num0.1

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

    6. Applied log-rec0.1

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

    if 8891.449503616011 < N

    1. Initial program 59.5

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

    3. Applied diff-log59.3

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

    5. Applied clear-num59.3

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

    6. Applied log-rec59.2

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

    7. Taylor expanded around inf 0.0

      \[\leadsto -\color{blue}{\left(0.5 \cdot \frac{1}{{N}^{2}} - \left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right)\right)}\]

    8. Simplified0.0

      \[\leadsto -\color{blue}{\left(\frac{\frac{0.5}{N}}{N} + \left(\frac{-0.333333333333333315}{{N}^{3}} - \frac{1}{N}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8891.4495036160115:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{\frac{0.5}{N}}{N} + \left(\frac{-0.333333333333333315}{{N}^{3}} - \frac{1}{N}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))