Average Error: 30.1 → 0.1
Time: 3.6s
Precision: binary64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \leq 8131.4453676694275:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \leq 8131.4453676694275:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

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

\end{array}
(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N)
 :precision binary64
 (if (<= N 8131.4453676694275)
   (log (/ (+ N 1.0) 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 <= 8131.4453676694275)) {
		tmp = ((double) log((((double) (N + 1.0)) / N)));
	} else {
		tmp = ((double) ((0.3333333333333333 / ((double) pow(N, 3.0))) + ((double) ((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 < 8131.44536766942747

    1. Initial program Error: 0.1 bits

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

    3. Applied diff-logError: 0.1 bits

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

    if 8131.44536766942747 < N

    1. Initial program Error: 59.4 bits

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

    2. Taylor expanded around inf Error: 0.0 bits

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

    3. SimplifiedError: 0.0 bits

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

  4. Final simplificationError: 0.1 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 8131.4453676694275:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))