Average Error: 29.8 → 0.1
Time: 3.8s
Precision: binary64
\[\log \left(N + 1\right) - \log N \]
↓
\[\begin{array}{l} \mathbf{if}\;N \leq 6013.484448010823:\\ \;\;\;\;\log \left(\frac{N + 1}{\sqrt{N}}\right) - \log \left(\sqrt{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333}{{N}^{3}}\\ \end{array} \]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \leq 6013.484448010823:\\
\;\;\;\;\log \left(\frac{N + 1}{\sqrt{N}}\right) - \log \left(\sqrt{N}\right)\\

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


\end{array}

(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N)
 :precision binary64
 (if (<= N 6013.484448010823)
   (- (log (/ (+ N 1.0) (sqrt N))) (log (sqrt N)))
   (+ (- (/ 1.0 N) (/ 0.5 (* N N))) (/ 0.3333333333333333 (pow N 3.0)))))
double code(double N) {
	return log(N + 1.0) - log(N);
}
↓
double code(double N) {
	double tmp;
	if (N <= 6013.484448010823) {
		tmp = log((N + 1.0) / sqrt(N)) - log(sqrt(N));
	} else {
		tmp = ((1.0 / N) - (0.5 / (N * N))) + (0.3333333333333333 / pow(N, 3.0));
	}
	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 < 6013.48444801082314

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]

    3. Applied add-sqr-sqrt_binary640.1

      \[\leadsto \mathsf{log1p}\left(N\right) - \log \color{blue}{\left(\sqrt{N} \cdot \sqrt{N}\right)} \]

    4. Applied log-prod_binary640.1

      \[\leadsto \mathsf{log1p}\left(N\right) - \color{blue}{\left(\log \left(\sqrt{N}\right) + \log \left(\sqrt{N}\right)\right)} \]

    5. Applied associate--r+_binary640.1

      \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(N\right) - \log \left(\sqrt{N}\right)\right) - \log \left(\sqrt{N}\right)} \]

    6. Applied log1p-udef_binary640.1

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

    7. Applied diff-log_binary640.1

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

    if 6013.48444801082314 < N

    1. Initial program 59.5

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

    2. Simplified59.5

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]

    3. Taylor expanded in N around inf 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 6013.484448010823:\\ \;\;\;\;\log \left(\frac{N + 1}{\sqrt{N}}\right) - \log \left(\sqrt{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333}{{N}^{3}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))