Average Error: 29.3 → 0.1
Time: 3.7s
Precision: binary64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 6.966541121755654 \cdot 10^{-08}:\\ \;\;\;\;\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;\log \left(N + 1\right) - \log N \leq 6.966541121755654 \cdot 10^{-08}:\\
\;\;\;\;\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\

\end{array}
(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 6.966541121755654e-08)
   (-
    (+ (* 0.3333333333333333 (/ 1.0 (pow N 3.0))) (/ 1.0 N))
    (* 0.5 (/ 1.0 (pow N 2.0))))
   (+ (log (sqrt (/ (+ N 1.0) N))) (log (sqrt (/ (+ N 1.0) N))))))
double code(double N) {
	return log(N + 1.0) - log(N);
}
↓
double code(double N) {
	double tmp;
	if ((log(N + 1.0) - log(N)) <= 6.966541121755654e-08) {
		tmp = ((0.3333333333333333 * (1.0 / pow(N, 3.0))) + (1.0 / N)) - (0.5 * (1.0 / pow(N, 2.0)));
	} else {
		tmp = log(sqrt((N + 1.0) / N)) + log(sqrt((N + 1.0) / 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 (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 6.96654112e-8

    1. Initial program 59.9

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

    2. Taylor expanded around inf 0.0

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

    if 6.96654112e-8 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

    1. Initial program 0.3

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

    3. Applied diff-log_binary64_1700.2

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

    5. Applied add-sqr-sqrt_binary64_1000.2

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

    6. Applied log-prod_binary64_1640.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 6.966541121755654 \cdot 10^{-08}:\\ \;\;\;\;\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021024 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))