Average Error: 29.4 → 0.1
Time: 5.1s
Precision: binary64
\[\log \left(N + 1\right) - \log N \]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2.818761394962621 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333}{{N}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(N\right) - \log N\\ \end{array} \]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2.818761394962621 \cdot 10^{-6}:\\
\;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333}{{N}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{log1p}\left(N\right) - \log N\\


\end{array}

(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 2.818761394962621e-6)
   (+ (- (/ 1.0 N) (/ 0.5 (* N N))) (/ 0.3333333333333333 (pow N 3.0)))
   (- (log1p N) (log 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)) <= 2.818761394962621e-6) {
		tmp = ((1.0 / N) - (0.5 / (N * N))) + (0.3333333333333333 / pow(N, 3.0));
	} else {
		tmp = log1p(N) - log(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)) < 2.818761395e-6

    1. Initial program 59.7

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

    2. Simplified59.7

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

    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

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

    if 2.818761395e-6 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

    1. Initial program 0.2

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

    2. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2.818761394962621 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.3333333333333333}{{N}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(N\right) - \log N\\ \end{array} \]

Reproduce

herbie shell --seed 2021224 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))