Average Error: 29.5 → 0.0
Time: 3.6s
Precision: binary64
\[\log \left(N + 1\right) - \log N \]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 8.387371250329068 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(N\right) - \log N\\ \end{array} \]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;\log \left(N + 1\right) - \log N \leq 8.387371250329068 \cdot 10^{-5}:\\
\;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\

\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)) 8.387371250329068e-5)
   (-
    (+ (/ 1.0 N) (/ 0.3333333333333333 (pow N 3.0)))
    (+ (/ 0.5 (* N N)) (/ 0.25 (pow N 4.0))))
   (- (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)) <= 8.387371250329068e-5) {
		tmp = ((1.0 / N) + (0.3333333333333333 / pow(N, 3.0))) - ((0.5 / (N * N)) + (0.25 / pow(N, 4.0)));
	} 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)) < 8.38737125033e-5

    1. Initial program 59.6

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

    2. Simplified59.6

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

    3. Taylor expanded in N around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]

    4. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)} \]

    if 8.38737125033e-5 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

    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. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 8.387371250329068 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(N\right) - \log N\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))