Average Error: 29.6 → 0.2
Time: 3.5s
Precision: binary64
\[\log \left(N + 1\right) - \log N \]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 8.861960054673546 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;\log \left(N + 1\right) - \log N \leq 8.861960054673546 \cdot 10^{-10}:\\
\;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\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)) 8.861960054673546e-10)
   (/ (- 1.0 (/ 0.5 N)) N)
   (log (/ (+ 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)) <= 8.861960054673546e-10) {
		tmp = (1.0 - (0.5 / N)) / N;
	} else {
		tmp = log(((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)) < 8.8619601e-10

    1. Initial program 60.1

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

    2. Taylor expanded in N around inf 0.0

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{N} - \frac{0.5}{N \cdot N}} \]

    4. Applied egg-rr0.0

      \[\leadsto \color{blue}{\frac{1 - \frac{0.5}{N}}{N}} \]

    if 8.8619601e-10 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

    1. Initial program 0.5

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

    2. Applied egg-rr0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 8.861960054673546 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))