Average Error: 29.5 → 0.1
Time: 2.4s
Precision: binary64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \leq 12532.10507989266:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \leq 12532.10507989266:\\
\;\;\;\;\log \left(1 + \frac{1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{N} + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N \cdot N}\\

\end{array}
(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))
↓
(FPCore (N)
 :precision binary64
 (if (<= N 12532.10507989266)
   (log (+ 1.0 (/ 1.0 N)))
   (+ (/ 1.0 N) (/ (+ -0.5 (/ 0.3333333333333333 N)) (* N N)))))
double code(double N) {
	return log(N + 1.0) - log(N);
}
↓
double code(double N) {
	double tmp;
	if (N <= 12532.10507989266) {
		tmp = log(1.0 + (1.0 / N));
	} else {
		tmp = (1.0 / N) + ((-0.5 + (0.3333333333333333 / N)) / (N * 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 N < 12532.1050798926608

    1. Initial program 0.1

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

    3. Applied diff-log_binary640.1

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

    5. Applied add-sqr-sqrt_binary640.1

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

    6. Applied associate-/r*_binary640.1

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

    7. Taylor expanded around 0 0.1

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

    8. Simplified0.1

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

    if 12532.1050798926608 < N

    1. Initial program 59.5

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

    3. Applied diff-log_binary6459.3

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

    5. Applied add-sqr-sqrt_binary6459.8

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

    6. Applied associate-/r*_binary6459.8

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

    7. Taylor expanded around 0 59.3

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

    8. Simplified59.3

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

    9. Taylor expanded around inf 0.0

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

    10. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 12532.10507989266:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))