2log (problem 3.3.6)

Average Error: 28.9 → 0.1

Time: 5.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 3345.38054805357797:\\ \;\;\;\;\log \left({N}^{3} + {1}^{3}\right) - \left(\log \left(N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)\right) + \log N\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

C

double code(double N) {
	return (log((N + 1.0)) - log(N));
}

↓

double code(double N) {
	double temp;
	if ((N <= 3345.380548053578)) {
		temp = (log((pow(N, 3.0) + pow(1.0, 3.0))) - (log(((N * N) + ((1.0 * 1.0) - (N * 1.0)))) + log(N)));
	} else {
		temp = (((1.0 / pow(N, 2.0)) * ((0.3333333333333333 / N) - 0.5)) + (1.0 / N));
	}
	return temp;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 3345.380548053578

   1. Initial program 0.1

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

   3. Applied flip3-+ 0.1

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

   4. Applied log-div 0.1

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

   5. Applied associate--l- 0.1

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

   if 3345.380548053578 < N

   1. Initial program 59.5

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

   2. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

   3. Simplified 0.1

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 3345.38054805357797:\\ \;\;\;\;\log \left({N}^{3} + {1}^{3}\right) - \left(\log \left(N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)\right) + \log N\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))