logs (example 3.8)

Average Error: 63.0 → 0

Time: 6.4s

Precision: binary64

\[n > 6.8 \cdot 10^{+15}\]

Math

\[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]

↓

\[\frac{0.5}{n} + \left(1 \cdot \log n - \frac{0.16666666666666669}{n \cdot n}\right)\]

C

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

↓

double code(double n) {
	return ((double) ((0.5 / n) + ((double) (((double) (1.0 * ((double) log(n)))) - (0.16666666666666669 / ((double) (n * n)))))));
}

Error

Bits error versus n

Target

Original	63.0
Target	0
Herbie	0

\[\log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right)\]

Derivation

1. Initial program 63.0

   \[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1\]

2. Simplified 62.0

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

3. Taylor expanded around inf 0.0

   \[\leadsto \color{blue}{\left(1 \cdot \log 1 + 0.5 \cdot \frac{1}{n}\right) - \left(1 \cdot \log \left(\frac{1}{n}\right) + 0.16666666666666669 \cdot \frac{1}{{n}^{2}}\right)}\]

4. Simplified 0

   \[\leadsto \color{blue}{\frac{0.5}{n} + \left(1 \cdot \log n - \frac{0.16666666666666669}{n \cdot n}\right)}\]

5. Final simplification 0

   \[\leadsto \frac{0.5}{n} + \left(1 \cdot \log n - \frac{0.16666666666666669}{n \cdot n}\right)\]

Reproduce

herbie shell --seed 2020196 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1.0)) (- (/ 1.0 (* 2.0 n)) (- (/ 1.0 (* 3.0 (* n n))) (/ 4.0 (pow n 3.0)))))

  (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))