logs (example 3.8)

Average Error: 63.0 → 0.0

Time: 4.5s

Precision: binary64

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

Math

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

↓

\[\left(\left(1 + \frac{0.5}{n}\right) - \left(\frac{0.16666666666666666}{n \cdot n} - \log n\right)\right) - 1\]

FPCore

(FPCore (n)
 :precision binary64
 (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))

↓

(FPCore (n)
 :precision binary64
 (- (- (+ 1.0 (/ 0.5 n)) (- (/ 0.16666666666666666 (* n n)) (log n))) 1.0))

C

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

↓

double code(double n) {
	return ((1.0 + (0.5 / n)) - ((0.16666666666666666 / (n * n)) - log(n))) - 1.0;
}

Error

Bits error versus n

Target

Original	63.0
Target	0
Herbie	0.0

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

Derivation

1. Initial program 63.0

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

2. Taylor expanded around inf 0.0

   \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{n} + 1\right) - \left(\log \left(\frac{1}{n}\right) + 0.16666666666666666 \cdot \frac{1}{{n}^{2}}\right)\right)} - 1\]

3. Simplified 0.0

   \[\leadsto \color{blue}{\left(\left(1 + \frac{0.5}{n}\right) - \left(\frac{0.16666666666666666}{n \cdot n} - \log n\right)\right)} - 1\]

4. Final simplification 0.0

   \[\leadsto \left(\left(1 + \frac{0.5}{n}\right) - \left(\frac{0.16666666666666666}{n \cdot n} - \log n\right)\right) - 1\]

Reproduce

herbie shell --seed 2020231 
(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))