logs (example 3.8)

Average Error: 63.0 → 0.0

Time: 5.0s

Precision: 64

\[n \gt 6.8 \cdot 10^{15}\]

Math

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

↓

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

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) (((double) (1.0 - ((double) (((double) log(((double) (((double) sqrt(1.0)) / 1.0)))) * 1.0)))) - ((double) (((double) (1.0 + ((double) (((double) (0.16666666666666669 / ((double) pow(n, 2.0)))) + ((double) (((double) log(((double) (1.0 / n)))) * 1.0)))))) - ((double) (0.5 / n))))));
}

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

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

3. Simplified 0.0

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

5. Applied *-un-lft-identity 0.0

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

6. Applied add-sqr-sqrt 0.0

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

7. Applied times-frac 0.0

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

8. Applied log-prod 0.0

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

9. Applied distribute-rgt-in 0.0

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

10. Applied associate-+l+ 0.0

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

11. Applied associate--r+ 0.0

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

12. Applied associate-+l- 0.0

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

13. Applied associate--l- 0.0

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

14. Simplified 0.0

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

15. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020113 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1)) (- (/ 1 (* 2 n)) (- (/ 1 (* 3 (* n n))) (/ 4 (pow n 3)))))

  (- (- (* (+ n 1) (log (+ n 1))) (* n (log n))) 1))