Error

Derivation

if x < -0.11817437388404825

1. Initial program 20.0b

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
2. Using strategy rm

3. Applied flip-- 53.5b

   \[\leadsto \color{blue}{\frac{{\left(\frac{1}{x + 1}\right)}^2 - {\left(\frac{2}{x}\right)}^2}{\frac{1}{x + 1} + \frac{2}{x}}} + \frac{1}{x - 1}\]

4. Applied frac-add 54.6b

   \[\leadsto \color{blue}{\frac{\left({\left(\frac{1}{x + 1}\right)}^2 - {\left(\frac{2}{x}\right)}^2\right) \cdot \left(x - 1\right) + \left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot 1}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(x - 1\right)}}\]

5. Applied simplify 26.4b

   \[\leadsto \frac{\color{blue}{\left(1 + \left(x - 1\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right)\right) \cdot \left(\frac{1}{1 + x} + \frac{2}{x}\right)}}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(x - 1\right)}\]

6. Applied taylor 0.2b

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

7. Taylor expanded around inf 0.2b

   \[\leadsto \frac{\color{blue}{\left(\left(2 \cdot \frac{1}{{x}^{4}} + 2 \cdot \frac{1}{{x}^2}\right) - 2 \cdot \frac{1}{{x}^{3}}\right)} \cdot \left(\frac{1}{1 + x} + \frac{2}{x}\right)}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(x - 1\right)}\]

8. Applied simplify 0.1b

   \[\leadsto \color{blue}{\frac{\frac{2}{{x}^{4}} + \left(\frac{\frac{2}{x}}{x} - \frac{2}{{x}^3}\right)}{x - 1}}\]

if -0.11817437388404825 < x < 906705844.034969

1. Initial program 0.5b

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
2. Using strategy rm

3. Applied flip-- 29.6b

   \[\leadsto \color{blue}{\frac{{\left(\frac{1}{x + 1}\right)}^2 - {\left(\frac{2}{x}\right)}^2}{\frac{1}{x + 1} + \frac{2}{x}}} + \frac{1}{x - 1}\]

4. Applied frac-add 29.6b

   \[\leadsto \color{blue}{\frac{\left({\left(\frac{1}{x + 1}\right)}^2 - {\left(\frac{2}{x}\right)}^2\right) \cdot \left(x - 1\right) + \left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot 1}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(x - 1\right)}}\]

5. Applied simplify 29.6b

   \[\leadsto \frac{\color{blue}{\left(1 + \left(x - 1\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right)\right) \cdot \left(\frac{1}{1 + x} + \frac{2}{x}\right)}}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(x - 1\right)}\]
6. Using strategy rm

7. Applied flip-- 29.6b

   \[\leadsto \frac{\left(1 + \left(x - 1\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right)\right) \cdot \left(\frac{1}{1 + x} + \frac{2}{x}\right)}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \color{blue}{\frac{{x}^2 - {1}^2}{x + 1}}}\]

8. Applied associate-*r/ 29.6b

   \[\leadsto \frac{\left(1 + \left(x - 1\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right)\right) \cdot \left(\frac{1}{1 + x} + \frac{2}{x}\right)}{\color{blue}{\frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left({x}^2 - {1}^2\right)}{x + 1}}}\]

9. Applied associate-/r/ 29.6b

   \[\leadsto \color{blue}{\frac{\left(1 + \left(x - 1\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right)\right) \cdot \left(\frac{1}{1 + x} + \frac{2}{x}\right)}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left({x}^2 - {1}^2\right)} \cdot \left(x + 1\right)}\]

10. Applied simplify 0.5b

    \[\leadsto \color{blue}{\frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} \cdot \left(x - 1\right)\right) + \frac{1}{1 + x}}{x - 1}} \cdot \left(x + 1\right)\]
11. Using strategy rm

12. Applied frac-sub 0.5b

    \[\leadsto \frac{\color{blue}{\frac{1 \cdot x - \left(1 + x\right) \cdot 2}{\left(1 + x\right) \cdot x}} \cdot \left(\frac{1}{1 + x} \cdot \left(x - 1\right)\right) + \frac{1}{1 + x}}{x - 1} \cdot \left(x + 1\right)\]

13. Applied associate-*l/ 0.5b

    \[\leadsto \frac{\color{blue}{\frac{\left(1 \cdot x - \left(1 + x\right) \cdot 2\right) \cdot \left(\frac{1}{1 + x} \cdot \left(x - 1\right)\right)}{\left(1 + x\right) \cdot x}} + \frac{1}{1 + x}}{x - 1} \cdot \left(x + 1\right)\]

14. Applied frac-add 0.3b

    \[\leadsto \frac{\color{blue}{\frac{\left(\left(1 \cdot x - \left(1 + x\right) \cdot 2\right) \cdot \left(\frac{1}{1 + x} \cdot \left(x - 1\right)\right)\right) \cdot \left(1 + x\right) + \left(\left(1 + x\right) \cdot x\right) \cdot 1}{\left(\left(1 + x\right) \cdot x\right) \cdot \left(1 + x\right)}}}{x - 1} \cdot \left(x + 1\right)\]

15. Applied associate-/l/ 0.3b

    \[\leadsto \color{blue}{\frac{\left(\left(1 \cdot x - \left(1 + x\right) \cdot 2\right) \cdot \left(\frac{1}{1 + x} \cdot \left(x - 1\right)\right)\right) \cdot \left(1 + x\right) + \left(\left(1 + x\right) \cdot x\right) \cdot 1}{\left(x - 1\right) \cdot \left(\left(\left(1 + x\right) \cdot x\right) \cdot \left(1 + x\right)\right)}} \cdot \left(x + 1\right)\]

if 906705844.034969 < x

1. Initial program 19.1b

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

2. Applied taylor 0.7b

   \[\leadsto 2 \cdot \frac{1}{{x}^{3}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{7}}\right)\]

3. Taylor expanded around inf 0.7b

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{7}}\right)}\]

4. Applied simplify 0.7b

   \[\leadsto \color{blue}{\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^3} + \frac{2}{{x}^{7}}\right)}\]
1. Removed slow pow expressions

Runtime

Total time: 1.1m Debug log

herbie --seed '#(3854178499 1134783363 3795021006 3630138376 302827505 2134793165)'
(FPCore (x)
  :name "NMSE problem 3.3.3"
  
  :target
  (/ 2 (* x (- (sqr x) 1)))(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))