Error

Derivation

if x < -7.621017336353511e-15

1. Initial program 3.3b

   \[\frac{1 - \cos x}{{x}^2}\]
2. Using strategy rm

3. Applied flip-- 3.5b

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

4. Applied simplify 1.2b

   \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{{x}^2}\]
5. Using strategy rm

6. Applied square-mult 1.2b

   \[\leadsto \frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{\color{blue}{x \cdot x}}\]

7. Applied *-un-lft-identity 1.2b

   \[\leadsto \frac{\color{blue}{1 \cdot \frac{\sin x \cdot \sin x}{1 + \cos x}}}{x \cdot x}\]

8. Applied times-frac 0.5b

   \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{x}}\]

9. Applied simplify 0.5b

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

if -7.621017336353511e-15 < x < 0.4439134345824553

1. Initial program 61.7b

   \[\frac{1 - \cos x}{{x}^2}\]

2. Applied taylor 0.0b

   \[\leadsto \left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^2\]

3. Taylor expanded around 0 0.0b

   \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^2}\]

if 0.4439134345824553 < x

1. Initial program 1.0b

   \[\frac{1 - \cos x}{{x}^2}\]
2. Using strategy rm

3. Applied flip-- 1.2b

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

4. Applied simplify 1.1b

   \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{{x}^2}\]
1. Removed slow pow expressions

Runtime

Total time: 33.9s Debug log

herbie --seed '#(3823920687 2130546804 1378687420 2818341529 732645619 212482852)'
(FPCore (x)
  :name "NMSE problem 3.4.1"
  (/ (- 1 (cos x)) (sqr x)))