Error

Derivation

if x < -1.8773586721409905e-08 or 108678362595.63406 < x

1. Initial program 1.4

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

3. Applied flip-- 1.6

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

4. Applied simplify 1.1

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

6. Applied *-un-lft-identity 1.1

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

7. Applied *-un-lft-identity 1.1

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

8. Applied times-frac 1.1

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

9. Applied times-frac 1.1

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

10. Applied simplify 1.1

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

if -1.8773586721409905e-08 < x < 108678362595.63406

1. Initial program 61.5

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

2. Applied taylor 0.0

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

3. Taylor expanded around 0 0.0

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

Runtime

Total time: 32.5s Debug log

Please report a bug with the following info:

herbie --seed '#(2392692419 2980109941 1597978070 756077822 3260956253 366005631)'
(FPCore (x)
  :name "NMSE problem 3.4.1"
  (/ (- 1 (cos x)) (sqr x)))