Error

Derivation

1. Split input into 3 regimes

2. if x < -0.030571418330079037

   1. Initial program 1.0

      \[\frac{1 - \cos x}{x \cdot x}\]

   2. Initial simplification1.0

      \[\leadsto \frac{1 - \cos x}{x \cdot x}\]
   3. Using strategy rm

   4. Applied div-inv1.0

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

   if -0.030571418330079037 < x < 0.005631025657119507

   1. Initial program 61.2

      \[\frac{1 - \cos x}{x \cdot x}\]

   2. Initial simplification61.2

      \[\leadsto \frac{1 - \cos x}{x \cdot x}\]
   3. Using strategy rm

   4. Applied flip--61.2

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

   5. Applied associate-/l/61.2

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

   6. Simplified30.6

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

   7. Taylor expanded around 0 0.0

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

   8. Simplified0.0

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

   if 0.005631025657119507 < x

   1. Initial program 1.1

      \[\frac{1 - \cos x}{x \cdot x}\]

   2. Initial simplification1.1

      \[\leadsto \frac{1 - \cos x}{x \cdot x}\]
   3. Using strategy rm

   4. Applied flip--1.4

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

   5. Applied associate-/l/1.4

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

   6. Simplified1.2

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

   7. Taylor expanded around inf 1.2

      \[\leadsto \frac{\sin x \cdot \sin x}{\left(x \cdot x\right) \cdot \color{blue}{\left(\cos x + 1\right)}}\]
   8. Using strategy rm

   9. Applied associate-*l*1.2

      \[\leadsto \frac{\sin x \cdot \sin x}{\color{blue}{x \cdot \left(x \cdot \left(\cos x + 1\right)\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.030571418330079037:\\ \;\;\;\;\frac{1}{x \cdot x} \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \le 0.005631025657119507:\\ \;\;\;\;(\frac{1}{720} \cdot \left({x}^{4}\right) + \left((\left(x \cdot \frac{-1}{24}\right) \cdot x + \frac{1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \sin x}{x \cdot \left(\left(1 + \cos x\right) \cdot x\right)}\\ \end{array}\]

Runtime

Time bar (total: 32.7s)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))