Error

Derivation

1. Split input into 2 regimes

2. if x < -0.03514239104203424 or 0.030140279439797614 < x

   1. Initial program 1.1

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

   3. Applied associate-/r*0.5

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

   if -0.03514239104203424 < x < 0.030140279439797614

   1. Initial program 61.4

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

   3. Applied flip--61.4

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

   4. Applied associate-/l/61.4

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

   5. Simplified29.7

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

   7. Applied flip3-+29.7

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

   8. Applied associate-*r/29.7

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

   9. Simplified29.7

      \[\leadsto \frac{\sin x \cdot \sin x}{\frac{\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)}{\color{blue}{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}\]

   10. Taylor expanded around 0 0.0

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

   11. Simplified0.0

       \[\leadsto \color{blue}{(\frac{1}{720} \cdot \left({x}^{4}\right) + \left((\left(x \cdot \frac{-1}{24}\right) \cdot x + \frac{1}{2})_*\right))_*}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03514239104203424 \lor \neg \left(x \le 0.030140279439797614\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;(\frac{1}{720} \cdot \left({x}^{4}\right) + \left((\left(\frac{-1}{24} \cdot x\right) \cdot x + \frac{1}{2})_*\right))_*\\ \end{array}\]

Runtime

Time bar (total: 34.7s)Debug logProfile

herbie shell --seed 2018290 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))