Error

Derivation

1. Split input into 3 regimes

2. if x < -0.01170238699226889

   1. Initial program 1.1

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

   3. Applied flip--1.3

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

   4. Applied simplify1.1

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

   6. Applied frac-2neg1.1

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

   7. Applied simplify0.8

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

   if -0.01170238699226889 < x < 0.012699332412503013

   1. Initial program 61.3

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

   2. Taylor expanded around 0 0.0

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

   3. Applied simplify0.0

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

   if 0.012699332412503013 < x

   1. Initial program 1.0

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

   3. Applied flip--1.2

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

   4. Applied simplify1.0

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

   6. Applied add-sqr-sqrt1.1

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

   7. Applied times-frac0.6

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

   8. Applied simplify0.5

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

   9. Applied simplify0.4

      \[\leadsto \frac{\sqrt{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x} \cdot \color{blue}{\frac{\sqrt{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x}}\]
3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))