Error

Try it out

Derivation

1. Split input into 3 regimes

2. if x < -0.005836946988217429

   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 add-sqr-sqrt1.2

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

   7. Applied simplify1.1

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

   8. Applied simplify0.9

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

   if -0.005836946988217429 < x < 0.030606424940057816

   1. Initial program 61.4

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

   2. Taylor expanded around 0 0.0

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

   if 0.030606424940057816 < x

   1. Initial program 1.1

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

   3. Applied *-un-lft-identity1.1

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

   4. Applied times-frac0.5

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 52.4s)Debug logProfile

herbie shell --seed 2018199 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))