Error

Derivation

1. Split input into 3 regimes

2. if x < -0.010981861525964667

   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 simplify1.2

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

   6. Applied frac-2neg1.2

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

   7. Applied simplify0.9

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

   if -0.010981861525964667 < x < 0.02922727487380168

   1. Initial program 61.3

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

   2. Taylor expanded around 0 0.0

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

   if 0.02922727487380168 < x

   1. Initial program 0.9

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

   3. Applied *-un-lft-identity0.9

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

   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: 41.4s)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))