Error

Derivation

1. Split input into 3 regimes

2. if x < -0.01299087312505388

   1. Initial program 1.3

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

   3. Applied flip--1.5

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

   4. Applied simplify1.4

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

   6. Applied div-inv1.4

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

   7. Applied times-frac0.6

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

   8. Applied simplify0.6

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

   if -0.01299087312505388 < x < 0.026023199604116663

   1. Initial program 61.4

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

   2. Taylor expanded around 0 0.0

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

   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 0.026023199604116663 < x

   1. Initial program 1.0

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

   3. Applied *-un-lft-identity1.0

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

   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: 56.8s)Debug logProfile

herbie shell --seed 2018195 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))