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Derivation

1. Initial program 30.7

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

2. Initial simplification30.7

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

4. Applied flip--30.9

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

5. Applied associate-/l/30.9

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

6. Simplified15.7

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

8. Applied associate-/r*15.7

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

10. Applied times-frac0.3

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

11. Final simplification0.3

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

Runtime

Time bar (total: 32.8s)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))