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Derivation

1. Initial program 31.2

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

3. Applied flip--31.3

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

4. Applied associate-/l/31.3

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

5. Simplified15.4

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

7. Applied times-frac15.8

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

8. Simplified15.6

   \[\leadsto \frac{\sin x}{x \cdot x} \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}\]
9. Using strategy rm

10. Applied associate-/r*0.2

    \[\leadsto \color{blue}{\frac{\frac{\sin x}{x}}{x}} \cdot \tan \left(\frac{x}{2}\right)\]
11. Using strategy rm

12. Applied log1p-expm1-u0.1

    \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{\frac{\sin x}{x}}{x} \cdot \tan \left(\frac{x}{2}\right)} - 1)^*)}\]

13. Final simplification0.1

    \[\leadsto \log_* (1 + (e^{\frac{\frac{\sin x}{x}}{x} \cdot \tan \left(\frac{x}{2}\right)} - 1)^*)\]

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))

Details

Time bar (total: 16.1s)Debug log