Average Error: 30.5 → 0.2
Time: 28.0s
Precision: 64
Internal Precision: 2368
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{1}{x} \cdot \left(\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x}\right)\]

Error

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Results

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Derivation

  1. Initial program 30.5

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

  3. Applied flip--30.6

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

  4. Applied associate-/l/30.6

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

  5. Simplified14.9

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

  7. Applied times-frac15.2

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

  8. Simplified15.1

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

  10. Applied *-un-lft-identity15.1

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

  11. Applied times-frac0.2

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

  12. Applied associate-*l*0.2

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

  13. Final simplification0.2

    \[\leadsto \frac{1}{x} \cdot \left(\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x}\right)\]

Runtime

Time bar (total: 28.0s)Debug logProfile

herbie shell --seed 2018217 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))