Average Error: 21.6 → 20.4
Time: 23.3s
Precision: 64
Internal Precision: 128
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\sqrt{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \cdot \sqrt{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)}\]

Error

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Results

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Derivation

  1. Initial program 21.6

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

  3. Applied flip--21.7

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

  4. Applied associate-/l/21.7

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

  5. Simplified20.5

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

  7. Applied times-frac20.5

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

  8. Simplified20.3

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

  10. Applied add-sqr-sqrt20.4

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

  11. Final simplification20.4

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

Runtime

Time bar (total: 23.3s)Debug logProfile

herbie shell --seed 2018277 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))