Average Error: 31.4 → 0.3
Time: 36.0s
Precision: 64
Internal Precision: 128
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\left((\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x\right) \cdot \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{{\left(\cos x\right)}^{3} + 1}\]

Error

Bits error versus x

Derivation

  1. Initial program 31.4

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

  2. Initial simplification31.4

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

  4. Applied flip--31.5

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

  5. Applied associate-/l/31.5

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

  6. Simplified15.5

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

  8. Applied flip3-+15.5

    \[\leadsto \frac{\sin x \cdot \sin x}{\left(x \cdot x\right) \cdot \color{blue}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x - 1 \cdot \cos x\right)}}}\]

  9. Applied associate-*r/15.5

    \[\leadsto \frac{\sin x \cdot \sin x}{\color{blue}{\frac{\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)}{1 \cdot 1 + \left(\cos x \cdot \cos x - 1 \cdot \cos x\right)}}}\]

  10. Applied associate-/r/15.5

    \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)} \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x - 1 \cdot \cos x\right)\right)}\]

  11. Simplified15.5

    \[\leadsto \frac{\sin x \cdot \sin x}{\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)} \cdot \color{blue}{\left((\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x\right)}\]
  12. Using strategy rm

  13. Applied associate-/r*15.5

    \[\leadsto \color{blue}{\frac{\frac{\sin x \cdot \sin x}{x \cdot x}}{{1}^{3} + {\left(\cos x\right)}^{3}}} \cdot \left((\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x\right)\]
  14. Using strategy rm

  15. Applied times-frac0.3

    \[\leadsto \frac{\color{blue}{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}}{{1}^{3} + {\left(\cos x\right)}^{3}} \cdot \left((\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x\right)\]

  16. Final simplification0.3

    \[\leadsto \left((\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x\right) \cdot \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{{\left(\cos x\right)}^{3} + 1}\]

Runtime

Time bar (total: 36.0s)Debug logProfile

herbie shell --seed 2018278 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))