Average Error: 30.5 → 0.4
Time: 39.5s
Precision: 64
Internal Precision: 2432
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \sin x}{1 + {\left(\cos x\right)}^{3}} \cdot \frac{\left(1 - \cos x\right) + \cos x \cdot \cos x}{x}\]

Error

Bits error versus x

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 simplify15.6

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

  6. Applied flip3-+15.6

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

  7. Applied associate-/r/15.6

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

  8. Applied times-frac15.5

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

  9. Applied simplify0.4

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

  10. Applied simplify0.4

    \[\leadsto \frac{\frac{\sin x}{x} \cdot \sin x}{1 + {\left(\cos x\right)}^{3}} \cdot \color{blue}{\frac{\left(1 - \cos x\right) + \cos x \cdot \cos x}{x}}\]
  11. Removed slow pow expressions.

Runtime

Time bar (total: 39.5s)Debug log

herbie shell --seed '#(151349756 408087815 228312487 2538703040 1980610373 1250971417)' 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))