Average Error: 31.6 → 0.3
Time: 14.0s
Precision: 64
Internal precision: 2432
\[\frac{1 - \cos x}{{x}^2}\]
⬇
\[\frac{\sin x}{x} \cdot \frac{\frac{\sin x}{1 + \cos x}}{x}\]

Error

Bits error versus x

Derivation

  1. Initial program 31.6

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

  3. Applied flip-- 31.7

    \[\leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\cos x\right)}^2}{1 + \cos x}}}{{x}^2}\]

  4. Applied simplify 15.5

    \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{{x}^2}\]
  5. Using strategy rm

  6. Applied square-mult 15.5

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

  7. Applied *-un-lft-identity 15.5

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

  8. Applied times-frac 15.5

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

  9. Applied times-frac 0.3

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

  10. Applied simplify 0.3

    \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot \frac{\frac{\sin x}{1 + \cos x}}{x}\]
  11. Removed slow pow expressions

Runtime

Time bar (total: 14.0s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1067615470 1817955187 3564058462 2385304812 1026693554 2746013648)'
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (sqr x)))