Average Error: 31.1 → 0.2
Time: 23.3s
Precision: 64
Internal Precision: 2432
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{1}{x} \cdot \frac{\tan \left(\frac{x}{2}\right)}{\frac{x}{\sin x}}\]

Error

Bits error versus x

Derivation

  1. Initial program 31.1

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

  3. Applied flip--31.2

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

  4. Applied simplify15.4

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

  6. Applied *-un-lft-identity15.4

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

  7. Applied times-frac15.4

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

  8. Applied simplify0.2

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

Runtime

Time bar (total: 23.3s)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))