Average Error: 31.3 → 0.2
Time: 46.0s
Precision: 64
Internal Precision: 2368
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{1}{x} \cdot \left(\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 31.3

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

  3. Applied flip--31.4

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

  4. Applied simplify15.7

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

  6. Applied *-un-lft-identity15.7

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

  7. Applied times-frac15.6

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

  8. Applied simplify0.2

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

Runtime

Time bar (total: 46.0s)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))