Average Error: 30.9 → 0.3
Time: 26.9s
Precision: 64
Internal Precision: 2432
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.013047469261293217:\\ \;\;\;\;\frac{\sqrt{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x} \cdot \frac{\sqrt{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x}\\ \mathbf{if}\;x \le 0.0003791239906132239:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\sin x\right) \cdot \tan \left(\frac{x}{2}\right)}{-x \cdot x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.013047469261293217

    1. Initial program 1.0

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

    3. Applied flip--1.2

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

    4. Applied simplify1.1

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

    6. Applied add-sqr-sqrt1.1

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

    7. Applied times-frac0.6

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

    8. Applied simplify0.5

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

    9. Applied simplify0.4

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

    if -0.013047469261293217 < x < 0.0003791239906132239

    1. Initial program 61.4

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    if 0.0003791239906132239 < x

    1. Initial program 1.2

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

    3. Applied flip--1.4

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

    4. Applied simplify1.2

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

    6. Applied frac-2neg1.2

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

    7. Applied simplify0.9

      \[\leadsto \frac{\color{blue}{\left(-\sin x\right) \cdot \tan \left(\frac{x}{2}\right)}}{-x \cdot x}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 26.9s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))