Average Error: 31.2 → 0.4
Time: 49.8s
Precision: 64
Internal Precision: 2368
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.012996503772427033:\\ \;\;\;\;\left(\sqrt[3]{\tan \left(\frac{x}{2}\right) \cdot \sin x} \cdot \frac{\sqrt[3]{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x}\right) \cdot \frac{\sqrt[3]{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x}\\ \mathbf{if}\;x \le 0.002296654857650485:\\ \;\;\;\;(\left({x}^{4}\right) \cdot \frac{1}{720} + \frac{1}{2})_* - x \cdot \left(\frac{1}{24} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\sin x\right) \cdot \tan \left(\frac{x}{2}\right)}{-x \cdot x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.012996503772427033

    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-cube-cbrt1.4

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

    7. Applied times-frac0.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{\sin x \cdot \sin x}{1 + \cos x}} \cdot \sqrt[3]{\frac{\sin x \cdot \sin x}{1 + \cos x}}}{x} \cdot \frac{\sqrt[3]{\frac{\sin x \cdot \sin x}{1 + \cos x}}}{x}}\]

    8. Applied simplify0.7

      \[\leadsto \color{blue}{\left(\sqrt[3]{\tan \left(\frac{x}{2}\right) \cdot \sin x} \cdot \frac{\sqrt[3]{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x}\right)} \cdot \frac{\sqrt[3]{\frac{\sin x \cdot \sin x}{1 + \cos x}}}{x}\]

    9. Applied simplify0.7

      \[\leadsto \left(\sqrt[3]{\tan \left(\frac{x}{2}\right) \cdot \sin x} \cdot \frac{\sqrt[3]{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x}\right) \cdot \color{blue}{\frac{\sqrt[3]{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x}}\]

    if -0.012996503772427033 < x < 0.002296654857650485

    1. Initial program 61.3

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

    2. Taylor expanded around 0 0.0

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

    3. Applied simplify0.0

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

    if 0.002296654857650485 < x

    1. Initial program 1.0

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

    3. Applied flip--1.3

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

    4. Applied simplify1.0

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

    6. Applied frac-2neg1.0

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

    7. Applied simplify0.7

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

Runtime

Time bar (total: 49.8s)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))