Average Error: 31.2 → 0.3
Time: 29.4s
Precision: 64
Internal precision: 2432
\[\frac{1 - \cos x}{{x}^2}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -0.00023767617918960045:\\ \;\;\;\;{\left(\frac{\sqrt{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}}{x}\right)}^2\\ \mathbf{if}\;x \le 659531696.815955:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes.

  2. if x < -0.00023767617918960045

    1. Initial program 1.1

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

    3. Applied flip-- 1.4

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

    4. Applied simplify 1.2

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

    6. Applied add-sqr-sqrt 1.2

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

    7. Applied square-undiv 0.7

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

    8. Applied simplify 0.7

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

    if -0.00023767617918960045 < x < 659531696.815955

    1. Initial program 61.2

      \[\frac{1 - \cos x}{{x}^2}\]

    2. Applied taylor 0.0

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

    3. Taylor expanded around 0 0.0

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

    if 659531696.815955 < x

    1. Initial program 1.2

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

    3. Applied square-mult 1.2

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

    4. Applied *-un-lft-identity 1.2

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

    5. Applied times-frac 0.5

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 29.4s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(644180380 3784176976 401987740 22459203 1940947670 3323606534)'
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (sqr x)))