Average Error: 30.6 → 0.4
Time: 1.5m
Precision: 64
Internal Precision: 2368
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.013136295844982911:\\ \;\;\;\;\frac{\sin x}{\frac{\left(1 + \cos x\right) \cdot \left(x \cdot x\right)}{\sin x}}\\ \mathbf{elif}\;x \le 0.03401460433135982:\\ \;\;\;\;\frac{\sin x}{(x \cdot \left((x \cdot \left(\frac{-1}{6} \cdot x\right) + 2)_*\right) + \left(\frac{-1}{360} \cdot {x}^{5}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.013136295844982911

    1. Initial program 1.2

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

    2. Initial simplification1.2

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

    4. Applied flip--1.4

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

    5. Applied associate-/l/1.4

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

    6. Simplified1.3

      \[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
    7. Using strategy rm

    8. Applied associate-/l*1.2

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

    if -0.013136295844982911 < x < 0.03401460433135982

    1. Initial program 61.3

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

    2. Initial simplification61.3

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

    4. Applied flip--61.3

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

    5. Applied associate-/l/61.3

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

    6. Simplified28.6

      \[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
    7. Using strategy rm

    8. Applied associate-/l*29.2

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

    9. Taylor expanded around 0 0.0

      \[\leadsto \frac{\sin x}{\color{blue}{2 \cdot x - \left(\frac{1}{6} \cdot {x}^{3} + \frac{1}{360} \cdot {x}^{5}\right)}}\]

    10. Simplified0.0

      \[\leadsto \frac{\sin x}{\color{blue}{(x \cdot \left((x \cdot \left(\frac{-1}{6} \cdot x\right) + 2)_*\right) + \left(\frac{-1}{360} \cdot {x}^{5}\right))_*}}\]

    if 0.03401460433135982 < x

    1. Initial program 1.1

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

    2. Initial simplification1.1

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

    4. Applied associate-/r*0.4

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.013136295844982911:\\ \;\;\;\;\frac{\sin x}{\frac{\left(1 + \cos x\right) \cdot \left(x \cdot x\right)}{\sin x}}\\ \mathbf{elif}\;x \le 0.03401460433135982:\\ \;\;\;\;\frac{\sin x}{(x \cdot \left((x \cdot \left(\frac{-1}{6} \cdot x\right) + 2)_*\right) + \left(\frac{-1}{360} \cdot {x}^{5}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018235 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))