Average Error: 31.4 → 0.3
Time: 40.3s
Precision: 64
Internal Precision: 128
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.026007560286832525:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \le 7.623138198087047 \cdot 10^{-08}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \left(\frac{x}{2}\right)}{\frac{x \cdot x}{\sin x}}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.026007560286832525

    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 *-un-lft-identity1.2

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

    5. Applied times-frac0.5

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

    if -0.026007560286832525 < x < 7.623138198087047e-08

    1. Initial program 61.7

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

    2. Initial simplification61.7

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

    4. Applied flip--61.7

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

    5. Applied associate-/l/61.7

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

    6. Simplified30.0

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

    7. 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 7.623138198087047e-08 < x

    1. Initial program 1.6

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

    2. Initial simplification1.6

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

    4. Applied flip--1.8

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

    5. Applied associate-/l/1.8

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

    6. Simplified1.1

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

    7. Taylor expanded around inf 1.1

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

    8. Simplified0.8

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.026007560286832525:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \le 7.623138198087047 \cdot 10^{-08}:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \left(\frac{x}{2}\right)}{\frac{x \cdot x}{\sin x}}\\ \end{array}\]

Runtime

Time bar (total: 40.3s)Debug logProfile

herbie shell --seed 2018278 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))