Error

Derivation

1. Split input into 3 regimes.

2. if x < -0.0004749914434913193

   1. Initial program 1.1

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

   3. Applied flip-- 1.3

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

   4. Applied simplify 1.1

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

   6. Applied square-mult 1.1

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

   7. Applied *-un-lft-identity 1.1

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

   8. Applied *-un-lft-identity 1.1

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

   9. Applied square-prod 1.1

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

   10. Applied times-frac 1.1

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

   11. Applied times-frac 0.6

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

   12. Applied simplify 0.6

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

   if -0.0004749914434913193 < x < 28.042920304023237

   1. Initial program 61.4

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

   2. Applied taylor 0.0

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

   3. 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 28.042920304023237 < x

   1. Initial program 1.0

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

   3. Applied flip-- 1.3

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

   4. Applied simplify 1.1

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

   6. Applied square-mult 1.1

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

   7. Applied *-un-lft-identity 1.1

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

   8. Applied *-un-lft-identity 1.1

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

   9. Applied square-prod 1.1

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

   10. Applied times-frac 1.1

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

   11. Applied times-frac 0.6

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

   12. Applied simplify 0.6

       \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{x}\]
3. Recombined 3 regimes into one program.
4. Removed slow pow expressions

Runtime

Time bar (total: 32.1s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(2277612311 2645429965 1090895633 2857793080 2144184008 3989768357)'
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (sqr x)))