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Derivation

1. Split input into 3 regimes

2. if x < -8.592662095200064e-07

   1. Initial program 1.5

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

   3. Applied flip--1.7

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

   4. Applied associate-/l/1.7

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

   5. Simplified1.1

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

   7. Applied expm1-log1p-u1.1

      \[\leadsto \frac{\color{blue}{(e^{\log_* (1 + \sin x \cdot \sin x)} - 1)^*}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity1.1

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

   10. Applied times-frac1.2

       \[\leadsto \color{blue}{\frac{1}{x \cdot x} \cdot \frac{(e^{\log_* (1 + \sin x \cdot \sin x)} - 1)^*}{1 + \cos x}}\]

   11. Simplified0.8

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

   if -8.592662095200064e-07 < x < 9.466399131825878e+23

   1. Initial program 59.1

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

   3. Applied flip--59.1

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

   4. Applied associate-/l/59.1

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

   5. Simplified28.4

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

   7. Applied expm1-log1p-u28.4

      \[\leadsto \frac{\color{blue}{(e^{\log_* (1 + \sin x \cdot \sin x)} - 1)^*}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
   8. Using strategy rm

   9. Applied associate-/r*28.4

      \[\leadsto \color{blue}{\frac{\frac{(e^{\log_* (1 + \sin x \cdot \sin x)} - 1)^*}{x \cdot x}}{1 + \cos x}}\]
   10. Using strategy rm

   11. Applied add-cbrt-cube49.7

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

   12. Applied add-cbrt-cube49.5

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

   13. Applied cbrt-undiv49.4

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

   14. Simplified0.1

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

   if 9.466399131825878e+23 < x

   1. Initial program 1.1

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

   3. 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.592662095200064 \cdot 10^{-07}:\\ \;\;\;\;\frac{1}{x \cdot x} \cdot \left(\tan \left(\frac{x}{2}\right) \cdot \sin x\right)\\ \mathbf{elif}\;x \le 9.466399131825878 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\sin x}{x} \cdot \frac{\sin x}{x}\right)}^{3}}}{1 + \cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))

Details

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