Error

Derivation

1. Split input into 2 regimes

2. if (* (/ (/ (sin x) (* (cbrt (+ 1 (cos x))) (cbrt (+ 1 (cos x))))) x) (/ (/ (sin x) (cbrt (+ 1 (cos x)))) x)) < 0.487119932301508

   1. Initial program 0.9

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

   3. Applied flip--1.1

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

   4. Applied simplify1.0

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

   6. Applied add-cube-cbrt1.3

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

   7. Applied times-frac1.3

      \[\leadsto \frac{\color{blue}{\frac{\sin x}{\sqrt[3]{1 + \cos x} \cdot \sqrt[3]{1 + \cos x}} \cdot \frac{\sin x}{\sqrt[3]{1 + \cos x}}}}{x \cdot x}\]

   8. Applied times-frac0.9

      \[\leadsto \color{blue}{\frac{\frac{\sin x}{\sqrt[3]{1 + \cos x} \cdot \sqrt[3]{1 + \cos x}}}{x} \cdot \frac{\frac{\sin x}{\sqrt[3]{1 + \cos x}}}{x}}\]

   if 0.487119932301508 < (* (/ (/ (sin x) (* (cbrt (+ 1 (cos x))) (cbrt (+ 1 (cos x))))) x) (/ (/ (sin x) (cbrt (+ 1 (cos x)))) x))

   1. Initial program 61.1

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

   2. Taylor expanded around 0 0.1

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

   3. Applied simplify0.1

      \[\leadsto \color{blue}{(\left({x}^{4}\right) \cdot \frac{1}{720} + \frac{1}{2})_* - \left(x \cdot x\right) \cdot \frac{1}{24}}\]
3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 31.2s)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))