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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.03583350383358707:\\ \;\;\;\;\frac{\sqrt{1 - \cos x} \cdot \sqrt{1 - \cos x}}{x \cdot x}\\ \mathbf{if}\;x \le 0.03009247951436836:\\ \;\;\;\;(\left({x}^{4}\right) \cdot \frac{1}{720} + \frac{1}{2})_* - \left(x \cdot x\right) \cdot \frac{1}{24}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \end{array}\]

Derivation

Split input into 3 regimes
if x < -0.03583350383358707
1. Initial program 1.0
  \[\frac{1 - \cos x}{x \cdot x}\]
2. Using strategy rm
3. Applied add-sqr-sqrt1.1
  \[\leadsto \frac{\color{blue}{\sqrt{1 - \cos x} \cdot \sqrt{1 - \cos x}}}{x \cdot x}\]
if -0.03583350383358707 < x < 0.03009247951436836
1. Initial program 61.3
  \[\frac{1 - \cos x}{x \cdot x}\]
2. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^{2}}\]
3. Applied simplify0.0
  \[\leadsto \color{blue}{(\left({x}^{4}\right) \cdot \frac{1}{720} + \frac{1}{2})_* - \left(x \cdot x\right) \cdot \frac{1}{24}}\]
if 0.03009247951436836 < x
1. Initial program 1.0
  \[\frac{1 - \cos x}{x \cdot x}\]
2. Using strategy rm
3. Applied *-un-lft-identity1.0
  \[\leadsto \frac{\color{blue}{1 \cdot \left(1 - \cos x\right)}}{x \cdot x}\]
4. Applied times-frac0.5
  \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
Recombined 3 regimes into one program.

Runtime

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))

Error

Derivation

`if x < -0.03583350383358707`

`if -0.03583350383358707 < x < 0.03009247951436836`

`if 0.03009247951436836 < x`

Runtime