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

⬇

\[\begin{array}{l} \mathbf{if}\;x \le -6.667433711053936 \cdot 10^{-08}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{e^{\log \left(1 - \cos x\right)}}{x}\\ \mathbf{if}\;x \le 1183236732512159.2:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{e^{\log \left(1 - \cos x\right)}}{x}\\ \end{array}\]

Derivation

Split input into 2 regimes.
if x < -6.667433711053936e-08 or 1183236732512159.2 < x
1. Initial program 1.3
  \[\frac{1 - \cos x}{x \cdot x}\]
2. Applied simplify 1.3
  \[\leadsto \color{blue}{\frac{1 - \cos x}{{x}^2}}\]
3. Using strategy rm
4. Applied square-mult 1.3
  \[\leadsto \frac{1 - \cos x}{\color{blue}{x \cdot x}}\]
5. Applied *-un-lft-identity 1.3
  \[\leadsto \frac{\color{blue}{1 \cdot \left(1 - \cos x\right)}}{x \cdot x}\]
6. Applied times-frac 0.7
  \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
7. Using strategy rm
8. Applied add-exp-log 0.7
  \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{e^{\log \left(1 - \cos x\right)}}}{x}\]
if -6.667433711053936e-08 < x < 1183236732512159.2
1. Initial program 61.5
  \[\frac{1 - \cos x}{x \cdot x}\]
2. Applied simplify 61.5
  \[\leadsto \color{blue}{\frac{1 - \cos x}{{x}^2}}\]
3. Applied taylor 0.0
  \[\leadsto \left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^2\]
4. Taylor expanded around 0 0.0
  
  \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^2}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie shell --seed '#(3052192724 3812927732 3686175817 630908657 2373248591 511094450)'
(FPCore (x)
  :name "NMSE problem 3.4.1"
  :pre (!= x 0)
  (/ (- 1 (cos x)) (* x x)))

Error

Derivation

`if x < -6.667433711053936e-08 or 1183236732512159.2 < x`

`if -6.667433711053936e-08 < x < 1183236732512159.2`

Runtime