\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 10.3 s
Input Error: 18.6
Output Error: 9.2
Log: ⚲
Profile: 🕒
\(\frac{\frac{{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^3\right)}^3 - {\left({\left(\cos x\right)}^3\right)}^3}{{\left({\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3\right)}^2 + \left({\left({\left(\cos x\right)}^3\right)}^2 + {\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 \cdot {\left(\cos x\right)}^3\right)}}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*}\)
  1. Started with
    \[\cos \left(x + \varepsilon\right) - \cos x\]
    18.6
  2. Using strategy rm
    18.6
  3. Applied cos-sum to get
    \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    9.2
  4. Using strategy rm
    9.2
  5. Applied flip3-- to get
    \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}}\]
    10.9
  6. Applied simplify to get
    \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}\]
    9.2
  7. Applied simplify to get
    \[\frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{red}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 + \left({\left(\cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x\right)}} \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}{\color{blue}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*}}\]
    9.2
  8. Using strategy rm
    9.2
  9. Applied flip3-- to get
    \[\frac{\color{red}{{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 - {\left(\cos x\right)}^3}}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*} \leadsto \frac{\color{blue}{\frac{{\left({\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3\right)}^{3} - {\left({\left(\cos x\right)}^3\right)}^{3}}{{\left({\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3\right)}^2 + \left({\left({\left(\cos x\right)}^3\right)}^2 + {\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 \cdot {\left(\cos x\right)}^3\right)}}}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*}\]
    10.9
  10. Applied simplify to get
    \[\frac{\frac{\color{red}{{\left({\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3\right)}^{3} - {\left({\left(\cos x\right)}^3\right)}^{3}}}{{\left({\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3\right)}^2 + \left({\left({\left(\cos x\right)}^3\right)}^2 + {\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 \cdot {\left(\cos x\right)}^3\right)}}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*} \leadsto \frac{\frac{\color{blue}{{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^3\right)}^3 - {\left({\left(\cos x\right)}^3\right)}^3}}{{\left({\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3\right)}^2 + \left({\left({\left(\cos x\right)}^3\right)}^2 + {\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)}^3 \cdot {\left(\cos x\right)}^3\right)}}{(\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) * \left((\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \cos x\right))_*}\]
    9.2

Original test:


(lambda ((x default) (eps default))
  #:name "NMSE problem 3.3.5"
  (- (cos (+ x eps)) (cos x)))