\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 19.5 s
Input Error: 18.2
Output Error: 9.3
Log: ⚲
Profile: 🕒
\(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}{\log \left(e^{{\left(\cos x \cdot \cos \varepsilon\right)}^2}\right) + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\)
  1. Started with
    \[\cos \left(x + \varepsilon\right) - \cos x\]
    18.2
  2. Using strategy rm
    18.2
  3. Applied cos-sum to get
    \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    9.1
  4. Applied associate--l- to get
    \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
    9.1
  5. Using strategy rm
    9.1
  6. Applied flip3-- to get
    \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}}\]
    10.8
  7. Applied simplify to get
    \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
    9.1
  8. Using strategy rm
    9.1
  9. Applied add-log-exp to get
    \[\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^2} + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left(\cos x + \sin x \cdot \sin \varepsilon\right)}^3}{\color{blue}{\log \left(e^{{\left(\cos x \cdot \cos \varepsilon\right)}^2}\right)} + \left({\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
    9.3

Original test:


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