\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 10.1 s
Input Error: 18.3
Output Error: 3.2
Log: ⚲
Profile: 🕒
\(1 \cdot \left(\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin x \cdot \sin \varepsilon\right)\)
  1. Started with
    \[\cos \left(x + \varepsilon\right) - \cos x\]
    18.3
  2. Using strategy rm
    18.3
  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 flip-- to get
    \[\color{red}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]
    9.1
  7. Using strategy rm
    9.1
  8. Applied *-un-lft-identity to get
    \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\color{red}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\color{blue}{1 \cdot \left(\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}}\]
    9.1
  9. Applied *-un-lft-identity to get
    \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}}{1 \cdot \left(\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \leadsto \frac{\color{blue}{1 \cdot \left({\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2\right)}}{1 \cdot \left(\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}\]
    9.1
  10. Applied times-frac to get
    \[\color{red}{\frac{1 \cdot \left({\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2\right)}{1 \cdot \left(\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)}} \leadsto \color{blue}{\frac{1}{1} \cdot \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]
    9.1
  11. Applied simplify to get
    \[\color{red}{\frac{1}{1}} \cdot \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \color{blue}{1} \cdot \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
    9.1
  12. Applied simplify to get
    \[1 \cdot \color{red}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}} \leadsto 1 \cdot \color{blue}{\left(\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin x \cdot \sin \varepsilon\right)}\]
    3.2
  13. Removed slow pow expressions

Original test:


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