\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 12.2 s
Input Error: 18.3
Output Error: 9.1
Log: ⚲
Profile: 🕒
\(\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - \frac{{\left({\left(\sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2\right)}^2}{{\left(\sin x \cdot \sin \varepsilon - \cos x\right)}^2}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\)
  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 flip-+ to get
    \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\color{red}{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\color{blue}{\left(\frac{{\left(\sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\sin x \cdot \sin \varepsilon - \cos x}\right)}}^2}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
    9.1
  9. Applied square-div to get
    \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - \color{red}{{\left(\frac{{\left(\sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\sin x \cdot \sin \varepsilon - \cos x}\right)}^2}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - \color{blue}{\frac{{\left({\left(\sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2\right)}^2}{{\left(\sin x \cdot \sin \varepsilon - \cos x\right)}^2}}}{\cos x \cdot \cos \varepsilon + \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
    9.1

Original test:


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