\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 24.5 s
Input Error: 39.7
Output Error: 24.7
Log: ⚲
Profile: 🕒
\(\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^3}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}\)
  1. Started with
    \[\cos \left(x + \varepsilon\right) - \cos x\]
    39.7
  2. Using strategy rm
    39.7
  3. Applied cos-sum to get
    \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    24.6
  4. Applied associate--l- to get
    \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
    24.6
  5. Applied simplify to get
    \[\cos x \cdot \cos \varepsilon - \color{red}{\left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \cos x \cdot \cos \varepsilon - \color{blue}{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\]
    24.6
  6. Using strategy rm
    24.6
  7. Applied flip3-- to get
    \[\color{red}{\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}}\]
    24.7
  8. Applied simplify to get
    \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}\]
    24.7

Original test:


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