\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 18.3 s
Input Error: 39.9
Output Error: 25.1
Log: ⚲
Profile: 🕒
\(\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\)
  1. Started with
    \[\cos \left(x + \varepsilon\right) - \cos x\]
    39.9
  2. Using strategy rm
    39.9
  3. Applied cos-sum to get
    \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    25.1
  4. Applied associate--l- to get
    \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
    25.1
  5. Applied simplify to get
    \[\cos x \cdot \cos \varepsilon - \color{red}{\left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \cos x \cdot \cos \varepsilon - \color{blue}{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\]
    25.1
  6. Using strategy rm
    25.1
  7. Applied flip-- to get
    \[\color{red}{\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{\cos x \cdot \cos \varepsilon + (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}}\]
    25.3
  8. Applied simplify to get
    \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{\color{red}{\cos x \cdot \cos \varepsilon + (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}}\]
    25.3
  9. Applied taylor to get
    \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}\]
    25.3
  10. Taylor expanded around 0 to get
    \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{\color{red}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}}\]
    25.3
  11. Applied simplify to get
    \[\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^2 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*} \leadsto \frac{(\left(\cos x\right) * \left(\cos \varepsilon\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}{\frac{(\left(\cos x\right) * \left(\cos \varepsilon\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}{\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}}\]
    25.1
  12. Applied final simplification
  13. Applied simplify to get
    \[\color{red}{\frac{(\left(\cos x\right) * \left(\cos \varepsilon\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}{\frac{(\left(\cos x\right) * \left(\cos \varepsilon\right) + \left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right))_*}{\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}}} \leadsto \color{blue}{\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\]
    25.1

Original test:


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