\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 9.4 s
Input Error: 18.6
Output Error: 9.3
Log: ⚲
Profile: 🕒
\(\cos x \cdot \cos \varepsilon - \left({\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3 + \cos x\right)\)
  1. Started with
    \[\cos \left(x + \varepsilon\right) - \cos x\]
    18.6
  2. Using strategy rm
    18.6
  3. Applied cos-sum to get
    \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    9.2
  4. Applied associate--l- to get
    \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
    9.2
  5. Using strategy rm
    9.2
  6. Applied add-cube-cbrt to get
    \[\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \color{red}{\sin \varepsilon} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\sin x \cdot \color{blue}{{\left(\sqrt[3]{\sin \varepsilon}\right)}^3} + \cos x\right)\]
    9.3
  7. Applied add-cube-cbrt to get
    \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sin x} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3 + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{{\left(\sqrt[3]{\sin x}\right)}^3} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3 + \cos x\right)\]
    9.3
  8. Applied cube-unprod to get
    \[\cos x \cdot \cos \varepsilon - \left(\color{red}{{\left(\sqrt[3]{\sin x}\right)}^3 \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3} + \cos x\right)\]
    9.3

Original test:


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