\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 25.3 s
Input Error: 39.7
Output Error: 24.2
Log: ⚲
Profile: 🕒
\(\cos x \cdot \cos \varepsilon - \left(\sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3} + \cos x\right)\)
  1. Started with
    \[\cos \left(x + \varepsilon\right) - \cos x\]
    39.7
  2. Using strategy rm
    39.7
  3. Applied cos-sum to get
    \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    24.1
  4. Applied associate--l- to get
    \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
    24.2
  5. Using strategy rm
    24.2
  6. Applied add-cbrt-cube to get
    \[\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \color{red}{\sin \varepsilon} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\sin x \cdot \color{blue}{\sqrt[3]{{\left(\sin \varepsilon\right)}^3}} + \cos x\right)\]
    24.2
  7. Applied add-cbrt-cube to get
    \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sin x} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\sqrt[3]{{\left(\sin x\right)}^3}} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3} + \cos x\right)\]
    24.2
  8. Applied cbrt-unprod to get
    \[\cos x \cdot \cos \varepsilon - \left(\color{red}{\sqrt[3]{{\left(\sin x\right)}^3} \cdot \sqrt[3]{{\left(\sin \varepsilon\right)}^3}} + \cos x\right) \leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\sqrt[3]{{\left(\sin x\right)}^3 \cdot {\left(\sin \varepsilon\right)}^3}} + \cos x\right)\]
    24.2

Original test:


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