\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 25.0 s
Input Error: 18.4
Output Error: 9.0
Log: ⚲
Profile: 🕒
\(\left(\cos x \cdot \cos \varepsilon - {\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3\right) - \cos x\)
  1. Started with
    \[\cos \left(x + \varepsilon\right) - \cos x\]
    18.4
  2. Using strategy rm
    18.4
  3. Applied cos-sum to get
    \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    8.9
  4. Using strategy rm
    8.9
  5. Applied add-cube-cbrt to get
    \[\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{red}{\sin \varepsilon}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{blue}{{\left(\sqrt[3]{\sin \varepsilon}\right)}^3}\right) - \cos x\]
    9.0
  6. Applied add-cube-cbrt to get
    \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{{\left(\sqrt[3]{\sin x}\right)}^3} \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3\right) - \cos x\]
    9.0
  7. Applied cube-unprod to get
    \[\left(\cos x \cdot \cos \varepsilon - \color{red}{{\left(\sqrt[3]{\sin x}\right)}^3 \cdot {\left(\sqrt[3]{\sin \varepsilon}\right)}^3}\right) - \cos x \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin \varepsilon}\right)}^3}\right) - \cos x\]
    9.0

Original test:


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