\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 24.0 s
Input Error: 39.4
Output Error: 19.8
Log: ⚲
Profile: 🕒
\(\sqrt[3]{{\left(\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin \varepsilon \cdot \sin x\right)}^3}\)
  1. Started with
    \[\cos \left(x + \varepsilon\right) - \cos x\]
    39.4
  2. Using strategy rm
    39.4
  3. Applied cos-sum to get
    \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    24.7
  4. Using strategy rm
    24.7
  5. Applied add-log-exp to get
    \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)}\]
    24.8
  6. Using strategy rm
    24.8
  7. Applied add-cbrt-cube to get
    \[\color{red}{\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)} \leadsto \color{blue}{\sqrt[3]{{\left(\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\right)}^3}}\]
    24.9
  8. Applied simplify to get
    \[\sqrt[3]{\color{red}{{\left(\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\right)}^3}} \leadsto \sqrt[3]{\color{blue}{{\left(\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin \varepsilon \cdot \sin x\right)}^3}}\]
    19.8

Original test:


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