\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 30.4 s
Input Error: 39.5
Output Error: 24.0
Log: ⚲
Profile: 🕒
\(\log \left(e^{\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\right)\)
  1. Started with
    \[\cos \left(x + \varepsilon\right) - \cos x\]
    39.5
  2. Using strategy rm
    39.5
  3. Applied cos-sum to get
    \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
    24.0
  4. Using strategy rm
    24.0
  5. Applied add-log-exp to get
    \[\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \color{red}{\cos x} \leadsto \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \color{blue}{\log \left(e^{\cos x}\right)}\]
    24.1
  6. Applied add-log-exp to get
    \[\left(\cos x \cdot \cos \varepsilon - \color{red}{\sin x \cdot \sin \varepsilon}\right) - \log \left(e^{\cos x}\right) \leadsto \left(\cos x \cdot \cos \varepsilon - \color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon}\right)}\right) - \log \left(e^{\cos x}\right)\]
    24.2
  7. Applied add-log-exp to get
    \[\left(\color{red}{\cos x \cdot \cos \varepsilon} - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right) - \log \left(e^{\cos x}\right) \leadsto \left(\color{blue}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right)} - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right) - \log \left(e^{\cos x}\right)\]
    24.1
  8. Applied diff-log to get
    \[\color{red}{\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \log \left(e^{\sin x \cdot \sin \varepsilon}\right)\right)} - \log \left(e^{\cos x}\right) \leadsto \color{blue}{\log \left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}\right)} - \log \left(e^{\cos x}\right)\]
    24.2
  9. Applied diff-log to get
    \[\color{red}{\log \left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}\right) - \log \left(e^{\cos x}\right)} \leadsto \color{blue}{\log \left(\frac{\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}}{e^{\cos x}}\right)}\]
    24.2
  10. Applied simplify to get
    \[\log \color{red}{\left(\frac{\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{\sin x \cdot \sin \varepsilon}}}{e^{\cos x}}\right)} \leadsto \log \color{blue}{\left(e^{\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\right)}\]
    24.0

Original test:


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