\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 10.3 s
Input Error: 18.6
Output Error: 9.3
Log: ⚲
Profile: 🕒
\(\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\log_* (1 + (e^{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_*} - 1)^*) - \sin \varepsilon \cdot \sin x}\)
  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. Using strategy rm
    9.2
  5. Applied flip-- to get
    \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}}\]
    9.2
  6. Applied simplify to get
    \[\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_* - \sin \varepsilon \cdot \sin x}}\]
    9.3
  7. Using strategy rm
    9.3
  8. Applied log1p-expm1-u to get
    \[\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\color{red}{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_*} - \sin \varepsilon \cdot \sin x} \leadsto \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^2 - {\left(\cos x\right)}^2}{\color{blue}{\log_* (1 + (e^{(\left(\cos \varepsilon\right) * \left(\cos x\right) + \left(\cos x\right))_*} - 1)^*)} - \sin \varepsilon \cdot \sin x}\]
    9.3

Original test:


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