\[\cos \left(x + \varepsilon\right) - \cos x\]
Test:
NMSE problem 3.3.5
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 22.3 s
Input Error: 39.1
Output Error: 24.2
Log: ⚲
Profile: 🕒
\(\cos x \cdot \cos \varepsilon - \log_* (1 + (e^{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*} - 1)^*)\)
  1. Started with
    \[\cos \left(x + \varepsilon\right) - \cos x\]
    39.1
  2. Using strategy rm
    39.1
  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
    \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
    24.1
  5. Applied simplify to get
    \[\cos x \cdot \cos \varepsilon - \color{red}{\left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \cos x \cdot \cos \varepsilon - \color{blue}{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\]
    24.1
  6. Using strategy rm
    24.1
  7. Applied log1p-expm1-u to get
    \[\cos x \cdot \cos \varepsilon - \color{red}{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*} \leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\log_* (1 + (e^{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*} - 1)^*)}\]
    24.2

Original test:


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