\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 26.6 s
Input Error: 36.9
Output Error: 22.3
Log: ⚲
Profile: 🕒
\(\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - \tan x\)
  1. Started with
    \[\tan \left(x + \varepsilon\right) - \tan x\]
    36.9
  2. Using strategy rm
    36.9
  3. Applied tan-quot to get
    \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x\]
    37.0
  4. Using strategy rm
    37.0
  5. Applied sin-sum to get
    \[\frac{\color{red}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \tan x \leadsto \frac{\color{blue}{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}}{\cos \left(x + \varepsilon\right)} - \tan x\]
    36.1
  6. Using strategy rm
    36.1
  7. Applied cos-sum to get
    \[\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\color{red}{\cos \left(x + \varepsilon\right)}} - \tan x \leadsto \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\color{blue}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}} - \tan x\]
    22.3

Original test:


(lambda ((x default) (eps default))
  #:name "NMSE problem 3.3.2"
  (- (tan (+ x eps)) (tan x))
  #:target
  (/ (sin eps) (* (cos x) (cos (+ x eps)))))