\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 36.0 s
Input Error: 36.9
Output Error: 13.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - \tan x & \text{when } \varepsilon \le -3.5438160870943954 \cdot 10^{-16} \\ \frac{(\left(\sin x\right) * \left(\cos \varepsilon \cdot \cos x - \cos \left(x + \varepsilon\right)\right) + \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos x\right)\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x} & \text{when } \varepsilon \le 0.0016565824112741959 \\ \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - \tan x & \text{otherwise} \end{cases}\)

    if eps < -3.5438160870943954e-16 or 0.0016565824112741959 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      29.5
    2. Using strategy rm
      29.5
    3. Applied tan-quot to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x\]
      29.6
    4. Using strategy rm
      29.6
    5. Applied sin-sum to get
      \[\frac{\color{red}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \tan x \leadsto \frac{\color{blue}{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}}{\cos \left(x + \varepsilon\right)} - \tan x\]
      27.7
    6. Using strategy rm
      27.7
    7. Applied cos-sum to get
      \[\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\color{red}{\cos \left(x + \varepsilon\right)}} - \tan x \leadsto \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\color{blue}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}} - \tan x\]
      0.9

    if -3.5438160870943954e-16 < eps < 0.0016565824112741959

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      44.6
    2. Using strategy rm
      44.6
    3. Applied tan-quot to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x\]
      44.7
    4. Using strategy rm
      44.7
    5. Applied sin-sum to get
      \[\frac{\color{red}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \tan x \leadsto \frac{\color{blue}{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}}{\cos \left(x + \varepsilon\right)} - \tan x\]
      44.7
    6. Using strategy rm
      44.7
    7. Applied tan-quot to get
      \[\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right)} - \color{red}{\tan x} \leadsto \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right)} - \color{blue}{\frac{\sin x}{\cos x}}\]
      44.5
    8. Applied frac-sub to get
      \[\color{red}{\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \cos x - \cos \left(x + \varepsilon\right) \cdot \sin x}{\cos \left(x + \varepsilon\right) \cdot \cos x}}\]
      44.5
    9. Applied simplify to get
      \[\frac{\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \cos x - \cos \left(x + \varepsilon\right) \cdot \sin x}}{\cos \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{(\left(\sin x\right) * \left(\cos \varepsilon \cdot \cos x - \cos \left(x + \varepsilon\right)\right) + \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos x\right)\right))_*}}{\cos \left(x + \varepsilon\right) \cdot \cos x}\]
      26.6
  1. Removed slow pow expressions

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)))))