\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 10.6 s
Input Error: 16.8
Output Error: 11.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x} & \text{when } \varepsilon \le -8.263889f-09 \\ \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right) & \text{when } \varepsilon \le 5.412501f-10 \\ \frac{\sin \left(x + \varepsilon\right)}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - \frac{1}{\cot x} & \text{otherwise} \end{cases}\)

    if eps < -8.263889f-09

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.4
    2. Using strategy rm
      14.4
    3. Applied tan-cotan to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
      14.2
    4. Using strategy rm
      14.2
    5. Applied tan-quot to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
      14.3
    6. Using strategy rm
      14.3
    7. Applied sin-sum to get
      \[\frac{\color{red}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \frac{\color{blue}{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}\]
      12.7

    if -8.263889f-09 < eps < 5.412501f-10

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      20.9
    2. Applied taylor to get
      \[\tan \left(x + \varepsilon\right) - \tan x \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)\]
      10.0
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)}\]
      10.0

    if 5.412501f-10 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.2
    2. Using strategy rm
      14.2
    3. Applied tan-cotan to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
      14.0
    4. Using strategy rm
      14.0
    5. Applied tan-quot to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
      14.1
    6. Using strategy rm
      14.1
    7. Applied cos-sum to get
      \[\frac{\sin \left(x + \varepsilon\right)}{\color{red}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x} \leadsto \frac{\sin \left(x + \varepsilon\right)}{\color{blue}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}} - \frac{1}{\cot x}\]
      12.6
  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)))))