\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 27.0 s
Input Error: 36.6
Output Error: 25.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2 - {\left(\tan x\right)}^2}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x} & \text{when } \varepsilon \le -3.2567851254892807 \cdot 10^{-46} \\ \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right) & \text{when } \varepsilon \le 1.7814613637570238 \cdot 10^{-06} \\ \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \sin \left(x + \varepsilon\right) - \tan x & \text{otherwise} \end{cases}\)

    if eps < -3.2567851254892807e-46

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      30.6
    2. Using strategy rm
      30.6
    3. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
      30.6
    4. Using strategy rm
      30.6
    5. Applied flip-- to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x} \leadsto \color{blue}{\frac{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2 - {\left(\tan x\right)}^2}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}}\]
      30.7

    if -3.2567851254892807e-46 < eps < 1.7814613637570238e-06

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      44.5
    2. Applied taylor to get
      \[\tan \left(x + \varepsilon\right) - \tan x \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)\]
      18.1
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)}\]
      18.1

    if 1.7814613637570238e-06 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      30.1
    2. Using strategy rm
      30.1
    3. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
      30.2
    4. Using strategy rm
      30.2
    5. Applied cotan-quot to get
      \[\frac{1}{\color{red}{\cot \left(x + \varepsilon\right)}} - \tan x \leadsto \frac{1}{\color{blue}{\frac{\cos \left(x + \varepsilon\right)}{\sin \left(x + \varepsilon\right)}}} - \tan x\]
      30.2
    6. Applied associate-/r/ to get
      \[\color{red}{\frac{1}{\frac{\cos \left(x + \varepsilon\right)}{\sin \left(x + \varepsilon\right)}}} - \tan x \leadsto \color{blue}{\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \sin \left(x + \varepsilon\right)} - \tan x\]
      30.2
  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)))))