\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 41.9 s
Input Error: 36.7
Output Error: 24.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{1}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} \cdot \sin \left(x + \varepsilon\right) - \tan x & \text{when } \varepsilon \le -3.2567851254892807 \cdot 10^{-46} \\ \left(\varepsilon + \left(x \cdot x\right) \cdot {\varepsilon}^3\right) + {\varepsilon}^{4} \cdot {x}^3 & \text{when } \varepsilon \le 2.380286695042028 \cdot 10^{-22} \\ \left(\frac{\sin x \cdot \cos \varepsilon}{\cos \left(\varepsilon + x\right)} + \frac{\cos x \cdot \sin \varepsilon}{\cos \left(\varepsilon + x\right)}\right) - \tan x & \text{otherwise} \end{cases}\)

    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 cotan-quot to get
      \[\frac{1}{\color{red}{\cot \left(x + \varepsilon\right)}} - \tan x \leadsto \frac{1}{\color{blue}{\frac{\cos \left(x + \varepsilon\right)}{\sin \left(x + \varepsilon\right)}}} - \tan x\]
      30.6
    6. Applied associate-/r/ to get
      \[\color{red}{\frac{1}{\frac{\cos \left(x + \varepsilon\right)}{\sin \left(x + \varepsilon\right)}}} - \tan x \leadsto \color{blue}{\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \sin \left(x + \varepsilon\right)} - \tan x\]
      30.5
    7. Using strategy rm
      30.5
    8. Applied cos-sum to get
      \[\frac{1}{\color{red}{\cos \left(x + \varepsilon\right)}} \cdot \sin \left(x + \varepsilon\right) - \tan x \leadsto \frac{1}{\color{blue}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}} \cdot \sin \left(x + \varepsilon\right) - \tan x\]
      28.8

    if -3.2567851254892807e-46 < eps < 2.380286695042028e-22

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      45.2
    2. Applied taylor to get
      \[\tan \left(x + \varepsilon\right) - \tan x \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)\]
      18.5
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)}\]
      18.5
    4. Applied simplify to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\left(\varepsilon + \left(x \cdot x\right) \cdot {\varepsilon}^3\right) + {\varepsilon}^{4} \cdot {x}^3}\]
      18.5

    if 2.380286695042028e-22 < 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
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
      30.1
    4. Using strategy rm
      30.1
    5. Applied cotan-quot to get
      \[\frac{1}{\color{red}{\cot \left(x + \varepsilon\right)}} - \tan x \leadsto \frac{1}{\color{blue}{\frac{\cos \left(x + \varepsilon\right)}{\sin \left(x + \varepsilon\right)}}} - \tan x\]
      30.1
    6. Applied associate-/r/ to get
      \[\color{red}{\frac{1}{\frac{\cos \left(x + \varepsilon\right)}{\sin \left(x + \varepsilon\right)}}} - \tan x \leadsto \color{blue}{\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \sin \left(x + \varepsilon\right)} - \tan x\]
      30.1
    7. Using strategy rm
      30.1
    8. Applied sin-sum to get
      \[\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \color{red}{\sin \left(x + \varepsilon\right)} - \tan x \leadsto \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \tan x\]
      28.3
    9. Applied distribute-lft-in to get
      \[\color{red}{\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \tan x \leadsto \color{blue}{\left(\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\sin x \cdot \cos \varepsilon\right) + \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\cos x \cdot \sin \varepsilon\right)\right)} - \tan x\]
      28.3
    10. Applied simplify to get
      \[\left(\color{red}{\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\sin x \cdot \cos \varepsilon\right)} + \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\cos x \cdot \sin \varepsilon\right)\right) - \tan x \leadsto \left(\color{blue}{\frac{\sin x \cdot \cos \varepsilon}{\cos \left(\varepsilon + x\right)}} + \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\cos x \cdot \sin \varepsilon\right)\right) - \tan x\]
      28.3
    11. Applied simplify to get
      \[\left(\frac{\sin x \cdot \cos \varepsilon}{\cos \left(\varepsilon + x\right)} + \color{red}{\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\cos x \cdot \sin \varepsilon\right)}\right) - \tan x \leadsto \left(\frac{\sin x \cdot \cos \varepsilon}{\cos \left(\varepsilon + x\right)} + \color{blue}{\frac{\cos x \cdot \sin \varepsilon}{\cos \left(\varepsilon + x\right)}}\right) - \tan x\]
      28.2
  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)))))