\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 51.2 s
Input Error: 37.5
Output Error: 14.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{1}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} \cdot \sin \left(x + \varepsilon\right) - \tan x & \text{when } \varepsilon \le -7.175399107131671 \cdot 10^{-12} \\ \left(\left(\frac{\varepsilon}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^2}{\cos x} \cdot \sin x\right) + \frac{{\left(\sin x\right)}^3}{\frac{{\left(\cos x\right)}^3}{{\varepsilon}^2}}\right) + \left(\left(\frac{1}{3} \cdot {\varepsilon}^3 + \varepsilon\right) + \left(\frac{\left(\frac{4}{3} \cdot \varepsilon\right) \cdot {\varepsilon}^2}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^3}{\frac{{\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}}\right)\right) & \text{when } \varepsilon \le 45926.76670985681 \\ \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \tan x & \text{otherwise} \end{cases}\)

    if eps < -7.175399107131671e-12

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      30.5
    2. Using strategy rm
      30.5
    3. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
      30.5
    4. Using strategy rm
      30.5
    5. Applied cotan-quot to get
      \[\frac{1}{\color{red}{\cot \left(x + \varepsilon\right)}} - \tan x \leadsto \frac{1}{\color{blue}{\frac{\cos \left(x + \varepsilon\right)}{\sin \left(x + \varepsilon\right)}}} - \tan x\]
      30.5
    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
      \[\frac{1}{\color{red}{\cos \left(x + \varepsilon\right)}} \cdot \sin \left(x + \varepsilon\right) - \tan x \leadsto \frac{1}{\color{blue}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}} \cdot \sin \left(x + \varepsilon\right) - \tan x\]
      28.6

    if -7.175399107131671e-12 < eps < 45926.76670985681

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      44.9
    2. Using strategy rm
      44.9
    3. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
      45.1
    4. Using strategy rm
      45.1
    5. Applied add-cube-cbrt to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x}\right)}^3}\]
      45.4
    6. Applied taylor to get
      \[{\left(\sqrt[3]{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x}\right)}^3 \leadsto \frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)\]
      0.2
    7. Taylor expanded around 0 to get
      \[\color{red}{\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)} \leadsto \color{blue}{\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)}\]
      0.2
    8. Applied simplify to get
      \[\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right) \leadsto \left(\left(\frac{\varepsilon}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^2}{\cos x} \cdot \sin x\right) + \frac{{\left(\sin x\right)}^3}{\frac{{\left(\cos x\right)}^3}{{\varepsilon}^2}}\right) + \left(\left(\frac{1}{3} \cdot {\varepsilon}^3 + \varepsilon\right) + \left(\frac{\left(\frac{4}{3} \cdot \varepsilon\right) \cdot {\varepsilon}^2}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^3}{\frac{{\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}}\right)\right)\]
      0.2
    9. Applied final simplification

    if 45926.76670985681 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      30.2
    2. Using strategy rm
      30.2
    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
    7. Using strategy rm
      30.2
    8. Applied sin-sum to get
      \[\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \color{red}{\sin \left(x + \varepsilon\right)} - \tan x \leadsto \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \tan x\]
      28.5
  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)))))