\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 46.8 s
Input Error: 40.8
Output Error: 16.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - \tan x & \text{when } \varepsilon \le -3.990253093358043 \cdot 10^{-26} \\ \frac{(\left(\sin x\right) * \left(\cos \varepsilon \cdot \cos x - \cos \left(\varepsilon + x\right)\right) + \left(\sin \varepsilon \cdot {\left(\cos \left(\frac{1}{x}\right)\right)}^2\right))_*}{\cos \left(\varepsilon + x\right) \cdot \cos x} & \text{when } \varepsilon \le -1.6766537535852387 \cdot 10^{-273} \\ \frac{(\left(\sin x\right) * \left(\varepsilon \cdot x\right) + \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos x\right)\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x} & \text{when } \varepsilon \le 2.2197886996052214 \cdot 10^{-176} \\ \frac{(\left(\sin x\right) * \left(\cos \varepsilon \cdot \cos x - \cos \left(\varepsilon + x\right)\right) + \left(\sin \varepsilon \cdot {\left(\cos \left(\frac{1}{x}\right)\right)}^2\right))_*}{\cos \left(\varepsilon + x\right) \cdot \cos x} & \text{when } \varepsilon \le 2.1700823787981142 \cdot 10^{-103} \\ \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon} - \tan x & \text{otherwise} \end{cases}\)

    if eps < -3.990253093358043e-26 or 2.1700823787981142e-103 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      30.4
    2. Using strategy rm
      30.4
    3. Applied tan-quot to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x\]
      30.4
    4. Using strategy rm
      30.4
    5. Applied sin-sum to get
      \[\frac{\color{red}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \tan x \leadsto \frac{\color{blue}{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}}{\cos \left(x + \varepsilon\right)} - \tan x\]
      28.6
    6. Using strategy rm
      28.6
    7. Applied cos-sum to get
      \[\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\color{red}{\cos \left(x + \varepsilon\right)}} - \tan x \leadsto \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\color{blue}{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}} - \tan x\]
      5.6

    if -3.990253093358043e-26 < eps < -1.6766537535852387e-273 or 2.2197886996052214e-176 < eps < 2.1700823787981142e-103

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      58.0
    2. Using strategy rm
      58.0
    3. Applied tan-quot to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x\]
      58.1
    4. Using strategy rm
      58.1
    5. Applied sin-sum to get
      \[\frac{\color{red}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \tan x \leadsto \frac{\color{blue}{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}}{\cos \left(x + \varepsilon\right)} - \tan x\]
      58.1
    6. Using strategy rm
      58.1
    7. Applied tan-quot to get
      \[\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right)} - \color{red}{\tan x} \leadsto \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right)} - \color{blue}{\frac{\sin x}{\cos x}}\]
      58.0
    8. Applied frac-sub to get
      \[\color{red}{\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \cos x - \cos \left(x + \varepsilon\right) \cdot \sin x}{\cos \left(x + \varepsilon\right) \cdot \cos x}}\]
      58.0
    9. Applied simplify to get
      \[\frac{\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \cos x - \cos \left(x + \varepsilon\right) \cdot \sin x}}{\cos \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{(\left(\sin x\right) * \left(\cos \varepsilon \cdot \cos x - \cos \left(x + \varepsilon\right)\right) + \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos x\right)\right))_*}}{\cos \left(x + \varepsilon\right) \cdot \cos x}\]
      51.6
    10. Applied taylor to get
      \[\frac{(\left(\sin x\right) * \left(\cos \varepsilon \cdot \cos x - \cos \left(x + \varepsilon\right)\right) + \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos x\right)\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{(\left(\sin x\right) * \left(\cos \varepsilon \cdot \cos x - \cos \left(x + \varepsilon\right)\right) + \left(\sin \varepsilon \cdot {\left(\cos \left(\frac{1}{x}\right)\right)}^2\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x}\]
      50.8
    11. Taylor expanded around inf to get
      \[\frac{(\left(\sin x\right) * \left(\cos \varepsilon \cdot \cos x - \cos \left(x + \varepsilon\right)\right) + \left(\sin \varepsilon \cdot \color{red}{{\left(\cos \left(\frac{1}{x}\right)\right)}^2}\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{(\left(\sin x\right) * \left(\cos \varepsilon \cdot \cos x - \cos \left(x + \varepsilon\right)\right) + \left(\sin \varepsilon \cdot \color{blue}{{\left(\cos \left(\frac{1}{x}\right)\right)}^2}\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x}\]
      50.8
    12. Applied simplify to get
      \[\frac{(\left(\sin x\right) * \left(\cos \varepsilon \cdot \cos x - \cos \left(x + \varepsilon\right)\right) + \left(\sin \varepsilon \cdot {\left(\cos \left(\frac{1}{x}\right)\right)}^2\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{(\left(\sin x\right) * \left(\cos x \cdot \cos \varepsilon - \cos \left(x + \varepsilon\right)\right) + \left(\cos \left(\frac{1}{x}\right) \cdot \left(\sin \varepsilon \cdot \cos \left(\frac{1}{x}\right)\right)\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x}\]
      50.8
    13. Applied final simplification
    14. Applied simplify to get
      \[\color{red}{\frac{(\left(\sin x\right) * \left(\cos x \cdot \cos \varepsilon - \cos \left(x + \varepsilon\right)\right) + \left(\cos \left(\frac{1}{x}\right) \cdot \left(\sin \varepsilon \cdot \cos \left(\frac{1}{x}\right)\right)\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x}} \leadsto \color{blue}{\frac{(\left(\sin x\right) * \left(\cos \varepsilon \cdot \cos x - \cos \left(\varepsilon + x\right)\right) + \left(\sin \varepsilon \cdot {\left(\cos \left(\frac{1}{x}\right)\right)}^2\right))_*}{\cos \left(\varepsilon + x\right) \cdot \cos x}}\]
      50.8

    if -1.6766537535852387e-273 < eps < 2.2197886996052214e-176

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      54.0
    2. Using strategy rm
      54.0
    3. Applied tan-quot to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x\]
      54.0
    4. Using strategy rm
      54.0
    5. Applied sin-sum to get
      \[\frac{\color{red}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \tan x \leadsto \frac{\color{blue}{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}}{\cos \left(x + \varepsilon\right)} - \tan x\]
      54.0
    6. Using strategy rm
      54.0
    7. Applied tan-quot to get
      \[\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right)} - \color{red}{\tan x} \leadsto \frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right)} - \color{blue}{\frac{\sin x}{\cos x}}\]
      54.0
    8. Applied frac-sub to get
      \[\color{red}{\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \cos x - \cos \left(x + \varepsilon\right) \cdot \sin x}{\cos \left(x + \varepsilon\right) \cdot \cos x}}\]
      54.0
    9. Applied simplify to get
      \[\frac{\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) \cdot \cos x - \cos \left(x + \varepsilon\right) \cdot \sin x}}{\cos \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{(\left(\sin x\right) * \left(\cos \varepsilon \cdot \cos x - \cos \left(x + \varepsilon\right)\right) + \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos x\right)\right))_*}}{\cos \left(x + \varepsilon\right) \cdot \cos x}\]
      30.6
    10. Applied taylor to get
      \[\frac{(\left(\sin x\right) * \left(\cos \varepsilon \cdot \cos x - \cos \left(x + \varepsilon\right)\right) + \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos x\right)\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{(\left(\sin x\right) * \left(\varepsilon \cdot x\right) + \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos x\right)\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x}\]
      0.3
    11. Taylor expanded around 0 to get
      \[\frac{(\left(\sin x\right) * \color{red}{\left(\varepsilon \cdot x\right)} + \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos x\right)\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{(\left(\sin x\right) * \color{blue}{\left(\varepsilon \cdot x\right)} + \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos x\right)\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x}\]
      0.3
    12. Applied simplify to get
      \[\frac{(\left(\sin x\right) * \left(\varepsilon \cdot x\right) + \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos x\right)\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{(\left(\sin x\right) * \left(\varepsilon \cdot x\right) + \left(\sin \varepsilon \cdot \left(\cos x \cdot \cos x\right)\right))_*}{\cos \left(x + \varepsilon\right) \cdot \cos x}\]
      0.3
    13. Applied final simplification
  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)))))