\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 49.9 s
Input Error: 34.2
Output Error: 13.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{(\left(\sin x\right) * \left((\frac{1}{3} * \left({\varepsilon}^3\right) + \varepsilon)_*\right) + \left((\frac{4}{3} * \left(\frac{{\varepsilon}^3}{\sin x} \cdot \left(\cos x \cdot \cos x\right)\right) + \left((\left({\left(\frac{\varepsilon}{\sin x}\right)}^3\right) * \left({\left(\cos x\right)}^{4}\right) + \left(\frac{\cos x \cdot \cos x}{\frac{\sin x}{\varepsilon}}\right))_*\right))_*\right))_* - (\left(\cos x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left({\left(\frac{\varepsilon}{\sin x}\right)}^2 \cdot {\left(\cos x\right)}^3\right))_*}{\cos x \cdot \cot \left(x + \varepsilon\right)} & \text{when } x \le -2.6247031050557197 \cdot 10^{-37} \\ \log_* (1 + (e^{\tan \left(x + \varepsilon\right)} - 1)^*) - \tan x & \text{when } x \le 5.484441058442028 \cdot 10^{-42} \\ \frac{(\left(\sin x\right) * \left((\frac{1}{3} * \left({\varepsilon}^3\right) + \varepsilon)_*\right) + \left((\frac{4}{3} * \left(\frac{{\varepsilon}^3}{\sin x} \cdot \left(\cos x \cdot \cos x\right)\right) + \left((\left({\left(\frac{\varepsilon}{\sin x}\right)}^3\right) * \left({\left(\cos x\right)}^{4}\right) + \left(\frac{\cos x \cdot \cos x}{\frac{\sin x}{\varepsilon}}\right))_*\right))_*\right))_* - (\left(\cos x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left({\left(\frac{\varepsilon}{\sin x}\right)}^2 \cdot {\left(\cos x\right)}^3\right))_*}{\cos x \cdot \cot \left(x + \varepsilon\right)} & \text{otherwise} \end{cases}\)

    if x < -2.6247031050557197e-37 or 5.484441058442028e-42 < x

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      60.8
    2. Using strategy rm
      60.8
    3. Applied tan-quot to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
      60.9
    4. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{\sin x}{\cos x}\]
      61.0
    5. Applied frac-sub to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}}\]
      61.0
    6. Applied simplify to get
      \[\frac{\color{red}{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
      61.0
    7. Using strategy rm
      61.0
    8. Applied add-cube-cbrt to get
      \[\frac{\color{red}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}\right)}^3}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
      61.0
    9. Applied taylor to get
      \[\frac{{\left(\sqrt[3]{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}\right)}^3}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(\varepsilon \cdot \sin x + \left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \left(\frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{3}} + \frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{\sin x}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^2} + {\varepsilon}^2 \cdot \cos x\right)}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
      14.8
    10. Taylor expanded around 0 to get
      \[\frac{\color{red}{\left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(\varepsilon \cdot \sin x + \left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \left(\frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{3}} + \frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{\sin x}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^2} + {\varepsilon}^2 \cdot \cos x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(\varepsilon \cdot \sin x + \left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \left(\frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{3}} + \frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{\sin x}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^2} + {\varepsilon}^2 \cdot \cos x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
      14.8
    11. Applied simplify to get
      \[\color{red}{\frac{\left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(\varepsilon \cdot \sin x + \left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \left(\frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{3}} + \frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{\sin x}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^2} + {\varepsilon}^2 \cdot \cos x\right)}{\cot \left(x + \varepsilon\right) \cdot \cos x}} \leadsto \color{blue}{\frac{(\left(\sin x\right) * \left((\frac{1}{3} * \left({\varepsilon}^3\right) + \varepsilon)_*\right) + \left((\frac{4}{3} * \left(\frac{{\varepsilon}^3}{\sin x} \cdot \left(\cos x \cdot \cos x\right)\right) + \left((\left(\frac{{\varepsilon}^3}{{\left(\sin x\right)}^3}\right) * \left({\left(\cos x\right)}^{4}\right) + \left(\left(\cos x \cdot \cos x\right) \cdot \frac{\varepsilon}{\sin x}\right))_*\right))_*\right))_* - (\left(\cos x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left(\left(\frac{\varepsilon}{\sin x} \cdot \frac{\varepsilon}{\sin x}\right) \cdot {\left(\cos x\right)}^3\right))_*}{\cos x \cdot \cot \left(x + \varepsilon\right)}}\]
      14.8
    12. Applied simplify to get
      \[\frac{\color{red}{(\left(\sin x\right) * \left((\frac{1}{3} * \left({\varepsilon}^3\right) + \varepsilon)_*\right) + \left((\frac{4}{3} * \left(\frac{{\varepsilon}^3}{\sin x} \cdot \left(\cos x \cdot \cos x\right)\right) + \left((\left(\frac{{\varepsilon}^3}{{\left(\sin x\right)}^3}\right) * \left({\left(\cos x\right)}^{4}\right) + \left(\left(\cos x \cdot \cos x\right) \cdot \frac{\varepsilon}{\sin x}\right))_*\right))_*\right))_* - (\left(\cos x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left(\left(\frac{\varepsilon}{\sin x} \cdot \frac{\varepsilon}{\sin x}\right) \cdot {\left(\cos x\right)}^3\right))_*}}{\cos x \cdot \cot \left(x + \varepsilon\right)} \leadsto \frac{\color{blue}{(\left(\sin x\right) * \left((\frac{1}{3} * \left({\varepsilon}^3\right) + \varepsilon)_*\right) + \left((\frac{4}{3} * \left(\frac{{\varepsilon}^3}{\sin x} \cdot \left(\cos x \cdot \cos x\right)\right) + \left((\left({\left(\frac{\varepsilon}{\sin x}\right)}^3\right) * \left({\left(\cos x\right)}^{4}\right) + \left(\frac{\cos x \cdot \cos x}{\frac{\sin x}{\varepsilon}}\right))_*\right))_*\right))_* - (\left(\cos x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left({\left(\frac{\varepsilon}{\sin x}\right)}^2 \cdot {\left(\cos x\right)}^3\right))_*}}{\cos x \cdot \cot \left(x + \varepsilon\right)}\]
      14.8

    if -2.6247031050557197e-37 < x < 5.484441058442028e-42

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      11.8
    2. Using strategy rm
      11.8
    3. Applied log1p-expm1-u to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\log_* (1 + (e^{\tan \left(x + \varepsilon\right)} - 1)^*)} - \tan x\]
      12.5
  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)))))