\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 34.8 s
Input Error: 17.1
Output Error: 7.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \tan \left(x + \varepsilon\right) - {\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\frac{1}{\cos x}}\right)}^3 & \text{when } \varepsilon \le -0.0035937247f0 \\ \left(\left(\frac{\varepsilon}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^2}{\cos x} \cdot \sin x\right) + \frac{{\left(\sin x\right)}^3}{\frac{{\left(\cos x\right)}^3}{{\varepsilon}^2}}\right) + \left(\left(\frac{1}{3} \cdot {\varepsilon}^3 + \varepsilon\right) + \left(\frac{\left(\frac{4}{3} \cdot \varepsilon\right) \cdot {\varepsilon}^2}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^3}{\frac{{\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}}\right)\right) & \text{when } \varepsilon \le 0.3662591f0 \\ \tan \left(x + \varepsilon\right) - {\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\frac{1}{\cos x}}\right)}^3 & \text{otherwise} \end{cases}\)

    if eps < -0.0035937247f0 or 0.3662591f0 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.4
    2. Using strategy rm
      14.4
    3. Applied tan-quot to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
      14.3
    4. Using strategy rm
      14.3
    5. Applied div-inv to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\frac{\sin x}{\cos x}} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\sin x \cdot \frac{1}{\cos x}}\]
      14.3
    6. Using strategy rm
      14.3
    7. Applied add-cube-cbrt to get
      \[\tan \left(x + \varepsilon\right) - \sin x \cdot \color{red}{\frac{1}{\cos x}} \leadsto \tan \left(x + \varepsilon\right) - \sin x \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{\cos x}}\right)}^3}\]
      14.3
    8. Applied add-cube-cbrt to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\sin x} \cdot {\left(\sqrt[3]{\frac{1}{\cos x}}\right)}^3 \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{{\left(\sqrt[3]{\sin x}\right)}^3} \cdot {\left(\sqrt[3]{\frac{1}{\cos x}}\right)}^3\]
      14.3
    9. Applied cube-unprod to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{{\left(\sqrt[3]{\sin x}\right)}^3 \cdot {\left(\sqrt[3]{\frac{1}{\cos x}}\right)}^3} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\frac{1}{\cos x}}\right)}^3}\]
      14.3

    if -0.0035937247f0 < eps < 0.3662591f0

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      19.9
    2. Using strategy rm
      19.9
    3. Applied add-cube-cbrt to get
      \[\color{red}{\tan \left(x + \varepsilon\right) - \tan x} \leadsto \color{blue}{{\left(\sqrt[3]{\tan \left(x + \varepsilon\right) - \tan x}\right)}^3}\]
      20.1
    4. Applied taylor to get
      \[{\left(\sqrt[3]{\tan \left(x + \varepsilon\right) - \tan x}\right)}^3 \leadsto \frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)\]
      0.2
    5. 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
    6. 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
    7. 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)))))