\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 22.7 s
Input Error: 16.8
Output Error: 7.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \tan \left(x + \varepsilon\right) - {\left(\frac{\sqrt[3]{\sin x}}{\sqrt[3]{\cos x}}\right)}^3 & \text{when } \tan \left(x + \varepsilon\right) - \tan x \le -0.0013098668f0 \\ \left(\left(\frac{\varepsilon}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^2}{\cos x} \cdot \sin x\right) + \frac{{\left(\sin x\right)}^3}{\frac{{\left(\cos x\right)}^3}{{\varepsilon}^2}}\right) + \left(\left(\frac{1}{3} \cdot {\varepsilon}^3 + \varepsilon\right) + \left(\frac{\left(\frac{4}{3} \cdot \varepsilon\right) \cdot {\varepsilon}^2}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^3}{\frac{{\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}}\right)\right) & \text{when } \tan \left(x + \varepsilon\right) - \tan x \le 0.012468386f0 \\ \tan \left(x + \varepsilon\right) - {\left(\frac{\sqrt[3]{\sin x}}{\sqrt[3]{\cos x}}\right)}^3 & \text{otherwise} \end{cases}\)

    if (- (tan (+ x eps)) (tan x)) < -0.0013098668f0 or 0.012468386f0 < (- (tan (+ x eps)) (tan x))

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      10.8
    2. Using strategy rm
      10.8
    3. Applied tan-quot to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
      10.8
    4. Using strategy rm
      10.8
    5. Applied add-cube-cbrt to get
      \[\tan \left(x + \varepsilon\right) - \frac{\sin x}{\color{red}{\cos x}} \leadsto \tan \left(x + \varepsilon\right) - \frac{\sin x}{\color{blue}{{\left(\sqrt[3]{\cos x}\right)}^3}}\]
      10.8
    6. Applied add-cube-cbrt to get
      \[\tan \left(x + \varepsilon\right) - \frac{\color{red}{\sin x}}{{\left(\sqrt[3]{\cos x}\right)}^3} \leadsto \tan \left(x + \varepsilon\right) - \frac{\color{blue}{{\left(\sqrt[3]{\sin x}\right)}^3}}{{\left(\sqrt[3]{\cos x}\right)}^3}\]
      10.8
    7. Applied cube-undiv to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\frac{{\left(\sqrt[3]{\sin x}\right)}^3}{{\left(\sqrt[3]{\cos x}\right)}^3}} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{{\left(\frac{\sqrt[3]{\sin x}}{\sqrt[3]{\cos x}}\right)}^3}\]
      10.8

    if -0.0013098668f0 < (- (tan (+ x eps)) (tan x)) < 0.012468386f0

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      21.6
    2. Using strategy rm
      21.6
    3. Applied tan-cotan to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
      21.7
    4. Using strategy rm
      21.7
    5. Applied add-cube-cbrt to get
      \[\color{red}{\tan \left(x + \varepsilon\right) - \frac{1}{\cot x}} \leadsto \color{blue}{{\left(\sqrt[3]{\tan \left(x + \varepsilon\right) - \frac{1}{\cot x}}\right)}^3}\]
      21.8
    6. Applied taylor to get
      \[{\left(\sqrt[3]{\tan \left(x + \varepsilon\right) - \frac{1}{\cot x}}\right)}^3 \leadsto \frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)\]
      5.0
    7. Taylor expanded around 0 to get
      \[\color{red}{\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)} \leadsto \color{blue}{\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)}\]
      5.0
    8. Applied simplify to get
      \[\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right) \leadsto \left(\left(\frac{\varepsilon}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^2}{\cos x} \cdot \sin x\right) + \frac{{\left(\sin x\right)}^3}{\frac{{\left(\cos x\right)}^3}{{\varepsilon}^2}}\right) + \left(\left(\frac{1}{3} \cdot {\varepsilon}^3 + \varepsilon\right) + \left(\frac{\left(\frac{4}{3} \cdot \varepsilon\right) \cdot {\varepsilon}^2}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^3}{\frac{{\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}}\right)\right)\]
      5.0
    9. 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)))))