\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 21.5 s
Input Error: 16.8
Output Error: 11.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \tan \left(x + \varepsilon\right) - {\left(\sqrt[3]{\tan x}\right)}^3 & \text{when } \varepsilon \le -0.00010876831f0 \\ (\left({x}^2\right) * \left({\varepsilon}^3\right) + \varepsilon)_* & \text{when } \varepsilon \le 2.7455978f-06 \\ \frac{\frac{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^3\right)}^2 - {\left({\left(\tan x\right)}^3\right)}^2}{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^2\right)}^2 + \left({\left({\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}^2 + {\left(\tan \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}}{\tan x + \tan \left(x + \varepsilon\right)} & \text{otherwise} \end{cases}\)

    if eps < -0.00010876831f0

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.0
    2. Using strategy rm
      14.0
    3. Applied add-cube-cbrt to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{{\left(\sqrt[3]{\tan x}\right)}^3}\]
      13.9

    if -0.00010876831f0 < eps < 2.7455978f-06

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      20.3
    2. Applied taylor to get
      \[\tan \left(x + \varepsilon\right) - \tan x \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)\]
      10.2
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)}\]
      10.2
    4. Applied taylor to get
      \[\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right) \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + 0\right)\]
      7.3
    5. Taylor expanded around inf to get
      \[\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + \color{red}{0}\right) \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + \color{blue}{0}\right)\]
      7.3
    6. Applied simplify to get
      \[\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + 0\right) \leadsto (\left(x \cdot x\right) * \left({\varepsilon}^3\right) + \varepsilon)_*\]
      7.3
    7. Applied final simplification
    8. Applied simplify to get
      \[\color{red}{(\left(x \cdot x\right) * \left({\varepsilon}^3\right) + \varepsilon)_*} \leadsto \color{blue}{(\left({x}^2\right) * \left({\varepsilon}^3\right) + \varepsilon)_*}\]
      7.3

    if 2.7455978f-06 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.4
    2. Using strategy rm
      14.4
    3. Applied log1p-expm1-u to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\log_* (1 + (e^{\tan x} - 1)^*)}\]
      14.4
    4. Using strategy rm
      14.4
    5. Applied flip-- to get
      \[\color{red}{\tan \left(x + \varepsilon\right) - \log_* (1 + (e^{\tan x} - 1)^*)} \leadsto \color{blue}{\frac{{\left(\tan \left(x + \varepsilon\right)\right)}^2 - {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2}{\tan \left(x + \varepsilon\right) + \log_* (1 + (e^{\tan x} - 1)^*)}}\]
      14.4
    6. Applied simplify to get
      \[\frac{{\left(\tan \left(x + \varepsilon\right)\right)}^2 - {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2}{\color{red}{\tan \left(x + \varepsilon\right) + \log_* (1 + (e^{\tan x} - 1)^*)}} \leadsto \frac{{\left(\tan \left(x + \varepsilon\right)\right)}^2 - {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2}{\color{blue}{\tan x + \tan \left(x + \varepsilon\right)}}\]
      14.4
    7. Using strategy rm
      14.4
    8. Applied flip3-- to get
      \[\frac{\color{red}{{\left(\tan \left(x + \varepsilon\right)\right)}^2 - {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2}}{\tan x + \tan \left(x + \varepsilon\right)} \leadsto \frac{\color{blue}{\frac{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^2\right)}^{3} - {\left({\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}^{3}}{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^2\right)}^2 + \left({\left({\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}^2 + {\left(\tan \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}}}{\tan x + \tan \left(x + \varepsilon\right)}\]
      14.7
    9. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^2\right)}^{3} - {\left({\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}^{3}}}{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^2\right)}^2 + \left({\left({\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}^2 + {\left(\tan \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}}{\tan x + \tan \left(x + \varepsilon\right)} \leadsto \frac{\frac{\color{blue}{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^3\right)}^2 - {\left({\left(\tan x\right)}^3\right)}^2}}{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^2\right)}^2 + \left({\left({\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}^2 + {\left(\tan \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}}{\tan x + \tan \left(x + \varepsilon\right)}\]
      14.5
  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)))))