Average Error: 37.4 → 13.6
Time: 1.3m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot {\varepsilon}^{2} + \left({x}^{2} \cdot {\varepsilon}^{3} + \varepsilon\right) \le -4.074291138435494 \cdot 10^{-15}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{if}\;x \cdot {\varepsilon}^{2} + \left({x}^{2} \cdot {\varepsilon}^{3} + \varepsilon\right) \le 1.3658761349758168 \cdot 10^{-42}:\\ \;\;\;\;x \cdot {\varepsilon}^{2} + \left({x}^{2} \cdot {\varepsilon}^{3} + \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \sin x\right) + \left(\sin x \cdot \tan x\right) \cdot \tan \varepsilon}{\cos x - \tan \varepsilon \cdot \left(\cos x \cdot \tan x\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.4
Target15.2
Herbie13.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ (* x (pow eps 2)) (+ (* (pow x 2) (pow eps 3)) eps)) < -4.074291138435494e-15

    1. Initial program 36.5

      \[\tan \left(x + \varepsilon\right) - \tan x\]
    2. Using strategy rm

    3. Applied tan-sum9.8

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
    4. Using strategy rm

    5. Applied tan-quot9.8

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}} - \tan x\]

    6. Applied associate-*r/9.8

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}} - \tan x\]

    if -4.074291138435494e-15 < (+ (* x (pow eps 2)) (+ (* (pow x 2) (pow eps 3)) eps)) < 1.3658761349758168e-42

    1. Initial program 40.4

      \[\tan \left(x + \varepsilon\right) - \tan x\]

    2. Taylor expanded around 0 16.1

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left({x}^{2} \cdot {\varepsilon}^{3} + \varepsilon\right)}\]

    if 1.3658761349758168e-42 < (+ (* x (pow eps 2)) (+ (* (pow x 2) (pow eps 3)) eps))

    1. Initial program 35.6

      \[\tan \left(x + \varepsilon\right) - \tan x\]
    2. Using strategy rm

    3. Applied tan-sum14.4

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
    4. Using strategy rm

    5. Applied add-cbrt-cube14.4

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\sqrt[3]{\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon}}} - \tan x\]

    6. Applied add-cbrt-cube14.4

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \tan x}} \cdot \sqrt[3]{\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon}} - \tan x\]

    7. Applied cbrt-unprod14.4

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\sqrt[3]{\left(\left(\tan x \cdot \tan x\right) \cdot \tan x\right) \cdot \left(\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon\right)}}} - \tan x\]

    8. Applied simplify14.4

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\color{blue}{{\left(\tan \varepsilon \cdot \tan x\right)}^{3}}}} - \tan x\]
    9. Using strategy rm

    10. Applied tan-quot14.5

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan \varepsilon \cdot \tan x\right)}^{3}}} - \color{blue}{\frac{\sin x}{\cos x}}\]

    11. Applied frac-sub14.6

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \sqrt[3]{{\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right) \cdot \sin x}{\left(1 - \sqrt[3]{{\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right) \cdot \cos x}}\]

    12. Applied simplify13.3

      \[\leadsto \frac{\color{blue}{\left(\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \sin x\right) + \left(\sin x \cdot \tan x\right) \cdot \tan \varepsilon}}{\left(1 - \sqrt[3]{{\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right) \cdot \cos x}\]

    13. Applied simplify13.3

      \[\leadsto \frac{\left(\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \sin x\right) + \left(\sin x \cdot \tan x\right) \cdot \tan \varepsilon}{\color{blue}{\cos x - \tan \varepsilon \cdot \left(\cos x \cdot \tan x\right)}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2020178 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))