Average Error: 37.3 → 14.5
Time: 1.6m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le -3.225476105605433 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \left({1}^{3} + {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)\right) \cdot \cos x - \sin x \cdot \left(\left(1 + \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) - \tan \varepsilon \cdot \tan x\right)\right) \cdot \left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)}{\cos x \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(1 - \tan \varepsilon \cdot \tan x\right)\right) \cdot \left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)}\\ \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le 2.281937818813476 \cdot 10^{-27}:\\ \;\;\;\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 + \tan \varepsilon \cdot \tan x\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos x \cdot \left(\cos x \cdot \cos \varepsilon\right)}}\right) \cdot \left(\left(1 + \tan \varepsilon \cdot \tan x\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos x \cdot \left(\cos x \cdot \cos \varepsilon\right)}}\right) - \tan x \cdot \tan x}{\left(1 + \tan \varepsilon \cdot \tan x\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos x \cdot \left(\cos x \cdot \cos \varepsilon\right)}} + \tan x}\\ \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

Original37.3
Target15.1
Herbie14.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < -3.225476105605433e-18

    1. Initial program 37.5

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

    3. Applied tan-sum10.7

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

    5. Applied flip--10.8

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

    6. Applied associate-/r/10.8

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

    7. Applied simplify10.8

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

    9. Applied tan-quot10.8

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

    10. Applied flip3-+10.8

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

    11. Applied frac-times10.8

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

    12. Applied frac-sub10.9

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

    13. Applied simplify10.9

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

    if -3.225476105605433e-18 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < 2.281937818813476e-27

    1. Initial program 39.4

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

    2. Taylor expanded around 0 16.1

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

    if 2.281937818813476e-27 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x)))

    1. Initial program 35.8

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

    3. Applied tan-sum14.7

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

    5. Applied flip--14.7

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

    6. Applied associate-/r/14.7

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

    7. Applied simplify14.7

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

    9. Applied tan-quot14.7

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

    10. Applied associate-*r/14.7

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

    11. Applied tan-quot14.7

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

    12. Applied tan-quot14.7

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

    13. Applied frac-times14.7

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

    14. Applied frac-times14.7

      \[\leadsto \frac{\tan \varepsilon + \tan x}{1 - \color{blue}{\frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\tan \varepsilon \cdot \sin x\right)}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \cos x}}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\]
    15. Using strategy rm

    16. Applied flip--14.8

      \[\leadsto \color{blue}{\frac{\left(\frac{\tan \varepsilon + \tan x}{1 - \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\tan \varepsilon \cdot \sin x\right)}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \cos x}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\frac{\tan \varepsilon + \tan x}{1 - \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\tan \varepsilon \cdot \sin x\right)}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \cos x}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right) - \tan x \cdot \tan x}{\frac{\tan \varepsilon + \tan x}{1 - \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\tan \varepsilon \cdot \sin x\right)}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \cos x}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) + \tan x}}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify14.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le -3.225476105605433 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \left({1}^{3} + {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)\right) \cdot \cos x - \sin x \cdot \left(\left(1 + \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) - \tan \varepsilon \cdot \tan x\right)\right) \cdot \left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)}{\cos x \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(1 - \tan \varepsilon \cdot \tan x\right)\right) \cdot \left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)}\\ \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le 2.281937818813476 \cdot 10^{-27}:\\ \;\;\;\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 + \tan \varepsilon \cdot \tan x\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos x \cdot \left(\cos x \cdot \cos \varepsilon\right)}}\right) \cdot \left(\left(1 + \tan \varepsilon \cdot \tan x\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos x \cdot \left(\cos x \cdot \cos \varepsilon\right)}}\right) - \tan x \cdot \tan x}{\left(1 + \tan \varepsilon \cdot \tan x\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos x \cdot \left(\cos x \cdot \cos \varepsilon\right)}} + \tan x}\\ \end{array}}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018198 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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