2tan (problem 3.3.2)

Average Error: 36.6 → 15.1

Time: 7.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.21724289424431991 \cdot 10^{-76}:\\ \;\;\;\;\frac{\left(-\sin x\right) \cdot \left(\left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right) - \tan x \cdot \tan \varepsilon\right) + \left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}} \cdot \left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)\right) \cdot \cos x}{\cos x \cdot \left(\left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right) - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \le 1.0191077428645162 \cdot 10^{-28}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original	36.6
Target	14.9
Herbie	15.1

\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

1. Split input into 3 regimes

2. if eps < -9.21724289424431991e-76

   1. Initial program 30.2

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

   3. Applied tan-sum 5.8

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

   5. Applied flip-- 5.8

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

   6. Applied associate-/r/ 5.8

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

   7. Simplified 5.8

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

   9. Applied pow1 5.8

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

   10. Applied pow1 5.8

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

   11. Applied pow-prod-down 5.8

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

   12. Applied pow1 5.8

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

   13. Applied pow1 5.8

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

   14. Applied pow-prod-down 5.8

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

   15. Applied pow-prod-up 5.8

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

   16. Simplified 5.8

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{\color{blue}{2}}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\]
   17. Using strategy rm

   18. Applied tan-quot 5.8

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]

   19. Applied flip3-+ 5.9

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}} \cdot \color{blue}{\frac{{1}^{3} + {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) - 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \frac{\sin x}{\cos x}\]

   20. Applied associate-*r/ 5.9

       \[\leadsto \color{blue}{\frac{\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}} \cdot \left({1}^{3} + {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) - 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \frac{\sin x}{\cos x}\]

   21. Applied frac-sub 5.9

       \[\leadsto \color{blue}{\frac{\left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}} \cdot \left({1}^{3} + {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)\right) \cdot \cos x - \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) - 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) \cdot \sin x}{\left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) - 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) \cdot \cos x}}\]

   22. Simplified 5.9

       \[\leadsto \frac{\color{blue}{\left(-\sin x\right) \cdot \left(\left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right) - \tan x \cdot \tan \varepsilon\right) + \left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}} \cdot \left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)\right) \cdot \cos x}}{\left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) - 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) \cdot \cos x}\]

   23. Simplified 5.9

       \[\leadsto \frac{\left(-\sin x\right) \cdot \left(\left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right) - \tan x \cdot \tan \varepsilon\right) + \left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}} \cdot \left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)\right) \cdot \cos x}{\color{blue}{\cos x \cdot \left(\left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right) - \tan x \cdot \tan \varepsilon\right)}}\]

   if -9.21724289424431991e-76 < eps < 1.0191077428645162e-28

   1. Initial program 45.9

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

   2. Taylor expanded around 0 30.4

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

   3. Simplified 30.2

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

   if 1.0191077428645162e-28 < eps

   1. Initial program 29.3

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

   3. Applied tan-sum 1.9

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

   5. Applied flip-- 2.0

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

   6. Applied associate-/r/ 1.9

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

   7. Simplified 1.9

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

   9. Applied pow1 1.9

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

   10. Applied pow1 1.9

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

   11. Applied pow-prod-down 1.9

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

   12. Applied pow1 1.9

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

   13. Applied pow1 1.9

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

   14. Applied pow-prod-down 1.9

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

   15. Applied pow-prod-up 1.9

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

   16. Simplified 1.9

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{\color{blue}{2}}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.21724289424431991 \cdot 10^{-76}:\\ \;\;\;\;\frac{\left(-\sin x\right) \cdot \left(\left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right) - \tan x \cdot \tan \varepsilon\right) + \left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}} \cdot \left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)\right) \cdot \cos x}{\cos x \cdot \left(\left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right) - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \le 1.0191077428645162 \cdot 10^{-28}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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