2tan (problem 3.3.2)

Average Error: 36.8 → 16.0

Time: 12.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5973706195668936 \cdot 10^{-61}:\\ \;\;\;\;\frac{\left(-\left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}\right)\right) \cdot \left(\sin x \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon - 1\right) + 1\right)\right) + \cos x \cdot \left(\left({\left(\tan x \cdot \tan \varepsilon\right)}^{3} + 1\right) \cdot \left(\tan x + \tan \varepsilon\right)\right)}{\cos x \cdot \left(\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon - 1\right) + 1\right) \cdot \left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}\right)\right)}\\ \mathbf{elif}\;\varepsilon \le 1.66269538116722889 \cdot 10^{-37}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) \cdot \cos x - \left(\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right) \cdot \sin x}{\left(\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	14.7
Herbie	16.0

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.5973706195668936e-61

   1. Initial program 29.7

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

   3. Applied tan-sum 5.1

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

   5. Applied flip-- 5.1

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

   6. Applied associate-/r/ 5.1

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

   7. Simplified 5.1

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

   9. Applied add-cbrt-cube 5.1

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

   10. Applied add-cbrt-cube 5.1

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

   11. Applied cbrt-unprod 5.1

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

   12. Applied add-cbrt-cube 5.2

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

   13. Applied add-cbrt-cube 5.2

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

   14. Applied cbrt-unprod 5.2

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

   15. Applied cbrt-unprod 5.1

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

   16. Simplified 5.1

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\color{blue}{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\]
   17. Using strategy rm

   18. Applied tan-quot 5.2

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]

   19. Applied flip3-+ 5.2

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}} \cdot \color{blue}{\frac{{1}^{3} + {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) - 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \frac{\sin x}{\cos x}\]

   20. Applied frac-times 5.2

       \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \left({1}^{3} + {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)}{\left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}\right) \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) - 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)}} - \frac{\sin x}{\cos x}\]

   21. Applied frac-sub 5.2

       \[\leadsto \color{blue}{\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \left({1}^{3} + {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)\right) \cdot \cos x - \left(\left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}\right) \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) - 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)\right) \cdot \sin x}{\left(\left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}\right) \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) - 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)\right) \cdot \cos x}}\]

   22. Simplified 5.2

       \[\leadsto \frac{\color{blue}{\left(-\left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}\right)\right) \cdot \left(\sin x \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon - 1\right) + 1\right)\right) + \cos x \cdot \left(\left({\left(\tan x \cdot \tan \varepsilon\right)}^{3} + 1\right) \cdot \left(\tan x + \tan \varepsilon\right)\right)}}{\left(\left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}\right) \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) - 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)\right) \cdot \cos x}\]

   23. Simplified 5.2

       \[\leadsto \frac{\left(-\left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}\right)\right) \cdot \left(\sin x \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon - 1\right) + 1\right)\right) + \cos x \cdot \left(\left({\left(\tan x \cdot \tan \varepsilon\right)}^{3} + 1\right) \cdot \left(\tan x + \tan \varepsilon\right)\right)}{\color{blue}{\cos x \cdot \left(\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon - 1\right) + 1\right) \cdot \left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}\right)\right)}}\]

   if -1.5973706195668936e-61 < eps < 1.66269538116722889e-37

   1. Initial program 46.9

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

   2. Taylor expanded around 0 32.7

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

   3. Simplified 32.5

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

   if 1.66269538116722889e-37 < eps

   1. Initial program 29.2

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

   3. Applied tan-sum 3.0

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

   5. Applied flip-- 3.0

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

   6. Applied associate-/r/ 3.1

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

   7. Simplified 3.1

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

   9. Applied tan-quot 3.1

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

   10. Applied flip-+ 3.1

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

   11. Applied frac-times 3.1

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

   12. Applied frac-sub 3.1

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

   13. Simplified 3.1

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

4. Final simplification 16.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5973706195668936 \cdot 10^{-61}:\\ \;\;\;\;\frac{\left(-\left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}\right)\right) \cdot \left(\sin x \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon - 1\right) + 1\right)\right) + \cos x \cdot \left(\left({\left(\tan x \cdot \tan \varepsilon\right)}^{3} + 1\right) \cdot \left(\tan x + \tan \varepsilon\right)\right)}{\cos x \cdot \left(\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon - 1\right) + 1\right) \cdot \left(1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{6}}\right)\right)}\\ \mathbf{elif}\;\varepsilon \le 1.66269538116722889 \cdot 10^{-37}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) \cdot \cos x - \left(\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right) \cdot \sin x}{\left(\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020164 
(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)))