2tan (problem 3.3.2)

Average Error: 36.9 → 0.6

Time: 34.6s

Precision: 64

Math

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

↓

\[\frac{\left(\frac{{\left(\sin x\right)}^{5} \cdot {\left(\sin \varepsilon\right)}^{4}}{{\left(\cos x\right)}^{4} \cdot {\left(\cos \varepsilon\right)}^{4}} + \left(\frac{{\left(\sin x\right)}^{8} \cdot {\left(\sin \varepsilon\right)}^{7}}{{\left(\cos \varepsilon\right)}^{7} \cdot {\left(\cos x\right)}^{7}} + \left(\left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + \frac{{\left(\sin x\right)}^{6} \cdot {\left(\sin \varepsilon\right)}^{7}}{{\left(\cos \varepsilon\right)}^{7} \cdot {\left(\cos x\right)}^{5}}\right) + \left(\left(3 \cdot \frac{{\left(\sin x\right)}^{4} \cdot {\left(\sin \varepsilon\right)}^{5}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{5}} + \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{4}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{4}}\right) + \left(3 \cdot \left(\frac{{\left(\sin x\right)}^{6} \cdot {\left(\sin \varepsilon\right)}^{5}}{{\left(\cos x\right)}^{5} \cdot {\left(\cos \varepsilon\right)}^{5}} + \frac{{\left(\sin x\right)}^{7} \cdot {\left(\sin \varepsilon\right)}^{6}}{{\left(\cos x\right)}^{6} \cdot {\left(\cos \varepsilon\right)}^{6}}\right) + 3 \cdot \frac{{\left(\sin x\right)}^{5} \cdot {\left(\sin \varepsilon\right)}^{6}}{{\left(\cos x\right)}^{4} \cdot {\left(\cos \varepsilon\right)}^{6}}\right)\right)\right)\right)\right) + \frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon}}{\left(1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right) \cdot \left(\cos x \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x + 1\right)\right) \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x + 1\right) - 1\right) + 1\right)\right)}\]

C

double f(double x, double eps) {
        double r94702 = x;
        double r94703 = eps;
        double r94704 = r94702 + r94703;
        double r94705 = tan(r94704);
        double r94706 = tan(r94702);
        double r94707 = r94705 - r94706;
        return r94707;
}

↓

double f(double x, double eps) {
        double r94708 = x;
        double r94709 = sin(r94708);
        double r94710 = 5.0;
        double r94711 = pow(r94709, r94710);
        double r94712 = eps;
        double r94713 = sin(r94712);
        double r94714 = 4.0;
        double r94715 = pow(r94713, r94714);
        double r94716 = r94711 * r94715;
        double r94717 = cos(r94708);
        double r94718 = pow(r94717, r94714);
        double r94719 = cos(r94712);
        double r94720 = pow(r94719, r94714);
        double r94721 = r94718 * r94720;
        double r94722 = r94716 / r94721;
        double r94723 = 8.0;
        double r94724 = pow(r94709, r94723);
        double r94725 = 7.0;
        double r94726 = pow(r94713, r94725);
        double r94727 = r94724 * r94726;
        double r94728 = pow(r94719, r94725);
        double r94729 = pow(r94717, r94725);
        double r94730 = r94728 * r94729;
        double r94731 = r94727 / r94730;
        double r94732 = 2.0;
        double r94733 = pow(r94709, r94732);
        double r94734 = r94733 * r94713;
        double r94735 = r94717 * r94719;
        double r94736 = r94734 / r94735;
        double r94737 = 6.0;
        double r94738 = pow(r94709, r94737);
        double r94739 = r94738 * r94726;
        double r94740 = pow(r94717, r94710);
        double r94741 = r94728 * r94740;
        double r94742 = r94739 / r94741;
        double r94743 = r94736 + r94742;
        double r94744 = 3.0;
        double r94745 = pow(r94709, r94714);
        double r94746 = pow(r94713, r94710);
        double r94747 = r94745 * r94746;
        double r94748 = pow(r94717, r94744);
        double r94749 = pow(r94719, r94710);
        double r94750 = r94748 * r94749;
        double r94751 = r94747 / r94750;
        double r94752 = r94744 * r94751;
        double r94753 = pow(r94709, r94744);
        double r94754 = r94753 * r94715;
        double r94755 = pow(r94717, r94732);
        double r94756 = r94755 * r94720;
        double r94757 = r94754 / r94756;
        double r94758 = r94752 + r94757;
        double r94759 = r94738 * r94746;
        double r94760 = r94740 * r94749;
        double r94761 = r94759 / r94760;
        double r94762 = pow(r94709, r94725);
        double r94763 = pow(r94713, r94737);
        double r94764 = r94762 * r94763;
        double r94765 = pow(r94717, r94737);
        double r94766 = pow(r94719, r94737);
        double r94767 = r94765 * r94766;
        double r94768 = r94764 / r94767;
        double r94769 = r94761 + r94768;
        double r94770 = r94744 * r94769;
        double r94771 = r94711 * r94763;
        double r94772 = r94718 * r94766;
        double r94773 = r94771 / r94772;
        double r94774 = r94744 * r94773;
        double r94775 = r94770 + r94774;
        double r94776 = r94758 + r94775;
        double r94777 = r94743 + r94776;
        double r94778 = r94731 + r94777;
        double r94779 = r94722 + r94778;
        double r94780 = r94713 * r94717;
        double r94781 = r94780 / r94719;
        double r94782 = r94779 + r94781;
        double r94783 = 1.0;
        double r94784 = tan(r94708);
        double r94785 = tan(r94712);
        double r94786 = r94784 * r94785;
        double r94787 = pow(r94786, r94744);
        double r94788 = r94783 - r94787;
        double r94789 = r94785 * r94784;
        double r94790 = r94789 + r94783;
        double r94791 = r94789 * r94790;
        double r94792 = r94791 - r94783;
        double r94793 = r94791 * r94792;
        double r94794 = r94793 + r94783;
        double r94795 = r94717 * r94794;
        double r94796 = r94788 * r94795;
        double r94797 = r94782 / r94796;
        return r94797;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	14.5
Herbie	0.6

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

Derivation

1. Initial program 36.9

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

3. Applied tan-sum 22.3

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

5. Applied flip3-- 22.3

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

6. Applied associate-/r/ 22.3

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

7. Simplified 22.3

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

9. Applied tan-quot 22.4

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

10. Applied flip3-+ 22.5

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

11. Applied frac-times 22.5

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

12. Applied frac-sub 22.5

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

13. Simplified 22.5

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

14. Simplified 22.5

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

15. Taylor expanded around -inf 0.6

    \[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \left(\frac{{\left(\sin x\right)}^{5} \cdot {\left(\sin \varepsilon\right)}^{4}}{{\left(\cos x\right)}^{4} \cdot {\left(\cos \varepsilon\right)}^{4}} + \left(\frac{{\left(\sin x\right)}^{8} \cdot {\left(\sin \varepsilon\right)}^{7}}{{\left(\cos \varepsilon\right)}^{7} \cdot {\left(\cos x\right)}^{7}} + \left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + \left(\frac{{\left(\sin x\right)}^{6} \cdot {\left(\sin \varepsilon\right)}^{7}}{{\left(\cos \varepsilon\right)}^{7} \cdot {\left(\cos x\right)}^{5}} + \left(3 \cdot \frac{{\left(\sin x\right)}^{4} \cdot {\left(\sin \varepsilon\right)}^{5}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{5}} + \left(\frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{4}}{{\left(\cos \varepsilon\right)}^{4} \cdot {\left(\cos x\right)}^{2}} + \left(3 \cdot \frac{{\left(\sin x\right)}^{5} \cdot {\left(\sin \varepsilon\right)}^{6}}{{\left(\cos x\right)}^{4} \cdot {\left(\cos \varepsilon\right)}^{6}} + \left(3 \cdot \frac{{\left(\sin x\right)}^{6} \cdot {\left(\sin \varepsilon\right)}^{5}}{{\left(\cos x\right)}^{5} \cdot {\left(\cos \varepsilon\right)}^{5}} + 3 \cdot \frac{{\left(\sin x\right)}^{7} \cdot {\left(\sin \varepsilon\right)}^{6}}{{\left(\cos x\right)}^{6} \cdot {\left(\cos \varepsilon\right)}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)}}{\left(1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right) \cdot \left(\cos x \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x + 1\right)\right) \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x + 1\right) - 1\right) + 1\right)\right)}\]

16. Simplified 0.6

    \[\leadsto \frac{\color{blue}{\left(\frac{{\left(\sin x\right)}^{5} \cdot {\left(\sin \varepsilon\right)}^{4}}{{\left(\cos x\right)}^{4} \cdot {\left(\cos \varepsilon\right)}^{4}} + \left(\frac{{\left(\sin x\right)}^{8} \cdot {\left(\sin \varepsilon\right)}^{7}}{{\left(\cos \varepsilon\right)}^{7} \cdot {\left(\cos x\right)}^{7}} + \left(\left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + \frac{{\left(\sin x\right)}^{6} \cdot {\left(\sin \varepsilon\right)}^{7}}{{\left(\cos \varepsilon\right)}^{7} \cdot {\left(\cos x\right)}^{5}}\right) + \left(\left(3 \cdot \frac{{\left(\sin x\right)}^{4} \cdot {\left(\sin \varepsilon\right)}^{5}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{5}} + \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{4}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{4}}\right) + \left(3 \cdot \left(\frac{{\left(\sin x\right)}^{6} \cdot {\left(\sin \varepsilon\right)}^{5}}{{\left(\cos x\right)}^{5} \cdot {\left(\cos \varepsilon\right)}^{5}} + \frac{{\left(\sin x\right)}^{7} \cdot {\left(\sin \varepsilon\right)}^{6}}{{\left(\cos x\right)}^{6} \cdot {\left(\cos \varepsilon\right)}^{6}}\right) + 3 \cdot \frac{{\left(\sin x\right)}^{5} \cdot {\left(\sin \varepsilon\right)}^{6}}{{\left(\cos x\right)}^{4} \cdot {\left(\cos \varepsilon\right)}^{6}}\right)\right)\right)\right)\right) + \frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon}}}{\left(1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right) \cdot \left(\cos x \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x + 1\right)\right) \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x + 1\right) - 1\right) + 1\right)\right)}\]

17. Final simplification 0.6

    \[\leadsto \frac{\left(\frac{{\left(\sin x\right)}^{5} \cdot {\left(\sin \varepsilon\right)}^{4}}{{\left(\cos x\right)}^{4} \cdot {\left(\cos \varepsilon\right)}^{4}} + \left(\frac{{\left(\sin x\right)}^{8} \cdot {\left(\sin \varepsilon\right)}^{7}}{{\left(\cos \varepsilon\right)}^{7} \cdot {\left(\cos x\right)}^{7}} + \left(\left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + \frac{{\left(\sin x\right)}^{6} \cdot {\left(\sin \varepsilon\right)}^{7}}{{\left(\cos \varepsilon\right)}^{7} \cdot {\left(\cos x\right)}^{5}}\right) + \left(\left(3 \cdot \frac{{\left(\sin x\right)}^{4} \cdot {\left(\sin \varepsilon\right)}^{5}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{5}} + \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{4}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{4}}\right) + \left(3 \cdot \left(\frac{{\left(\sin x\right)}^{6} \cdot {\left(\sin \varepsilon\right)}^{5}}{{\left(\cos x\right)}^{5} \cdot {\left(\cos \varepsilon\right)}^{5}} + \frac{{\left(\sin x\right)}^{7} \cdot {\left(\sin \varepsilon\right)}^{6}}{{\left(\cos x\right)}^{6} \cdot {\left(\cos \varepsilon\right)}^{6}}\right) + 3 \cdot \frac{{\left(\sin x\right)}^{5} \cdot {\left(\sin \varepsilon\right)}^{6}}{{\left(\cos x\right)}^{4} \cdot {\left(\cos \varepsilon\right)}^{6}}\right)\right)\right)\right)\right) + \frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon}}{\left(1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right) \cdot \left(\cos x \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x + 1\right)\right) \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x + 1\right) - 1\right) + 1\right)\right)}\]

Reproduce

herbie shell --seed 2019208 
(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)))