2tan (problem 3.3.2)

Average Error: 37.3 → 6.4

Time: 30.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.578153766106150173447679472971435469694 \cdot 10^{-73} \lor \neg \left(\varepsilon \le 5.03155655901486863758090640091667272774 \cdot 10^{-64}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{3}} \cdot \frac{{\left(\sin x\right)}^{2}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}} + \left(\mathsf{fma}\left(\frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, {\left(\frac{\sin x}{\cos x}\right)}^{3} + \frac{\sin x}{\cos x}, \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}\right) + \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)} - \frac{\sin x}{\cos x}\right)\right)\right) + \tan x \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}}, \mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left(\frac{1}{3}, {\varepsilon}^{3}, \varepsilon\right)\right)\right) + \tan x \cdot 0\\ \end{array}\]

C

double f(double x, double eps) {
        double r111736 = x;
        double r111737 = eps;
        double r111738 = r111736 + r111737;
        double r111739 = tan(r111738);
        double r111740 = tan(r111736);
        double r111741 = r111739 - r111740;
        return r111741;
}

↓

double f(double x, double eps) {
        double r111742 = eps;
        double r111743 = -5.57815376610615e-73;
        bool r111744 = r111742 <= r111743;
        double r111745 = 5.0315565590148686e-64;
        bool r111746 = r111742 <= r111745;
        double r111747 = !r111746;
        bool r111748 = r111744 || r111747;
        double r111749 = x;
        double r111750 = sin(r111749);
        double r111751 = 2.0;
        double r111752 = pow(r111750, r111751);
        double r111753 = cos(r111742);
        double r111754 = r111752 / r111753;
        double r111755 = sin(r111742);
        double r111756 = 1.0;
        double r111757 = r111750 * r111755;
        double r111758 = 3.0;
        double r111759 = pow(r111757, r111758);
        double r111760 = cos(r111749);
        double r111761 = r111760 * r111753;
        double r111762 = pow(r111761, r111758);
        double r111763 = r111759 / r111762;
        double r111764 = r111756 - r111763;
        double r111765 = pow(r111760, r111751);
        double r111766 = r111764 * r111765;
        double r111767 = r111755 / r111766;
        double r111768 = pow(r111755, r111758);
        double r111769 = pow(r111753, r111758);
        double r111770 = r111765 * r111769;
        double r111771 = r111768 / r111770;
        double r111772 = r111752 / r111764;
        double r111773 = r111771 * r111772;
        double r111774 = pow(r111755, r111751);
        double r111775 = pow(r111753, r111751);
        double r111776 = r111775 * r111764;
        double r111777 = r111774 / r111776;
        double r111778 = r111750 / r111760;
        double r111779 = pow(r111778, r111758);
        double r111780 = r111779 + r111778;
        double r111781 = r111778 / r111764;
        double r111782 = fma(r111777, r111780, r111781);
        double r111783 = r111753 * r111764;
        double r111784 = r111755 / r111783;
        double r111785 = r111784 - r111778;
        double r111786 = r111782 + r111785;
        double r111787 = r111773 + r111786;
        double r111788 = fma(r111754, r111767, r111787);
        double r111789 = tan(r111749);
        double r111790 = 0.0;
        double r111791 = r111789 * r111790;
        double r111792 = r111788 + r111791;
        double r111793 = pow(r111742, r111751);
        double r111794 = 0.3333333333333333;
        double r111795 = pow(r111742, r111758);
        double r111796 = fma(r111794, r111795, r111742);
        double r111797 = fma(r111749, r111793, r111796);
        double r111798 = fma(r111754, r111767, r111797);
        double r111799 = r111798 + r111791;
        double r111800 = r111748 ? r111792 : r111799;
        return r111800;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.3
Target	15.1
Herbie	6.4

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

Derivation

1. Split input into 2 regimes

2. if eps < -5.57815376610615e-73 or 5.0315565590148686e-64 < eps

   1. Initial program 30.6

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

   3. Applied tan-sum 5.6

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

   5. Applied add-cube-cbrt 5.9

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

   6. Applied flip3-- 5.9

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

   7. Applied associate-/r/ 5.9

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

   8. Applied prod-diff 5.9

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

   9. Simplified 5.5

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

   10. Simplified 5.6

       \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right) + \color{blue}{\tan x \cdot 0}\]

   11. Taylor expanded around -inf 5.8

       \[\leadsto \color{blue}{\left(\left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}\right)} + \left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \left(\frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \left({\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{3}\right)} + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{3}\right)} + \frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \left(\cos x \cdot {\left(\cos \varepsilon\right)}^{2}\right)}\right)\right)\right)\right)\right) - \frac{\sin x}{\cos x}\right)} + \tan x \cdot 0\]

   12. Simplified 5.1

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}}, \mathsf{fma}\left(\frac{{\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{3}}, \frac{{\left(\sin x\right)}^{2}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}, \mathsf{fma}\left(\frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, \frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}, \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}\right) + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}\right) - \frac{\sin x}{\cos x}\right)} + \tan x \cdot 0\]
   13. Using strategy rm

   14. Applied fma-udef 5.1

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}}, \color{blue}{\left(\frac{{\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{3}} \cdot \frac{{\left(\sin x\right)}^{2}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}} + \left(\mathsf{fma}\left(\frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, \frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}, \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}\right) + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}\right)\right)} - \frac{\sin x}{\cos x}\right) + \tan x \cdot 0\]

   15. Applied associate--l+ 5.1

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}}, \color{blue}{\frac{{\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{3}} \cdot \frac{{\left(\sin x\right)}^{2}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}} + \left(\left(\mathsf{fma}\left(\frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, \frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}, \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}\right) + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}\right) - \frac{\sin x}{\cos x}\right)}\right) + \tan x \cdot 0\]

   16. Simplified 5.1

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{3}} \cdot \frac{{\left(\sin x\right)}^{2}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}} + \color{blue}{\left(\mathsf{fma}\left(\frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, {\left(\frac{\sin x}{\cos x}\right)}^{3} + \frac{\sin x}{\cos x}, \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}\right) + \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)} - \frac{\sin x}{\cos x}\right)\right)}\right) + \tan x \cdot 0\]

   if -5.57815376610615e-73 < eps < 5.0315565590148686e-64

   1. Initial program 47.6

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

   3. Applied tan-sum 47.6

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

   5. Applied add-cube-cbrt 48.5

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

   6. Applied flip3-- 48.5

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

   7. Applied associate-/r/ 48.5

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

   8. Applied prod-diff 48.7

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

   9. Simplified 48.7

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

   10. Simplified 47.6

       \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right) + \color{blue}{\tan x \cdot 0}\]

   11. Taylor expanded around -inf 47.6

       \[\leadsto \color{blue}{\left(\left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}\right)} + \left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \left(\frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \left({\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{3}\right)} + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{3}\right)} + \frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \left(\cos x \cdot {\left(\cos \varepsilon\right)}^{2}\right)}\right)\right)\right)\right)\right) - \frac{\sin x}{\cos x}\right)} + \tan x \cdot 0\]

   12. Simplified 42.9

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}}, \mathsf{fma}\left(\frac{{\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{3}}, \frac{{\left(\sin x\right)}^{2}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}, \mathsf{fma}\left(\frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, \frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}, \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}\right) + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}\right) - \frac{\sin x}{\cos x}\right)} + \tan x \cdot 0\]

   13. Taylor expanded around 0 8.4

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}}, \color{blue}{x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)}\right) + \tan x \cdot 0\]

   14. Simplified 8.4

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}}, \color{blue}{\mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left(\frac{1}{3}, {\varepsilon}^{3}, \varepsilon\right)\right)}\right) + \tan x \cdot 0\]
3. Recombined 2 regimes into one program.

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.578153766106150173447679472971435469694 \cdot 10^{-73} \lor \neg \left(\varepsilon \le 5.03155655901486863758090640091667272774 \cdot 10^{-64}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{3}} \cdot \frac{{\left(\sin x\right)}^{2}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}} + \left(\mathsf{fma}\left(\frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, {\left(\frac{\sin x}{\cos x}\right)}^{3} + \frac{\sin x}{\cos x}, \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}\right) + \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)} - \frac{\sin x}{\cos x}\right)\right)\right) + \tan x \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}}, \mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left(\frac{1}{3}, {\varepsilon}^{3}, \varepsilon\right)\right)\right) + \tan x \cdot 0\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(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)))