2tan (problem 3.3.2)

Average Error: 36.7 → 6.1

Time: 30.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.692188506596356957702783826346824106171 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - {\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}}, \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot \left(\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon\right)}, \frac{{\left(\sin \varepsilon\right)}^{3}}{1 - {\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}}, \mathsf{fma}\left(\frac{\frac{{\left(\sin \varepsilon\right)}^{2}}{1 - {\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}}}{{\left(\cos \varepsilon\right)}^{2}}, {\left(\frac{\sin x}{\cos x}\right)}^{3} + \frac{\sin x}{\cos x}, \frac{\sin x}{\left(1 - {\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}\right) \cdot \cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - {\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}}\right) - \frac{\sin x}{\cos x}\right)\\ \mathbf{elif}\;\varepsilon \le 1.342409282000481224159305634141017423042 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - {\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}}, \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left(\frac{1}{3}, {\varepsilon}^{3}, \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \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)}, 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \tan x \cdot \tan \varepsilon\right), -\tan x\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r106613 = x;
        double r106614 = eps;
        double r106615 = r106613 + r106614;
        double r106616 = tan(r106615);
        double r106617 = tan(r106613);
        double r106618 = r106616 - r106617;
        return r106618;
}

↓

double f(double x, double eps) {
        double r106619 = eps;
        double r106620 = -2.692188506596357e-51;
        bool r106621 = r106619 <= r106620;
        double r106622 = sin(r106619);
        double r106623 = cos(r106619);
        double r106624 = r106622 / r106623;
        double r106625 = 1.0;
        double r106626 = x;
        double r106627 = sin(r106626);
        double r106628 = r106627 * r106622;
        double r106629 = cos(r106626);
        double r106630 = r106623 * r106629;
        double r106631 = r106628 / r106630;
        double r106632 = 3.0;
        double r106633 = pow(r106631, r106632);
        double r106634 = r106625 - r106633;
        double r106635 = r106624 / r106634;
        double r106636 = 2.0;
        double r106637 = pow(r106627, r106636);
        double r106638 = pow(r106629, r106636);
        double r106639 = r106637 / r106638;
        double r106640 = r106623 * r106623;
        double r106641 = r106640 * r106623;
        double r106642 = r106638 * r106641;
        double r106643 = r106637 / r106642;
        double r106644 = pow(r106622, r106632);
        double r106645 = r106644 / r106634;
        double r106646 = pow(r106622, r106636);
        double r106647 = r106646 / r106634;
        double r106648 = pow(r106623, r106636);
        double r106649 = r106647 / r106648;
        double r106650 = r106627 / r106629;
        double r106651 = pow(r106650, r106632);
        double r106652 = r106651 + r106650;
        double r106653 = r106634 * r106629;
        double r106654 = r106627 / r106653;
        double r106655 = fma(r106649, r106652, r106654);
        double r106656 = r106655 + r106635;
        double r106657 = fma(r106643, r106645, r106656);
        double r106658 = r106657 - r106650;
        double r106659 = fma(r106635, r106639, r106658);
        double r106660 = 1.3424092820004812e-20;
        bool r106661 = r106619 <= r106660;
        double r106662 = pow(r106619, r106636);
        double r106663 = 0.3333333333333333;
        double r106664 = pow(r106619, r106632);
        double r106665 = fma(r106663, r106664, r106619);
        double r106666 = fma(r106626, r106662, r106665);
        double r106667 = fma(r106635, r106639, r106666);
        double r106668 = tan(r106626);
        double r106669 = tan(r106619);
        double r106670 = r106668 + r106669;
        double r106671 = r106668 * r106668;
        double r106672 = r106671 * r106668;
        double r106673 = r106669 * r106669;
        double r106674 = r106673 * r106669;
        double r106675 = r106672 * r106674;
        double r106676 = r106625 - r106675;
        double r106677 = r106670 / r106676;
        double r106678 = r106668 * r106669;
        double r106679 = r106678 * r106678;
        double r106680 = r106679 + r106678;
        double r106681 = r106625 + r106680;
        double r106682 = -r106668;
        double r106683 = fma(r106677, r106681, r106682);
        double r106684 = r106661 ? r106667 : r106683;
        double r106685 = r106621 ? r106659 : r106684;
        return r106685;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.7
Target	14.9
Herbie	6.1

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

Derivation

1. Split input into 3 regimes

2. if eps < -2.692188506596357e-51

   1. Initial program 30.6

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

   3. Applied tan-sum 4.1

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

   5. Applied flip3-- 4.1

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

   6. Applied associate-/r/ 4.1

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

   7. Applied fma-neg 4.1

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

   8. Taylor expanded around -inf 4.4

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

   9. Simplified 3.8

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

   11. Applied add-cbrt-cube 3.9

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

   12. Applied rem-cube-cbrt 3.8

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

   if -2.692188506596357e-51 < eps < 1.3424092820004812e-20

   1. Initial program 45.8

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

   3. Applied tan-sum 45.8

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

   5. Applied flip3-- 45.8

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

   6. Applied associate-/r/ 45.8

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

   7. Applied fma-neg 45.8

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

   8. Taylor expanded around -inf 45.8

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

   9. Simplified 41.1

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

   10. Taylor expanded around 0 10.4

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

   11. Simplified 10.4

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

   if 1.3424092820004812e-20 < eps

   1. Initial program 28.0

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

   3. Applied tan-sum 1.4

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

   5. Applied flip3-- 1.4

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

   6. Applied associate-/r/ 1.4

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

   7. Applied fma-neg 1.4

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

   9. Applied add-cbrt-cube 1.5

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

   10. Applied add-cbrt-cube 1.5

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

   11. Applied cbrt-unprod 1.5

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

   12. Applied rem-cube-cbrt 1.4

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

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.692188506596356957702783826346824106171 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - {\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}}, \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot \left(\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon\right)}, \frac{{\left(\sin \varepsilon\right)}^{3}}{1 - {\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}}, \mathsf{fma}\left(\frac{\frac{{\left(\sin \varepsilon\right)}^{2}}{1 - {\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}}}{{\left(\cos \varepsilon\right)}^{2}}, {\left(\frac{\sin x}{\cos x}\right)}^{3} + \frac{\sin x}{\cos x}, \frac{\sin x}{\left(1 - {\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}\right) \cdot \cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - {\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}}\right) - \frac{\sin x}{\cos x}\right)\\ \mathbf{elif}\;\varepsilon \le 1.342409282000481224159305634141017423042 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - {\left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}^{3}}, \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left(\frac{1}{3}, {\varepsilon}^{3}, \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \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)}, 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \tan x \cdot \tan \varepsilon\right), -\tan x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)))