2tan (problem 3.3.2)

Average Error: 36.7 → 6.1

Time: 31.2s

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(\cos \varepsilon \cdot {\left(\cos \varepsilon\right)}^{2}\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 r82674 = x;
        double r82675 = eps;
        double r82676 = r82674 + r82675;
        double r82677 = tan(r82676);
        double r82678 = tan(r82674);
        double r82679 = r82677 - r82678;
        return r82679;
}

↓

double f(double x, double eps) {
        double r82680 = eps;
        double r82681 = -2.692188506596357e-51;
        bool r82682 = r82680 <= r82681;
        double r82683 = sin(r82680);
        double r82684 = cos(r82680);
        double r82685 = r82683 / r82684;
        double r82686 = 1.0;
        double r82687 = x;
        double r82688 = sin(r82687);
        double r82689 = r82688 * r82683;
        double r82690 = cos(r82687);
        double r82691 = r82684 * r82690;
        double r82692 = r82689 / r82691;
        double r82693 = 3.0;
        double r82694 = pow(r82692, r82693);
        double r82695 = r82686 - r82694;
        double r82696 = r82685 / r82695;
        double r82697 = 2.0;
        double r82698 = pow(r82688, r82697);
        double r82699 = pow(r82690, r82697);
        double r82700 = r82698 / r82699;
        double r82701 = pow(r82684, r82697);
        double r82702 = r82684 * r82701;
        double r82703 = r82699 * r82702;
        double r82704 = r82698 / r82703;
        double r82705 = pow(r82683, r82693);
        double r82706 = r82705 / r82695;
        double r82707 = pow(r82683, r82697);
        double r82708 = r82707 / r82695;
        double r82709 = r82708 / r82701;
        double r82710 = r82688 / r82690;
        double r82711 = pow(r82710, r82693);
        double r82712 = r82711 + r82710;
        double r82713 = r82695 * r82690;
        double r82714 = r82688 / r82713;
        double r82715 = fma(r82709, r82712, r82714);
        double r82716 = r82715 + r82696;
        double r82717 = fma(r82704, r82706, r82716);
        double r82718 = r82717 - r82710;
        double r82719 = fma(r82696, r82700, r82718);
        double r82720 = 1.3424092820004812e-20;
        bool r82721 = r82680 <= r82720;
        double r82722 = pow(r82680, r82697);
        double r82723 = 0.3333333333333333;
        double r82724 = pow(r82680, r82693);
        double r82725 = fma(r82723, r82724, r82680);
        double r82726 = fma(r82687, r82722, r82725);
        double r82727 = fma(r82696, r82700, r82726);
        double r82728 = tan(r82687);
        double r82729 = tan(r82680);
        double r82730 = r82728 + r82729;
        double r82731 = r82728 * r82728;
        double r82732 = r82731 * r82728;
        double r82733 = r82729 * r82729;
        double r82734 = r82733 * r82729;
        double r82735 = r82732 * r82734;
        double r82736 = r82686 - r82735;
        double r82737 = r82730 / r82736;
        double r82738 = r82728 * r82729;
        double r82739 = r82738 * r82738;
        double r82740 = r82739 + r82738;
        double r82741 = r82686 + r82740;
        double r82742 = -r82728;
        double r82743 = fma(r82737, r82741, r82742);
        double r82744 = r82721 ? r82727 : r82743;
        double r82745 = r82682 ? r82719 : r82744;
        return r82745;
}

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 cube-mult 3.8

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

   12. Simplified 3.8

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

   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(\cos \varepsilon \cdot {\left(\cos \varepsilon\right)}^{2}\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)))