2tan (problem 3.3.2)

Average Error: 37.0 → 15.5

Time: 34.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.151064551553271235467897851690081498157 \cdot 10^{-59}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon\right)\right) \cdot \left(\tan x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}, \mathsf{fma}\left(\frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon, \frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon, \frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon\right), \frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon\right)\right) \cdot \left(\tan x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \tan x\right) + \mathsf{fma}\left(\tan x, -1, \tan x\right)\right) + \mathsf{fma}\left(\tan x, -1, \tan x\right)\\ \mathbf{elif}\;\varepsilon \le 1.432165063698696594827716660641469430253 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \varepsilon, \varepsilon + x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\tan \varepsilon + \tan x}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \left(\frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}\right)} - \tan x\right) + \frac{\tan \varepsilon + \tan x}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \left(\frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}\right)} \cdot \mathsf{fma}\left(\frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}, \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}, \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}\right)\right) + \mathsf{fma}\left(\tan x, -1, \tan x\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r5801174 = x;
        double r5801175 = eps;
        double r5801176 = r5801174 + r5801175;
        double r5801177 = tan(r5801176);
        double r5801178 = tan(r5801174);
        double r5801179 = r5801177 - r5801178;
        return r5801179;
}

↓

double f(double x, double eps) {
        double r5801180 = eps;
        double r5801181 = -1.1510645515532712e-59;
        bool r5801182 = r5801180 <= r5801181;
        double r5801183 = tan(r5801180);
        double r5801184 = x;
        double r5801185 = tan(r5801184);
        double r5801186 = r5801183 + r5801185;
        double r5801187 = 1.0;
        double r5801188 = sin(r5801180);
        double r5801189 = cos(r5801180);
        double r5801190 = r5801188 / r5801189;
        double r5801191 = r5801185 * r5801190;
        double r5801192 = r5801185 / r5801189;
        double r5801193 = r5801192 * r5801188;
        double r5801194 = r5801191 * r5801193;
        double r5801195 = r5801194 * r5801191;
        double r5801196 = r5801187 - r5801195;
        double r5801197 = r5801186 / r5801196;
        double r5801198 = fma(r5801193, r5801193, r5801193);
        double r5801199 = r5801197 - r5801185;
        double r5801200 = fma(r5801197, r5801198, r5801199);
        double r5801201 = -1.0;
        double r5801202 = fma(r5801185, r5801201, r5801185);
        double r5801203 = r5801200 + r5801202;
        double r5801204 = r5801203 + r5801202;
        double r5801205 = 1.4321650636986966e-141;
        bool r5801206 = r5801180 <= r5801205;
        double r5801207 = r5801184 * r5801180;
        double r5801208 = r5801180 + r5801184;
        double r5801209 = fma(r5801207, r5801208, r5801180);
        double r5801210 = r5801189 / r5801188;
        double r5801211 = r5801185 / r5801210;
        double r5801212 = r5801188 * r5801185;
        double r5801213 = r5801212 / r5801189;
        double r5801214 = r5801213 * r5801213;
        double r5801215 = r5801211 * r5801214;
        double r5801216 = r5801187 - r5801215;
        double r5801217 = r5801186 / r5801216;
        double r5801218 = r5801217 - r5801185;
        double r5801219 = fma(r5801211, r5801211, r5801211);
        double r5801220 = r5801217 * r5801219;
        double r5801221 = r5801218 + r5801220;
        double r5801222 = r5801221 + r5801202;
        double r5801223 = r5801206 ? r5801209 : r5801222;
        double r5801224 = r5801182 ? r5801204 : r5801223;
        return r5801224;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
Target	15.1
Herbie	15.5

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.1510645515532712e-59

   1. Initial program 30.0

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

   3. Applied tan-sum 4.9

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

   5. Applied tan-quot 5.0

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

   6. Applied associate-*r/ 5.0

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

   8. Applied add-cube-cbrt 5.2

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

   9. Applied flip3-- 5.2

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

   10. Applied associate-/r/ 5.2

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

   11. Applied prod-diff 5.2

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

   12. Simplified 4.9

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

   13. Simplified 5.0

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

   15. Applied add-cube-cbrt 5.2

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

   16. Applied add-sqr-sqrt 36.7

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

   17. Applied prod-diff 36.7

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

   18. Simplified 4.9

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

   19. Simplified 4.3

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

   if -1.1510645515532712e-59 < eps < 1.4321650636986966e-141

   1. Initial program 48.2

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

   2. Taylor expanded around 0 32.0

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

   3. Simplified 31.8

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

   if 1.4321650636986966e-141 < eps

   1. Initial program 32.4

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

   3. Applied tan-sum 11.3

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

   5. Applied tan-quot 11.3

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

   6. Applied associate-*r/ 11.3

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

   8. Applied add-cube-cbrt 11.5

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

   9. Applied flip3-- 11.5

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

   10. Applied associate-/r/ 11.5

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

   11. Applied prod-diff 11.5

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

   12. Simplified 11.2

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

   13. Simplified 11.3

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

   15. Applied fma-udef 11.3

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

   16. Applied associate--l+ 9.7

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

4. Final simplification 15.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.151064551553271235467897851690081498157 \cdot 10^{-59}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon\right)\right) \cdot \left(\tan x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}, \mathsf{fma}\left(\frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon, \frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon, \frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon\right), \frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\tan x}{\cos \varepsilon} \cdot \sin \varepsilon\right)\right) \cdot \left(\tan x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} - \tan x\right) + \mathsf{fma}\left(\tan x, -1, \tan x\right)\right) + \mathsf{fma}\left(\tan x, -1, \tan x\right)\\ \mathbf{elif}\;\varepsilon \le 1.432165063698696594827716660641469430253 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \varepsilon, \varepsilon + x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\tan \varepsilon + \tan x}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \left(\frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}\right)} - \tan x\right) + \frac{\tan \varepsilon + \tan x}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \left(\frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}\right)} \cdot \mathsf{fma}\left(\frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}, \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}, \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}\right)\right) + \mathsf{fma}\left(\tan x, -1, \tan x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))