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 r96432 = x;
        double r96433 = eps;
        double r96434 = r96432 + r96433;
        double r96435 = tan(r96434);
        double r96436 = tan(r96432);
        double r96437 = r96435 - r96436;
        return r96437;
}

↓

double f(double x, double eps) {
        double r96438 = eps;
        double r96439 = -2.692188506596357e-51;
        bool r96440 = r96438 <= r96439;
        double r96441 = sin(r96438);
        double r96442 = cos(r96438);
        double r96443 = r96441 / r96442;
        double r96444 = 1.0;
        double r96445 = x;
        double r96446 = sin(r96445);
        double r96447 = r96446 * r96441;
        double r96448 = cos(r96445);
        double r96449 = r96442 * r96448;
        double r96450 = r96447 / r96449;
        double r96451 = 3.0;
        double r96452 = pow(r96450, r96451);
        double r96453 = r96444 - r96452;
        double r96454 = r96443 / r96453;
        double r96455 = 2.0;
        double r96456 = pow(r96446, r96455);
        double r96457 = pow(r96448, r96455);
        double r96458 = r96456 / r96457;
        double r96459 = pow(r96442, r96455);
        double r96460 = r96442 * r96459;
        double r96461 = r96457 * r96460;
        double r96462 = r96456 / r96461;
        double r96463 = pow(r96441, r96451);
        double r96464 = r96463 / r96453;
        double r96465 = pow(r96441, r96455);
        double r96466 = r96465 / r96453;
        double r96467 = r96466 / r96459;
        double r96468 = r96446 / r96448;
        double r96469 = pow(r96468, r96451);
        double r96470 = r96469 + r96468;
        double r96471 = r96453 * r96448;
        double r96472 = r96446 / r96471;
        double r96473 = fma(r96467, r96470, r96472);
        double r96474 = r96473 + r96454;
        double r96475 = fma(r96462, r96464, r96474);
        double r96476 = r96475 - r96468;
        double r96477 = fma(r96454, r96458, r96476);
        double r96478 = 1.3424092820004812e-20;
        bool r96479 = r96438 <= r96478;
        double r96480 = pow(r96438, r96455);
        double r96481 = 0.3333333333333333;
        double r96482 = pow(r96438, r96451);
        double r96483 = fma(r96481, r96482, r96438);
        double r96484 = fma(r96445, r96480, r96483);
        double r96485 = fma(r96454, r96458, r96484);
        double r96486 = tan(r96445);
        double r96487 = tan(r96438);
        double r96488 = r96486 + r96487;
        double r96489 = r96486 * r96486;
        double r96490 = r96489 * r96486;
        double r96491 = r96487 * r96487;
        double r96492 = r96491 * r96487;
        double r96493 = r96490 * r96492;
        double r96494 = r96444 - r96493;
        double r96495 = r96488 / r96494;
        double r96496 = r96486 * r96487;
        double r96497 = r96496 * r96496;
        double r96498 = r96497 + r96496;
        double r96499 = r96444 + r96498;
        double r96500 = -r96486;
        double r96501 = fma(r96495, r96499, r96500);
        double r96502 = r96479 ? r96485 : r96501;
        double r96503 = r96440 ? r96477 : r96502;
        return r96503;
}

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)))