Average Error: 36.7 → 0.6
Time: 1.6m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon}}\right) + \left(\left(\left((\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}\right) + \left(\frac{\frac{\sin x}{\cos x} \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right)}\right))_* + \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right) + \frac{\sin \varepsilon}{\left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right) \cdot \cos \varepsilon}\right))_*\]
double f(double x, double eps) {
        double r19192536 = x;
        double r19192537 = eps;
        double r19192538 = r19192536 + r19192537;
        double r19192539 = tan(r19192538);
        double r19192540 = tan(r19192536);
        double r19192541 = r19192539 - r19192540;
        return r19192541;
}
↓
double f(double x, double eps) {
        double r19192542 = x;
        double r19192543 = sin(r19192542);
        double r19192544 = cos(r19192542);
        double r19192545 = r19192543 / r19192544;
        double r19192546 = r19192545 * r19192545;
        double r19192547 = eps;
        double r19192548 = sin(r19192547);
        double r19192549 = cos(r19192547);
        double r19192550 = r19192548 / r19192549;
        double r19192551 = 1.0;
        double r19192552 = r19192548 * r19192548;
        double r19192553 = r19192552 * r19192548;
        double r19192554 = r19192545 * r19192546;
        double r19192555 = r19192553 * r19192554;
        double r19192556 = r19192549 * r19192549;
        double r19192557 = r19192556 * r19192549;
        double r19192558 = r19192555 / r19192557;
        double r19192559 = r19192551 - r19192558;
        double r19192560 = r19192550 / r19192559;
        double r19192561 = r19192553 / r19192556;
        double r19192562 = r19192561 / r19192549;
        double r19192563 = r19192554 * r19192562;
        double r19192564 = r19192551 - r19192563;
        double r19192565 = r19192562 / r19192564;
        double r19192566 = r19192545 * r19192552;
        double r19192567 = r19192556 * r19192564;
        double r19192568 = r19192566 / r19192567;
        double r19192569 = fma(r19192546, r19192565, r19192568);
        double r19192570 = r19192545 / r19192564;
        double r19192571 = r19192569 + r19192570;
        double r19192572 = r19192554 * r19192552;
        double r19192573 = r19192572 / r19192567;
        double r19192574 = r19192573 - r19192545;
        double r19192575 = r19192571 + r19192574;
        double r19192576 = r19192564 * r19192549;
        double r19192577 = r19192548 / r19192576;
        double r19192578 = r19192575 + r19192577;
        double r19192579 = fma(r19192546, r19192560, r19192578);
        return r19192579;
}


\tan \left(x + \varepsilon\right) - \tan x
↓
(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon}}\right) + \left(\left(\left((\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}\right) + \left(\frac{\frac{\sin x}{\cos x} \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right)}\right))_* + \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right) + \frac{\sin \varepsilon}{\left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right) \cdot \cos \varepsilon}\right))_*

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target15.1
Herbie0.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.7

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

  3. Applied tan-sum21.5

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

  5. Applied flip3--21.5

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

  6. Applied associate-/r/21.5

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

  7. Applied fma-neg21.5

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

  8. Simplified21.5

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

  9. Taylor expanded around -inf 21.6

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

  10. Simplified19.4

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

  11. Taylor expanded around inf 19.4

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

  12. Simplified0.6

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

  13. Final simplification0.6

    \[\leadsto (\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon}}\right) + \left(\left(\left((\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}\right) + \left(\frac{\frac{\sin x}{\cos x} \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right)}\right))_* + \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right) + \frac{\sin \varepsilon}{\left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right) \cdot \cos \varepsilon}\right))_*\]

Reproduce

herbie shell --seed 2019102 +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)))