2tan (problem 3.3.2)

Average Error: 37.1 → 0.3

Time: 19.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.23118950967794759 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\frac{\sin \varepsilon}{\cos \varepsilon} + \frac{\sin x}{\cos x}\right) \cdot \left(\frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}} + \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)}{1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}} + \left(1 \cdot \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \le 1.0062887731628292 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} + \left(\varepsilon + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \frac{2}{15} \cdot {\varepsilon}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} + \frac{\left(1 \cdot \left(-\left(\tan x + \tan \varepsilon\right)\right)\right) \cdot \cos x - \left(-\left(1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)\right) \cdot \sin x}{\left(-\left(1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)\right) \cdot \cos x}\\ \end{array}\]

C

double code(double x, double eps) {
	return (tan((x + eps)) - tan(x));
}

↓

double code(double x, double eps) {
	double temp;
	if ((eps <= -4.231189509677948e-06)) {
		temp = (((((sin(eps) / cos(eps)) + (sin(x) / cos(x))) * (((pow(sin(x), 2.0) * pow(sin(eps), 2.0)) / (pow(cos(x), 2.0) * pow(cos(eps), 2.0))) + ((sin(x) * sin(eps)) / (cos(x) * cos(eps))))) / (1.0 - ((pow(sin(x), 3.0) * pow(sin(eps), 3.0)) / (pow(cos(x), 3.0) * pow(cos(eps), 3.0))))) + ((1.0 * ((tan(x) + tan(eps)) / (1.0 - pow((tan(eps) * tan(x)), 3.0)))) - tan(x)));
	} else {
		double temp_1;
		if ((eps <= 1.0062887731628292e-05)) {
			temp_1 = (((((tan(eps) * tan(x)) * (tan(eps) * tan(x))) + (1.0 * (tan(eps) * tan(x)))) * ((tan(x) + tan(eps)) / (1.0 - pow((tan(eps) * tan(x)), 3.0)))) + (eps + ((0.3333333333333333 * pow(eps, 3.0)) + (0.13333333333333333 * pow(eps, 5.0)))));
		} else {
			temp_1 = (((((tan(eps) * tan(x)) * (tan(eps) * tan(x))) + (1.0 * (tan(eps) * tan(x)))) * ((tan(x) + tan(eps)) / (1.0 - pow((tan(eps) * tan(x)), 3.0)))) + ((((1.0 * -(tan(x) + tan(eps))) * cos(x)) - (-(1.0 - pow((tan(eps) * tan(x)), 3.0)) * sin(x))) / (-(1.0 - pow((tan(eps) * tan(x)), 3.0)) * cos(x))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	14.5
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -4.231189509677948e-06

   1. Initial program 30.0

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

   3. Applied tan-sum 0.4

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

   4. Simplified 0.4

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

   6. Applied flip3-- 0.4

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

   7. Applied associate-/r/ 0.4

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

   8. Simplified 0.4

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

   10. Applied +-commutative 0.4

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

   11. Applied distribute-rgt-in 0.4

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

   12. Applied associate--l+ 0.4

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

   13. Simplified 0.4

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

   14. Taylor expanded around inf 0.4

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

   if -4.231189509677948e-06 < eps < 1.0062887731628292e-05

   1. Initial program 44.4

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

   3. Applied tan-sum 43.9

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

   4. Simplified 43.9

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

   6. Applied flip3-- 43.9

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

   7. Applied associate-/r/ 43.9

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

   8. Simplified 43.9

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

   10. Applied +-commutative 43.9

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

   11. Applied distribute-rgt-in 43.9

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

   12. Applied associate--l+ 39.3

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

   13. Simplified 39.3

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

   14. Taylor expanded around 0 0.2

       \[\leadsto \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} + \color{blue}{\left(\varepsilon + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \frac{2}{15} \cdot {\varepsilon}^{5}\right)\right)}\]

   if 1.0062887731628292e-05 < eps

   1. Initial program 28.9

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

   3. Applied tan-sum 0.3

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

   4. Simplified 0.3

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

   6. Applied flip3-- 0.4

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

   7. Applied associate-/r/ 0.4

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

   8. Simplified 0.4

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

   10. Applied +-commutative 0.4

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

   11. Applied distribute-rgt-in 0.4

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

   12. Applied associate--l+ 0.4

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

   13. Simplified 0.4

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

   15. Applied tan-quot 0.4

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

   16. Applied frac-2neg 0.4

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

   17. Applied associate-*r/ 0.4

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

   18. Applied frac-sub 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.23118950967794759 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\frac{\sin \varepsilon}{\cos \varepsilon} + \frac{\sin x}{\cos x}\right) \cdot \left(\frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}} + \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)}{1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}} + \left(1 \cdot \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \le 1.0062887731628292 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} + \left(\varepsilon + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \frac{2}{15} \cdot {\varepsilon}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + 1 \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} + \frac{\left(1 \cdot \left(-\left(\tan x + \tan \varepsilon\right)\right)\right) \cdot \cos x - \left(-\left(1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)\right) \cdot \sin x}{\left(-\left(1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}\right)\right) \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(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)))