2tan (problem 3.3.2)

Average Error: 37.6 → 15.8

Time: 21.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.5307996958657947 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)} - \tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \mathbf{elif}\;\varepsilon \le 1.04782928541883701 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \frac{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}} \cdot \frac{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}}, 1 + \frac{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r491 = x;
        double r492 = eps;
        double r493 = r491 + r492;
        double r494 = tan(r493);
        double r495 = tan(r491);
        double r496 = r494 - r495;
        return r496;
}

↓

double f(double x, double eps) {
        double r497 = eps;
        double r498 = -4.530799695865795e-32;
        bool r499 = r497 <= r498;
        double r500 = x;
        double r501 = tan(r500);
        double r502 = tan(r497);
        double r503 = r501 + r502;
        double r504 = 1.0;
        double r505 = fma(r501, r502, r504);
        double r506 = r503 * r505;
        double r507 = r501 * r502;
        double r508 = r507 * r507;
        double r509 = log1p(r508);
        double r510 = expm1(r509);
        double r511 = r504 - r510;
        double r512 = r506 / r511;
        double r513 = r512 - r501;
        double r514 = -r501;
        double r515 = fma(r514, r504, r501);
        double r516 = r513 + r515;
        double r517 = 1.047829285418837e-23;
        bool r518 = r497 <= r517;
        double r519 = 2.0;
        double r520 = pow(r497, r519);
        double r521 = pow(r500, r519);
        double r522 = fma(r497, r521, r497);
        double r523 = fma(r520, r500, r522);
        double r524 = r523 + r515;
        double r525 = r501 * r501;
        double r526 = sin(r497);
        double r527 = pow(r526, r519);
        double r528 = r525 * r527;
        double r529 = cos(r497);
        double r530 = pow(r529, r519);
        double r531 = r528 / r530;
        double r532 = r531 * r531;
        double r533 = r504 - r532;
        double r534 = r506 / r533;
        double r535 = r504 + r531;
        double r536 = fma(r534, r535, r514);
        double r537 = r536 + r515;
        double r538 = r518 ? r524 : r537;
        double r539 = r499 ? r516 : r538;
        return r539;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.6
Target	15.4
Herbie	15.8

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

Derivation

1. Split input into 3 regimes

2. if eps < -4.530799695865795e-32

   1. Initial program 31.0

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

   3. Applied tan-sum 2.8

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

   5. Applied add-cube-cbrt 3.2

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

   6. Applied flip-- 3.2

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

   7. Applied associate-/r/ 3.2

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

   8. Applied prod-diff 3.2

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

   9. Simplified 2.8

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

   10. Simplified 2.8

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

   12. Applied expm1-log1p-u 2.9

       \[\leadsto \left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)}} - \tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\]

   if -4.530799695865795e-32 < eps < 1.047829285418837e-23

   1. Initial program 46.1

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

   3. Applied tan-sum 46.1

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

   5. Applied add-cube-cbrt 46.9

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

   6. Applied flip-- 46.9

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

   7. Applied associate-/r/ 46.9

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

   8. Applied prod-diff 47.0

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

   9. Simplified 47.0

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

   10. Simplified 46.1

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

   11. Taylor expanded around 0 32.3

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

   12. Simplified 32.3

       \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)} + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\]

   if 1.047829285418837e-23 < eps

   1. Initial program 30.1

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

   3. Applied tan-sum 1.6

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

   5. Applied add-cube-cbrt 1.9

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

   6. Applied flip-- 1.9

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

   7. Applied associate-/r/ 1.9

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

   8. Applied prod-diff 1.9

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

   9. Simplified 1.6

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

   10. Simplified 1.6

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

   12. Applied tan-quot 1.6

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

   13. Applied associate-*r/ 1.6

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

   14. Applied tan-quot 1.6

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

   15. Applied associate-*r/ 1.7

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

   16. Applied frac-times 1.7

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

   17. Simplified 1.7

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

   18. Simplified 1.7

       \[\leadsto \left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \frac{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}{\color{blue}{{\left(\cos \varepsilon\right)}^{2}}}} - \tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\]
   19. Using strategy rm

   20. Applied flip-- 1.7

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

   21. Applied associate-/r/ 1.7

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

   22. Applied fma-neg 1.7

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 \cdot 1 - \frac{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}} \cdot \frac{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}}, 1 + \frac{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}, -\tan x\right)} + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.5307996958657947 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)} - \tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \mathbf{elif}\;\varepsilon \le 1.04782928541883701 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\tan x + \tan \varepsilon\right) \cdot \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right)}{1 - \frac{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}} \cdot \frac{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}}, 1 + \frac{\left(\tan x \cdot \tan x\right) \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2}}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))