Toniolo and Linder, Equation (10+)

Average Error: 32.1 → 7.7

Time: 1.2m

Precision: 64

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4.5062174373973056 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{-2}{\sin k \cdot t}}{\frac{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\tan k \cdot \frac{t}{\ell}\right)\right) \cdot \left(-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)\right)}{\frac{\ell}{\sqrt[3]{t}}}}\\ \mathbf{elif}\;t \le 0.006592551945511662:\\ \;\;\;\;\frac{\frac{-2}{\frac{t}{\ell} \cdot \sin k}}{-\left(\frac{\sin k}{\frac{\cos k \cdot \ell}{k \cdot k}} + 2 \cdot \frac{t \cdot t}{\frac{\cos k \cdot \ell}{\sin k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{\frac{t}{\ell} \cdot \sin k}}{\frac{-1}{\frac{1}{\tan k \cdot t}} \cdot \left(\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\ell} \cdot t\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r3385520 = 2.0;
        double r3385521 = t;
        double r3385522 = 3.0;
        double r3385523 = pow(r3385521, r3385522);
        double r3385524 = l;
        double r3385525 = r3385524 * r3385524;
        double r3385526 = r3385523 / r3385525;
        double r3385527 = k;
        double r3385528 = sin(r3385527);
        double r3385529 = r3385526 * r3385528;
        double r3385530 = tan(r3385527);
        double r3385531 = r3385529 * r3385530;
        double r3385532 = 1.0;
        double r3385533 = r3385527 / r3385521;
        double r3385534 = pow(r3385533, r3385520);
        double r3385535 = r3385532 + r3385534;
        double r3385536 = r3385535 + r3385532;
        double r3385537 = r3385531 * r3385536;
        double r3385538 = r3385520 / r3385537;
        return r3385538;
}

↓

double f(double t, double l, double k) {
        double r3385539 = t;
        double r3385540 = -4.5062174373973056e+105;
        bool r3385541 = r3385539 <= r3385540;
        double r3385542 = -2.0;
        double r3385543 = k;
        double r3385544 = sin(r3385543);
        double r3385545 = r3385544 * r3385539;
        double r3385546 = r3385542 / r3385545;
        double r3385547 = cbrt(r3385539);
        double r3385548 = r3385547 * r3385547;
        double r3385549 = tan(r3385543);
        double r3385550 = l;
        double r3385551 = r3385539 / r3385550;
        double r3385552 = r3385549 * r3385551;
        double r3385553 = r3385548 * r3385552;
        double r3385554 = r3385543 / r3385539;
        double r3385555 = r3385554 * r3385554;
        double r3385556 = 2.0;
        double r3385557 = r3385555 + r3385556;
        double r3385558 = -r3385557;
        double r3385559 = r3385553 * r3385558;
        double r3385560 = r3385550 / r3385547;
        double r3385561 = r3385559 / r3385560;
        double r3385562 = r3385546 / r3385561;
        double r3385563 = 0.006592551945511662;
        bool r3385564 = r3385539 <= r3385563;
        double r3385565 = r3385551 * r3385544;
        double r3385566 = r3385542 / r3385565;
        double r3385567 = cos(r3385543);
        double r3385568 = r3385567 * r3385550;
        double r3385569 = r3385543 * r3385543;
        double r3385570 = r3385568 / r3385569;
        double r3385571 = r3385544 / r3385570;
        double r3385572 = r3385539 * r3385539;
        double r3385573 = r3385568 / r3385544;
        double r3385574 = r3385572 / r3385573;
        double r3385575 = r3385556 * r3385574;
        double r3385576 = r3385571 + r3385575;
        double r3385577 = -r3385576;
        double r3385578 = r3385566 / r3385577;
        double r3385579 = -1.0;
        double r3385580 = 1.0;
        double r3385581 = r3385549 * r3385539;
        double r3385582 = r3385580 / r3385581;
        double r3385583 = r3385579 / r3385582;
        double r3385584 = r3385557 / r3385550;
        double r3385585 = r3385584 * r3385539;
        double r3385586 = r3385583 * r3385585;
        double r3385587 = r3385566 / r3385586;
        double r3385588 = r3385564 ? r3385578 : r3385587;
        double r3385589 = r3385541 ? r3385562 : r3385588;
        return r3385589;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -4.5062174373973056e+105

   1. Initial program 22.8

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 8.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot t\right)}}{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]
   3. Using strategy rm

   4. Applied frac-2neg 8.3

      \[\leadsto \color{blue}{\frac{-\frac{\frac{2}{\tan k}}{\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot t\right)}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}}\]

   5. Simplified 4.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{-2}{\tan k}}{\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}}{t}}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 4.3

      \[\leadsto \frac{\frac{\frac{\frac{-2}{\tan k}}{\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}}{\color{blue}{1 \cdot t}}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]

   8. Applied div-inv 4.3

      \[\leadsto \frac{\frac{\frac{\color{blue}{-2 \cdot \frac{1}{\tan k}}}{\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}}{1 \cdot t}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]

   9. Applied times-frac 4.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{-2}{\sin k \cdot \frac{t}{\ell}} \cdot \frac{\frac{1}{\tan k}}{\frac{t}{\ell}}}}{1 \cdot t}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]

   10. Applied times-frac 0.7

       \[\leadsto \frac{\color{blue}{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1} \cdot \frac{\frac{\frac{1}{\tan k}}{\frac{t}{\ell}}}{t}}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]

   11. Applied associate-/l* 1.1

       \[\leadsto \color{blue}{\frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\frac{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}{\frac{\frac{\frac{1}{\tan k}}{\frac{t}{\ell}}}{t}}}}\]

   12. Simplified 1.1

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\color{blue}{\frac{\left(-\frac{k}{t} \cdot \frac{k}{t}\right) + -2}{\frac{1}{\frac{t}{\ell} \cdot \tan k}} \cdot t}}\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 1.3

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\frac{\left(-\frac{k}{t} \cdot \frac{k}{t}\right) + -2}{\frac{1}{\frac{t}{\ell} \cdot \tan k}} \cdot \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}\]

   15. Applied associate-*r* 1.3

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\color{blue}{\left(\frac{\left(-\frac{k}{t} \cdot \frac{k}{t}\right) + -2}{\frac{1}{\frac{t}{\ell} \cdot \tan k}} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \sqrt[3]{t}}}\]

   16. Simplified 1.4

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\color{blue}{\left(\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(-\frac{k}{t}\right) \cdot \frac{k}{t} + -2\right)\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right)} \cdot \sqrt[3]{t}}\]
   17. Using strategy rm

   18. Applied *-un-lft-identity 1.4

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{\color{blue}{1 \cdot 1}}}{\left(\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(-\frac{k}{t}\right) \cdot \frac{k}{t} + -2\right)\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right) \cdot \sqrt[3]{t}}\]

   19. Applied associate-*r/ 1.4

       \[\leadsto \frac{\frac{\frac{-2}{\color{blue}{\frac{\sin k \cdot t}{\ell}}}}{1 \cdot 1}}{\left(\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(-\frac{k}{t}\right) \cdot \frac{k}{t} + -2\right)\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right) \cdot \sqrt[3]{t}}\]

   20. Applied associate-/r/ 1.4

       \[\leadsto \frac{\frac{\color{blue}{\frac{-2}{\sin k \cdot t} \cdot \ell}}{1 \cdot 1}}{\left(\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(-\frac{k}{t}\right) \cdot \frac{k}{t} + -2\right)\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right) \cdot \sqrt[3]{t}}\]

   21. Applied times-frac 1.4

       \[\leadsto \frac{\color{blue}{\frac{\frac{-2}{\sin k \cdot t}}{1} \cdot \frac{\ell}{1}}}{\left(\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(-\frac{k}{t}\right) \cdot \frac{k}{t} + -2\right)\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right) \cdot \sqrt[3]{t}}\]

   22. Applied associate-/l* 1.9

       \[\leadsto \color{blue}{\frac{\frac{\frac{-2}{\sin k \cdot t}}{1}}{\frac{\left(\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(-\frac{k}{t}\right) \cdot \frac{k}{t} + -2\right)\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right) \cdot \sqrt[3]{t}}{\frac{\ell}{1}}}}\]

   23. Simplified 1.5

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot t}}{1}}{\color{blue}{\frac{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\tan k \cdot \frac{t}{\ell}\right)\right) \cdot \left(-2 + \frac{k}{t} \cdot \left(-\frac{k}{t}\right)\right)}{\frac{\ell}{\sqrt[3]{t}}}}}\]

   if -4.5062174373973056e+105 < t < 0.006592551945511662

   1. Initial program 43.0

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 33.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot t\right)}}{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]
   3. Using strategy rm

   4. Applied frac-2neg 33.2

      \[\leadsto \color{blue}{\frac{-\frac{\frac{2}{\tan k}}{\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot t\right)}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}}\]

   5. Simplified 30.4

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{-2}{\tan k}}{\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}}{t}}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 30.4

      \[\leadsto \frac{\frac{\frac{\frac{-2}{\tan k}}{\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}}{\color{blue}{1 \cdot t}}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]

   8. Applied div-inv 30.4

      \[\leadsto \frac{\frac{\frac{\color{blue}{-2 \cdot \frac{1}{\tan k}}}{\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}}{1 \cdot t}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]

   9. Applied times-frac 29.6

      \[\leadsto \frac{\frac{\color{blue}{\frac{-2}{\sin k \cdot \frac{t}{\ell}} \cdot \frac{\frac{1}{\tan k}}{\frac{t}{\ell}}}}{1 \cdot t}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]

   10. Applied times-frac 29.1

       \[\leadsto \frac{\color{blue}{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1} \cdot \frac{\frac{\frac{1}{\tan k}}{\frac{t}{\ell}}}{t}}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]

   11. Applied associate-/l* 25.6

       \[\leadsto \color{blue}{\frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\frac{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}{\frac{\frac{\frac{1}{\tan k}}{\frac{t}{\ell}}}{t}}}}\]

   12. Simplified 22.7

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\color{blue}{\frac{\left(-\frac{k}{t} \cdot \frac{k}{t}\right) + -2}{\frac{1}{\frac{t}{\ell} \cdot \tan k}} \cdot t}}\]

   13. Taylor expanded around inf 16.3

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\color{blue}{-\left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{\sin k \cdot {k}^{2}}{\cos k \cdot \ell}\right)}}\]

   14. Simplified 14.2

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\color{blue}{-\left(\frac{t \cdot t}{\frac{\cos k \cdot \ell}{\sin k}} \cdot 2 + \frac{\sin k}{\frac{\ell \cdot \cos k}{k \cdot k}}\right)}}\]

   if 0.006592551945511662 < t

   1. Initial program 21.9

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 10.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot t\right)}}{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]
   3. Using strategy rm

   4. Applied frac-2neg 10.5

      \[\leadsto \color{blue}{\frac{-\frac{\frac{2}{\tan k}}{\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot t\right)}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}}\]

   5. Simplified 6.0

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{-2}{\tan k}}{\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}}{t}}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 6.0

      \[\leadsto \frac{\frac{\frac{\frac{-2}{\tan k}}{\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}}{\color{blue}{1 \cdot t}}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]

   8. Applied div-inv 6.0

      \[\leadsto \frac{\frac{\frac{\color{blue}{-2 \cdot \frac{1}{\tan k}}}{\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}}{1 \cdot t}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]

   9. Applied times-frac 5.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{-2}{\sin k \cdot \frac{t}{\ell}} \cdot \frac{\frac{1}{\tan k}}{\frac{t}{\ell}}}}{1 \cdot t}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]

   10. Applied times-frac 2.4

       \[\leadsto \frac{\color{blue}{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1} \cdot \frac{\frac{\frac{1}{\tan k}}{\frac{t}{\ell}}}{t}}}{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}\]

   11. Applied associate-/l* 2.3

       \[\leadsto \color{blue}{\frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\frac{-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)}{\frac{\frac{\frac{1}{\tan k}}{\frac{t}{\ell}}}{t}}}}\]

   12. Simplified 2.3

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\color{blue}{\frac{\left(-\frac{k}{t} \cdot \frac{k}{t}\right) + -2}{\frac{1}{\frac{t}{\ell} \cdot \tan k}} \cdot t}}\]
   13. Using strategy rm

   14. Applied associate-*l/ 2.2

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\frac{\left(-\frac{k}{t} \cdot \frac{k}{t}\right) + -2}{\frac{1}{\color{blue}{\frac{t \cdot \tan k}{\ell}}}} \cdot t}\]

   15. Applied associate-/r/ 2.2

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\frac{\left(-\frac{k}{t} \cdot \frac{k}{t}\right) + -2}{\color{blue}{\frac{1}{t \cdot \tan k} \cdot \ell}} \cdot t}\]

   16. Applied *-un-lft-identity 2.2

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\frac{\color{blue}{1 \cdot \left(\left(-\frac{k}{t} \cdot \frac{k}{t}\right) + -2\right)}}{\frac{1}{t \cdot \tan k} \cdot \ell} \cdot t}\]

   17. Applied times-frac 2.2

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\color{blue}{\left(\frac{1}{\frac{1}{t \cdot \tan k}} \cdot \frac{\left(-\frac{k}{t} \cdot \frac{k}{t}\right) + -2}{\ell}\right)} \cdot t}\]

   18. Applied associate-*l* 2.2

       \[\leadsto \frac{\frac{\frac{-2}{\sin k \cdot \frac{t}{\ell}}}{1}}{\color{blue}{\frac{1}{\frac{1}{t \cdot \tan k}} \cdot \left(\frac{\left(-\frac{k}{t} \cdot \frac{k}{t}\right) + -2}{\ell} \cdot t\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.5062174373973056 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{-2}{\sin k \cdot t}}{\frac{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\tan k \cdot \frac{t}{\ell}\right)\right) \cdot \left(-\left(\frac{k}{t} \cdot \frac{k}{t} + 2\right)\right)}{\frac{\ell}{\sqrt[3]{t}}}}\\ \mathbf{elif}\;t \le 0.006592551945511662:\\ \;\;\;\;\frac{\frac{-2}{\frac{t}{\ell} \cdot \sin k}}{-\left(\frac{\sin k}{\frac{\cos k \cdot \ell}{k \cdot k}} + 2 \cdot \frac{t \cdot t}{\frac{\cos k \cdot \ell}{\sin k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{\frac{t}{\ell} \cdot \sin k}}{\frac{-1}{\frac{1}{\tan k \cdot t}} \cdot \left(\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\ell} \cdot t\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))