Toniolo and Linder, Equation (10+)

Average Error: 32.8 → 22.0

Time: 18.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 2.44576412721687134253434364277071819613 \cdot 10^{275}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \left(\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\sqrt[3]{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r129505 = 2.0;
        double r129506 = t;
        double r129507 = 3.0;
        double r129508 = pow(r129506, r129507);
        double r129509 = l;
        double r129510 = r129509 * r129509;
        double r129511 = r129508 / r129510;
        double r129512 = k;
        double r129513 = sin(r129512);
        double r129514 = r129511 * r129513;
        double r129515 = tan(r129512);
        double r129516 = r129514 * r129515;
        double r129517 = 1.0;
        double r129518 = r129512 / r129506;
        double r129519 = pow(r129518, r129505);
        double r129520 = r129517 + r129519;
        double r129521 = r129520 + r129517;
        double r129522 = r129516 * r129521;
        double r129523 = r129505 / r129522;
        return r129523;
}

↓

double f(double t, double l, double k) {
        double r129524 = l;
        double r129525 = r129524 * r129524;
        double r129526 = 2.4457641272168713e+275;
        bool r129527 = r129525 <= r129526;
        double r129528 = 1.0;
        double r129529 = t;
        double r129530 = cbrt(r129529);
        double r129531 = r129530 * r129530;
        double r129532 = 3.0;
        double r129533 = pow(r129531, r129532);
        double r129534 = r129528 / r129533;
        double r129535 = k;
        double r129536 = tan(r129535);
        double r129537 = cbrt(r129536);
        double r129538 = r129537 * r129537;
        double r129539 = r129534 / r129538;
        double r129540 = 2.0;
        double r129541 = pow(r129530, r129532);
        double r129542 = sin(r129535);
        double r129543 = r129541 * r129542;
        double r129544 = r129540 / r129543;
        double r129545 = r129544 * r129524;
        double r129546 = r129545 / r129537;
        double r129547 = 2.0;
        double r129548 = 1.0;
        double r129549 = r129535 / r129529;
        double r129550 = pow(r129549, r129540);
        double r129551 = fma(r129547, r129548, r129550);
        double r129552 = r129524 / r129551;
        double r129553 = r129546 * r129552;
        double r129554 = r129539 * r129553;
        double r129555 = cbrt(r129528);
        double r129556 = r129555 * r129555;
        double r129557 = r129532 / r129547;
        double r129558 = pow(r129531, r129557);
        double r129559 = r129556 / r129558;
        double r129560 = r129555 / r129558;
        double r129561 = r129560 * r129545;
        double r129562 = r129559 * r129561;
        double r129563 = r129562 / r129536;
        double r129564 = r129563 * r129552;
        double r129565 = r129527 ? r129554 : r129564;
        return r129565;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if (* l l) < 2.4457641272168713e+275

   1. Initial program 26.0

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

   2. Simplified 26.4

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 26.4

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   5. Applied times-frac 26.4

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

   6. Applied associate-*r* 25.3

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell}{1}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   7. Simplified 23.9

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k} \cdot \ell}{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 24.2

      \[\leadsto \frac{\frac{2}{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   10. Applied unpow-prod-down 24.2

       \[\leadsto \frac{\frac{2}{\color{blue}{\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   11. Applied associate-*l* 22.9

       \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k\right)}} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   12. Using strategy rm

   13. Applied *-un-lft-identity 22.9

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot 2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   14. Applied times-frac 22.9

       \[\leadsto \frac{\color{blue}{\left(\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   15. Applied associate-*l* 20.6

       \[\leadsto \frac{\color{blue}{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   16. Using strategy rm

   17. Applied add-cube-cbrt 20.6

       \[\leadsto \frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}{\color{blue}{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}\right) \cdot \sqrt[3]{\tan k}}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   18. Applied times-frac 20.5

       \[\leadsto \color{blue}{\left(\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\sqrt[3]{\tan k}}\right)} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   19. Applied associate-*l* 19.2

       \[\leadsto \color{blue}{\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \left(\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\sqrt[3]{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

   if 2.4457641272168713e+275 < (* l l)

   1. Initial program 61.2

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

   2. Simplified 61.1

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 61.1

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   5. Applied times-frac 56.6

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

   6. Applied associate-*r* 46.5

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell}{1}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   7. Simplified 46.5

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k} \cdot \ell}{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 46.7

      \[\leadsto \frac{\frac{2}{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   10. Applied unpow-prod-down 46.7

       \[\leadsto \frac{\frac{2}{\color{blue}{\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   11. Applied associate-*l* 45.8

       \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k\right)}} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   12. Using strategy rm

   13. Applied *-un-lft-identity 45.8

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot 2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   14. Applied times-frac 45.3

       \[\leadsto \frac{\color{blue}{\left(\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   15. Applied associate-*l* 40.6

       \[\leadsto \frac{\color{blue}{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   16. Using strategy rm

   17. Applied sqr-pow 40.6

       \[\leadsto \frac{\frac{1}{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}} \cdot \left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   18. Applied add-cube-cbrt 40.6

       \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   19. Applied times-frac 39.8

       \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)} \cdot \left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   20. Applied associate-*l* 33.7

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

4. Final simplification 22.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 2.44576412721687134253434364277071819613 \cdot 10^{275}:\\ \;\;\;\;\frac{\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \left(\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\sqrt[3]{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt[3]{1}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))