Toniolo and Linder, Equation (10-)

Average Error: 48.4 → 10.9

Time: 1.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -3.35534735876246717 \cdot 10^{146} \lor \neg \left(k \le 1.3318466035921655 \cdot 10^{154}\right):\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{1}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}\right)\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r302467 = 2.0;
        double r302468 = t;
        double r302469 = 3.0;
        double r302470 = pow(r302468, r302469);
        double r302471 = l;
        double r302472 = r302471 * r302471;
        double r302473 = r302470 / r302472;
        double r302474 = k;
        double r302475 = sin(r302474);
        double r302476 = r302473 * r302475;
        double r302477 = tan(r302474);
        double r302478 = r302476 * r302477;
        double r302479 = 1.0;
        double r302480 = r302474 / r302468;
        double r302481 = pow(r302480, r302467);
        double r302482 = r302479 + r302481;
        double r302483 = r302482 - r302479;
        double r302484 = r302478 * r302483;
        double r302485 = r302467 / r302484;
        return r302485;
}

↓

double f(double t, double l, double k) {
        double r302486 = k;
        double r302487 = -3.355347358762467e+146;
        bool r302488 = r302486 <= r302487;
        double r302489 = 1.3318466035921655e+154;
        bool r302490 = r302486 <= r302489;
        double r302491 = !r302490;
        bool r302492 = r302488 || r302491;
        double r302493 = 2.0;
        double r302494 = 1.0;
        double r302495 = 2.0;
        double r302496 = r302493 / r302495;
        double r302497 = pow(r302486, r302496);
        double r302498 = t;
        double r302499 = 1.0;
        double r302500 = pow(r302498, r302499);
        double r302501 = r302497 * r302500;
        double r302502 = r302497 * r302501;
        double r302503 = r302494 / r302502;
        double r302504 = pow(r302503, r302499);
        double r302505 = cos(r302486);
        double r302506 = sin(r302486);
        double r302507 = cbrt(r302506);
        double r302508 = 4.0;
        double r302509 = pow(r302507, r302508);
        double r302510 = l;
        double r302511 = r302509 / r302510;
        double r302512 = r302505 / r302511;
        double r302513 = r302512 / r302494;
        double r302514 = r302504 * r302513;
        double r302515 = pow(r302507, r302495);
        double r302516 = r302510 / r302515;
        double r302517 = r302514 * r302516;
        double r302518 = r302493 * r302517;
        double r302519 = pow(r302486, r302493);
        double r302520 = r302494 / r302519;
        double r302521 = pow(r302520, r302499);
        double r302522 = r302494 / r302500;
        double r302523 = pow(r302522, r302499);
        double r302524 = r302523 * r302512;
        double r302525 = r302521 * r302524;
        double r302526 = r302525 * r302516;
        double r302527 = r302493 * r302526;
        double r302528 = r302492 ? r302518 : r302527;
        return r302528;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -3.355347358762467e+146 or 1.3318466035921655e+154 < k

   1. Initial program 40.1

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

   2. Simplified 34.6

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

   3. Taylor expanded around inf 24.9

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 24.9

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

   6. Applied unpow-prod-down 24.9

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

   7. Applied associate-/r* 24.9

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

   8. Simplified 24.8

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
   9. Using strategy rm

   10. Applied *-un-lft-identity 24.8

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\color{blue}{\left(1 \cdot \sqrt[3]{\sin k}\right)}}^{2}}\right)\]

   11. Applied unpow-prod-down 24.8

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{\color{blue}{{1}^{2} \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\]

   12. Applied associate-/r/ 24.8

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}} \cdot \ell}}{{1}^{2} \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   13. Applied times-frac 24.8

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \color{blue}{\left(\frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{1}^{2}} \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)}\right)\]

   14. Applied associate-*r* 22.9

       \[\leadsto 2 \cdot \color{blue}{\left(\left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{1}^{2}}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)}\]

   15. Simplified 22.9

       \[\leadsto 2 \cdot \left(\color{blue}{\left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{1}\right)} \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
   16. Using strategy rm

   17. Applied sqr-pow 22.9

       \[\leadsto 2 \cdot \left(\left({\left(\frac{1}{\color{blue}{\left({k}^{\left(\frac{2}{2}\right)} \cdot {k}^{\left(\frac{2}{2}\right)}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{1}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   18. Applied associate-*l* 15.1

       \[\leadsto 2 \cdot \left(\left({\left(\frac{1}{\color{blue}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{1}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   if -3.355347358762467e+146 < k < 1.3318466035921655e+154

   1. Initial program 54.3

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

   2. Simplified 44.7

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

   3. Taylor expanded around inf 20.6

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 21.1

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

   6. Applied unpow-prod-down 21.1

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

   7. Applied associate-/r* 21.1

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

   8. Simplified 19.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
   9. Using strategy rm

   10. Applied *-un-lft-identity 19.0

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\color{blue}{\left(1 \cdot \sqrt[3]{\sin k}\right)}}^{2}}\right)\]

   11. Applied unpow-prod-down 19.0

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{\color{blue}{{1}^{2} \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\]

   12. Applied associate-/r/ 18.8

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}} \cdot \ell}}{{1}^{2} \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   13. Applied times-frac 17.7

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \color{blue}{\left(\frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{1}^{2}} \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)}\right)\]

   14. Applied associate-*r* 11.8

       \[\leadsto 2 \cdot \color{blue}{\left(\left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{1}^{2}}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)}\]

   15. Simplified 11.8

       \[\leadsto 2 \cdot \left(\color{blue}{\left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{1}\right)} \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
   16. Using strategy rm

   17. Applied *-un-lft-identity 11.8

       \[\leadsto 2 \cdot \left(\left({\left(\frac{\color{blue}{1 \cdot 1}}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{1}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   18. Applied times-frac 11.7

       \[\leadsto 2 \cdot \left(\left({\color{blue}{\left(\frac{1}{{k}^{2}} \cdot \frac{1}{{t}^{1}}\right)}}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{1}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   19. Applied unpow-prod-down 11.7

       \[\leadsto 2 \cdot \left(\left(\color{blue}{\left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right)} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{1}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   20. Applied associate-*l* 8.0

       \[\leadsto 2 \cdot \left(\color{blue}{\left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{1}\right)\right)} \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   21. Simplified 8.0

       \[\leadsto 2 \cdot \left(\left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \color{blue}{\left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}\right)}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -3.35534735876246717 \cdot 10^{146} \lor \neg \left(k \le 1.3318466035921655 \cdot 10^{154}\right):\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{1}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}\right)\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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))))