Toniolo and Linder, Equation (10-)

Average Error: 48.4 → 12.8

Time: 54.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 36341823884774118312709128287999229952:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\left(\ell \cdot \frac{\cos k}{\sin k}\right) \cdot \frac{\ell}{\sin k}\right)\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 2.702815358468802609888516667751806875166 \cdot 10^{307}:\\ \;\;\;\;\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left(\frac{\cos k}{\sin k} \cdot \frac{{\ell}^{2}}{\sin k}\right) \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r80632 = 2.0;
        double r80633 = t;
        double r80634 = 3.0;
        double r80635 = pow(r80633, r80634);
        double r80636 = l;
        double r80637 = r80636 * r80636;
        double r80638 = r80635 / r80637;
        double r80639 = k;
        double r80640 = sin(r80639);
        double r80641 = r80638 * r80640;
        double r80642 = tan(r80639);
        double r80643 = r80641 * r80642;
        double r80644 = 1.0;
        double r80645 = r80639 / r80633;
        double r80646 = pow(r80645, r80632);
        double r80647 = r80644 + r80646;
        double r80648 = r80647 - r80644;
        double r80649 = r80643 * r80648;
        double r80650 = r80632 / r80649;
        return r80650;
}

↓

double f(double t, double l, double k) {
        double r80651 = l;
        double r80652 = r80651 * r80651;
        double r80653 = 3.634182388477412e+37;
        bool r80654 = r80652 <= r80653;
        double r80655 = 2.0;
        double r80656 = 1.0;
        double r80657 = k;
        double r80658 = 2.0;
        double r80659 = r80655 / r80658;
        double r80660 = pow(r80657, r80659);
        double r80661 = t;
        double r80662 = 1.0;
        double r80663 = pow(r80661, r80662);
        double r80664 = r80660 * r80663;
        double r80665 = r80660 * r80664;
        double r80666 = r80656 / r80665;
        double r80667 = pow(r80666, r80662);
        double r80668 = cos(r80657);
        double r80669 = sin(r80657);
        double r80670 = r80668 / r80669;
        double r80671 = r80651 * r80670;
        double r80672 = r80651 / r80669;
        double r80673 = r80671 * r80672;
        double r80674 = r80667 * r80673;
        double r80675 = r80655 * r80674;
        double r80676 = 2.7028153584688026e+307;
        bool r80677 = r80652 <= r80676;
        double r80678 = r80656 / r80660;
        double r80679 = pow(r80678, r80662);
        double r80680 = pow(r80651, r80658);
        double r80681 = r80680 / r80669;
        double r80682 = r80670 * r80681;
        double r80683 = r80678 / r80663;
        double r80684 = pow(r80683, r80662);
        double r80685 = r80682 * r80684;
        double r80686 = r80679 * r80685;
        double r80687 = r80686 * r80655;
        double r80688 = cbrt(r80661);
        double r80689 = r80688 * r80688;
        double r80690 = 3.0;
        double r80691 = pow(r80689, r80690);
        double r80692 = r80691 / r80651;
        double r80693 = pow(r80688, r80690);
        double r80694 = r80693 / r80651;
        double r80695 = r80692 * r80694;
        double r80696 = r80695 * r80669;
        double r80697 = tan(r80657);
        double r80698 = r80696 * r80697;
        double r80699 = r80655 / r80698;
        double r80700 = r80657 / r80661;
        double r80701 = pow(r80700, r80655);
        double r80702 = r80699 / r80701;
        double r80703 = r80677 ? r80687 : r80702;
        double r80704 = r80654 ? r80675 : r80703;
        return r80704;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (* l l) < 3.634182388477412e+37

   1. Initial program 44.9

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

   2. Simplified 35.4

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

   3. Taylor expanded around inf 12.4

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

   5. Applied sqr-pow 12.4

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

   6. Applied associate-*l* 11.2

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

   8. Applied add-sqr-sqrt 37.3

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

   9. Applied unpow-prod-down 37.3

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

   10. Applied times-frac 37.1

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

   11. Simplified 37.1

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

   12. Simplified 10.8

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

   14. Applied *-un-lft-identity 10.8

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

   15. Applied sqr-pow 10.8

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

   16. Applied times-frac 8.1

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

   17. Applied associate-*r* 6.1

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

   18. Simplified 6.1

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

   if 3.634182388477412e+37 < (* l l) < 2.7028153584688026e+307

   1. Initial program 48.2

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

   2. Simplified 41.1

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

   3. Taylor expanded around inf 21.6

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

   5. Applied sqr-pow 21.6

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

   6. Applied associate-*l* 15.0

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

   8. Applied add-sqr-sqrt 39.2

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

   9. Applied unpow-prod-down 39.2

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

   10. Applied times-frac 39.2

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

   11. Simplified 39.1

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

   12. Simplified 15.0

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

   14. Applied add-cube-cbrt 15.0

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

   15. Applied times-frac 14.3

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

   16. Applied unpow-prod-down 14.3

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

   17. Applied associate-*l* 7.5

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

   18. Simplified 7.3

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

   if 2.7028153584688026e+307 < (* l l)

   1. Initial program 63.9

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

   2. Simplified 63.9

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

   4. Applied add-cube-cbrt 63.9

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

   5. Applied unpow-prod-down 63.9

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

   6. Applied times-frac 48.9

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 36341823884774118312709128287999229952:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\left(\ell \cdot \frac{\cos k}{\sin k}\right) \cdot \frac{\ell}{\sin k}\right)\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 2.702815358468802609888516667751806875166 \cdot 10^{307}:\\ \;\;\;\;\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left(\frac{\cos k}{\sin k} \cdot \frac{{\ell}^{2}}{\sin k}\right) \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +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))))