Toniolo and Linder, Equation (10-)

Average Error: 48.7 → 11.5

Time: 1.8m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 0.0:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{{\left(\sin k\right)}^{2}}{\ell \cdot \cos k}}{\ell}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 2.429270327298531141168205995229653525329 \cdot 10^{-217}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{\ell}{\frac{\sin k}{\ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\ell \cdot \cos k\right)\right)}{{\left(\sin k\right)}^{2}} \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r186701 = 2.0;
        double r186702 = t;
        double r186703 = 3.0;
        double r186704 = pow(r186702, r186703);
        double r186705 = l;
        double r186706 = r186705 * r186705;
        double r186707 = r186704 / r186706;
        double r186708 = k;
        double r186709 = sin(r186708);
        double r186710 = r186707 * r186709;
        double r186711 = tan(r186708);
        double r186712 = r186710 * r186711;
        double r186713 = 1.0;
        double r186714 = r186708 / r186702;
        double r186715 = pow(r186714, r186701);
        double r186716 = r186713 + r186715;
        double r186717 = r186716 - r186713;
        double r186718 = r186712 * r186717;
        double r186719 = r186701 / r186718;
        return r186719;
}

↓

double f(double t, double l, double k) {
        double r186720 = l;
        double r186721 = r186720 * r186720;
        double r186722 = 0.0;
        bool r186723 = r186721 <= r186722;
        double r186724 = 2.0;
        double r186725 = 1.0;
        double r186726 = k;
        double r186727 = 2.0;
        double r186728 = r186724 / r186727;
        double r186729 = pow(r186726, r186728);
        double r186730 = t;
        double r186731 = 1.0;
        double r186732 = pow(r186730, r186731);
        double r186733 = r186729 * r186732;
        double r186734 = r186725 / r186733;
        double r186735 = pow(r186734, r186731);
        double r186736 = r186725 / r186729;
        double r186737 = pow(r186736, r186731);
        double r186738 = sin(r186726);
        double r186739 = pow(r186738, r186727);
        double r186740 = cos(r186726);
        double r186741 = r186720 * r186740;
        double r186742 = r186739 / r186741;
        double r186743 = r186742 / r186720;
        double r186744 = r186737 / r186743;
        double r186745 = r186735 * r186744;
        double r186746 = r186724 * r186745;
        double r186747 = 2.429270327298531e-217;
        bool r186748 = r186721 <= r186747;
        double r186749 = pow(r186726, r186724);
        double r186750 = r186749 * r186732;
        double r186751 = r186725 / r186750;
        double r186752 = pow(r186751, r186731);
        double r186753 = r186740 / r186738;
        double r186754 = r186738 / r186720;
        double r186755 = r186720 / r186754;
        double r186756 = r186753 * r186755;
        double r186757 = r186752 * r186756;
        double r186758 = r186724 * r186757;
        double r186759 = r186737 * r186741;
        double r186760 = r186720 * r186759;
        double r186761 = r186760 / r186739;
        double r186762 = r186736 / r186732;
        double r186763 = pow(r186762, r186731);
        double r186764 = r186761 * r186763;
        double r186765 = r186724 * r186764;
        double r186766 = r186748 ? r186758 : r186765;
        double r186767 = r186723 ? r186746 : r186766;
        return r186767;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (* l l) < 0.0

   1. Initial program 46.5

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

   2. Simplified 36.9

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

   3. Taylor expanded around inf 20.5

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

   5. Applied sqr-pow 20.5

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

   6. Applied associate-*r* 20.5

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

   7. Simplified 20.5

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

   9. Applied *-un-lft-identity 20.5

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

   10. Applied times-frac 20.5

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

   11. Applied unpow-prod-down 20.5

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

   12. Applied associate-*l* 20.4

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

   13. Simplified 20.4

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

   15. Applied associate-/l* 20.4

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

   16. Simplified 13.3

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

   if 0.0 < (* l l) < 2.429270327298531e-217

   1. Initial program 45.5

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

   2. Simplified 34.6

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

   3. Taylor expanded around inf 5.3

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

   5. Applied add-sqr-sqrt 35.1

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

   6. Applied unpow-prod-down 35.1

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

   7. Applied times-frac 35.1

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

   8. Simplified 35.1

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

   9. Simplified 4.4

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

   if 2.429270327298531e-217 < (* l l)

   1. Initial program 50.4

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

   2. Simplified 43.6

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

   3. Taylor expanded around inf 26.2

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

   5. Applied sqr-pow 26.2

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

   6. Applied associate-*r* 22.2

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

   7. Simplified 22.2

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

   9. Applied *-un-lft-identity 22.2

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

   10. Applied times-frac 22.0

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

   11. Applied unpow-prod-down 22.0

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

   12. Applied associate-*l* 19.7

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

   13. Simplified 19.6

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

   15. Applied *-un-lft-identity 19.6

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

   16. Applied associate-*l* 19.6

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

   17. Simplified 11.8

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

4. Final simplification 11.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 0.0:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{{\left(\sin k\right)}^{2}}{\ell \cdot \cos k}}{\ell}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 2.429270327298531141168205995229653525329 \cdot 10^{-217}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \frac{\ell}{\frac{\sin k}{\ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\ell \cdot \cos k\right)\right)}{{\left(\sin k\right)}^{2}} \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))