Toniolo and Linder, Equation (10-)

Average Error: 48.5 → 16.1

Time: 52.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -2.987429350833538979849181671258036586182 \cdot 10^{-155} \lor \neg \left(\ell \le 2.033640709365090133619923281088995101044 \cdot 10^{-153}\right):\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left(\cos k \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right) \cdot \frac{{\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r80693 = 2.0;
        double r80694 = t;
        double r80695 = 3.0;
        double r80696 = pow(r80694, r80695);
        double r80697 = l;
        double r80698 = r80697 * r80697;
        double r80699 = r80696 / r80698;
        double r80700 = k;
        double r80701 = sin(r80700);
        double r80702 = r80699 * r80701;
        double r80703 = tan(r80700);
        double r80704 = r80702 * r80703;
        double r80705 = 1.0;
        double r80706 = r80700 / r80694;
        double r80707 = pow(r80706, r80693);
        double r80708 = r80705 + r80707;
        double r80709 = r80708 - r80705;
        double r80710 = r80704 * r80709;
        double r80711 = r80693 / r80710;
        return r80711;
}

↓

double f(double t, double l, double k) {
        double r80712 = l;
        double r80713 = -2.987429350833539e-155;
        bool r80714 = r80712 <= r80713;
        double r80715 = 2.0336407093650901e-153;
        bool r80716 = r80712 <= r80715;
        double r80717 = !r80716;
        bool r80718 = r80714 || r80717;
        double r80719 = 2.0;
        double r80720 = 1.0;
        double r80721 = k;
        double r80722 = 2.0;
        double r80723 = r80719 / r80722;
        double r80724 = pow(r80721, r80723);
        double r80725 = r80720 / r80724;
        double r80726 = 1.0;
        double r80727 = pow(r80725, r80726);
        double r80728 = cos(r80721);
        double r80729 = t;
        double r80730 = pow(r80729, r80726);
        double r80731 = r80724 * r80730;
        double r80732 = r80720 / r80731;
        double r80733 = pow(r80732, r80726);
        double r80734 = r80728 * r80733;
        double r80735 = pow(r80712, r80722);
        double r80736 = sin(r80721);
        double r80737 = pow(r80736, r80722);
        double r80738 = r80735 / r80737;
        double r80739 = r80734 * r80738;
        double r80740 = r80727 * r80739;
        double r80741 = r80719 * r80740;
        double r80742 = r80724 * r80731;
        double r80743 = r80720 / r80742;
        double r80744 = pow(r80743, r80726);
        double r80745 = cbrt(r80736);
        double r80746 = 4.0;
        double r80747 = pow(r80745, r80746);
        double r80748 = r80747 / r80712;
        double r80749 = r80748 / r80712;
        double r80750 = r80728 / r80749;
        double r80751 = pow(r80745, r80722);
        double r80752 = r80750 / r80751;
        double r80753 = r80744 * r80752;
        double r80754 = r80719 * r80753;
        double r80755 = r80718 ? r80741 : r80754;
        return r80755;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if l < -2.987429350833539e-155

   1. Initial program 48.8

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

   2. Simplified 41.9

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

   3. Taylor expanded around inf 23.0

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

   5. Applied sqr-pow 23.0

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

   6. Applied associate-*l* 19.6

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

   8. Applied *-un-lft-identity 19.6

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

   9. Applied times-frac 19.3

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

   10. Applied unpow-prod-down 19.3

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

   11. Applied associate-*l* 17.6

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

   13. Applied *-un-lft-identity 17.6

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

   14. Applied unpow-prod-down 17.6

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

   15. Applied times-frac 17.6

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

   16. Applied associate-*r* 17.6

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

   17. Simplified 17.6

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

   if -2.987429350833539e-155 < l < 2.0336407093650901e-153

   1. Initial program 47.0

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

   2. Simplified 38.3

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

   3. Taylor expanded around inf 19.7

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

   5. Applied sqr-pow 19.7

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

   6. Applied associate-*l* 19.7

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

   8. Applied add-cube-cbrt 19.7

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

   9. Applied unpow-prod-down 19.7

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

   10. Applied associate-/r* 19.5

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

   11. Simplified 13.2

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

   if 2.0336407093650901e-153 < l

   1. Initial program 49.6

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

   2. Simplified 41.7

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

   3. Taylor expanded around inf 23.2

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

   5. Applied sqr-pow 23.2

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

   6. Applied associate-*l* 20.2

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

   8. Applied *-un-lft-identity 20.2

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

   9. Applied times-frac 19.8

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

   10. Applied unpow-prod-down 19.8

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

   11. Applied associate-*l* 17.5

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

   13. Applied associate-*r/ 17.7

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

   14. Applied associate-*r/ 17.7

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

4. Final simplification 16.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -2.987429350833538979849181671258036586182 \cdot 10^{-155} \lor \neg \left(\ell \le 2.033640709365090133619923281088995101044 \cdot 10^{-153}\right):\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left(\cos k \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right) \cdot \frac{{\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(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))))