Toniolo and Linder, Equation (10-)

Average Error: 48.4 → 11.1

Time: 2.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -7.83518389073447 \cdot 10^{170} \lor \neg \left(k \le 6.69600377793563804 \cdot 10^{153}\right):\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{1}\right)}^{2}}\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\frac{1}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{1}\right)}^{2}}\right)\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r508704 = 2.0;
        double r508705 = t;
        double r508706 = 3.0;
        double r508707 = pow(r508705, r508706);
        double r508708 = l;
        double r508709 = r508708 * r508708;
        double r508710 = r508707 / r508709;
        double r508711 = k;
        double r508712 = sin(r508711);
        double r508713 = r508710 * r508712;
        double r508714 = tan(r508711);
        double r508715 = r508713 * r508714;
        double r508716 = 1.0;
        double r508717 = r508711 / r508705;
        double r508718 = pow(r508717, r508704);
        double r508719 = r508716 + r508718;
        double r508720 = r508719 - r508716;
        double r508721 = r508715 * r508720;
        double r508722 = r508704 / r508721;
        return r508722;
}

↓

double f(double t, double l, double k) {
        double r508723 = k;
        double r508724 = -7.83518389073447e+170;
        bool r508725 = r508723 <= r508724;
        double r508726 = 6.696003777935638e+153;
        bool r508727 = r508723 <= r508726;
        double r508728 = !r508727;
        bool r508729 = r508725 || r508728;
        double r508730 = 2.0;
        double r508731 = 1.0;
        double r508732 = 2.0;
        double r508733 = r508730 / r508732;
        double r508734 = pow(r508723, r508733);
        double r508735 = t;
        double r508736 = 1.0;
        double r508737 = pow(r508735, r508736);
        double r508738 = r508734 * r508737;
        double r508739 = r508734 * r508738;
        double r508740 = r508731 / r508739;
        double r508741 = pow(r508740, r508736);
        double r508742 = sin(r508723);
        double r508743 = cbrt(r508742);
        double r508744 = 4.0;
        double r508745 = pow(r508743, r508744);
        double r508746 = l;
        double r508747 = r508745 / r508746;
        double r508748 = r508731 / r508747;
        double r508749 = cbrt(r508731);
        double r508750 = pow(r508749, r508732);
        double r508751 = r508748 / r508750;
        double r508752 = r508741 * r508751;
        double r508753 = cos(r508723);
        double r508754 = r508731 / r508746;
        double r508755 = r508753 / r508754;
        double r508756 = pow(r508743, r508732);
        double r508757 = r508755 / r508756;
        double r508758 = r508752 * r508757;
        double r508759 = r508730 * r508758;
        double r508760 = pow(r508723, r508730);
        double r508761 = r508731 / r508760;
        double r508762 = pow(r508761, r508736);
        double r508763 = r508731 / r508737;
        double r508764 = pow(r508763, r508736);
        double r508765 = r508764 * r508751;
        double r508766 = r508762 * r508765;
        double r508767 = r508766 * r508757;
        double r508768 = r508730 * r508767;
        double r508769 = r508729 ? r508759 : r508768;
        return r508769;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -7.83518389073447e+170 or 6.696003777935638e+153 < k

   1. Initial program 39.9

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

   2. Simplified 34.5

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

   3. Taylor expanded around inf 24.7

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

   5. Applied add-cube-cbrt 24.7

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

   6. Applied unpow-prod-down 24.7

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

   7. Applied associate-/r* 24.7

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

   8. Simplified 24.7

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

   10. Applied *-un-lft-identity 24.7

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

   11. Applied cbrt-prod 24.7

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

   12. Applied unpow-prod-down 24.7

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

   13. Applied div-inv 24.7

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

   14. Applied *-un-lft-identity 24.7

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

   15. Applied times-frac 24.7

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

   16. Applied times-frac 24.7

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

   17. Applied associate-*r* 22.7

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

   19. Applied sqr-pow 22.7

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

   20. Applied associate-*l* 15.0

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

   if -7.83518389073447e+170 < k < 6.696003777935638e+153

   1. Initial program 53.7

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

   2. Simplified 44.3

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

   3. Taylor expanded around inf 20.9

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

   5. Applied add-cube-cbrt 21.4

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

   6. Applied unpow-prod-down 21.4

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

   7. Applied associate-/r* 21.4

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

   8. Simplified 19.4

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

   10. Applied *-un-lft-identity 19.4

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

   11. Applied cbrt-prod 19.4

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

   12. Applied unpow-prod-down 19.4

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

   13. Applied div-inv 19.4

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

   14. Applied *-un-lft-identity 19.4

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

   15. Applied times-frac 19.2

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

   16. Applied times-frac 18.2

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

   17. Applied associate-*r* 12.5

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

   19. Applied *-un-lft-identity 12.5

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

   20. Applied times-frac 12.4

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

   21. Applied unpow-prod-down 12.4

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

   22. Applied associate-*l* 8.7

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -7.83518389073447 \cdot 10^{170} \lor \neg \left(k \le 6.69600377793563804 \cdot 10^{153}\right):\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{1}\right)}^{2}}\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\frac{1}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{1}\right)}^{2}}\right)\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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))))