Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 10.7

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.6704179460034538 \cdot 10^{-223}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}}{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 3.08766200373374408 \cdot 10^{216}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{1}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{1}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r385907 = 2.0;
        double r385908 = t;
        double r385909 = 3.0;
        double r385910 = pow(r385908, r385909);
        double r385911 = l;
        double r385912 = r385911 * r385911;
        double r385913 = r385910 / r385912;
        double r385914 = k;
        double r385915 = sin(r385914);
        double r385916 = r385913 * r385915;
        double r385917 = tan(r385914);
        double r385918 = r385916 * r385917;
        double r385919 = 1.0;
        double r385920 = r385914 / r385908;
        double r385921 = pow(r385920, r385907);
        double r385922 = r385919 + r385921;
        double r385923 = r385922 - r385919;
        double r385924 = r385918 * r385923;
        double r385925 = r385907 / r385924;
        return r385925;
}

↓

double f(double t, double l, double k) {
        double r385926 = l;
        double r385927 = r385926 * r385926;
        double r385928 = 4.670417946003454e-223;
        bool r385929 = r385927 <= r385928;
        double r385930 = 2.0;
        double r385931 = 1.0;
        double r385932 = k;
        double r385933 = 2.0;
        double r385934 = r385930 / r385933;
        double r385935 = pow(r385932, r385934);
        double r385936 = t;
        double r385937 = 1.0;
        double r385938 = pow(r385936, r385937);
        double r385939 = r385935 * r385938;
        double r385940 = r385935 * r385939;
        double r385941 = r385931 / r385940;
        double r385942 = pow(r385941, r385937);
        double r385943 = sin(r385932);
        double r385944 = cbrt(r385943);
        double r385945 = pow(r385944, r385933);
        double r385946 = cbrt(r385926);
        double r385947 = r385946 * r385946;
        double r385948 = r385945 / r385947;
        double r385949 = r385948 / r385947;
        double r385950 = r385931 / r385949;
        double r385951 = sqrt(r385945);
        double r385952 = r385950 / r385951;
        double r385953 = r385942 * r385952;
        double r385954 = cos(r385932);
        double r385955 = r385945 / r385946;
        double r385956 = r385955 / r385946;
        double r385957 = r385954 / r385956;
        double r385958 = r385957 / r385951;
        double r385959 = r385953 * r385958;
        double r385960 = r385930 * r385959;
        double r385961 = 3.087662003733744e+216;
        bool r385962 = r385927 <= r385961;
        double r385963 = cbrt(r385931);
        double r385964 = r385963 * r385963;
        double r385965 = r385964 / r385935;
        double r385966 = pow(r385965, r385937);
        double r385967 = r385963 / r385939;
        double r385968 = pow(r385967, r385937);
        double r385969 = pow(r385926, r385933);
        double r385970 = r385954 * r385969;
        double r385971 = pow(r385943, r385933);
        double r385972 = r385970 / r385971;
        double r385973 = r385968 * r385972;
        double r385974 = r385966 * r385973;
        double r385975 = r385930 * r385974;
        double r385976 = cbrt(r385954);
        double r385977 = r385976 * r385976;
        double r385978 = 4.0;
        double r385979 = pow(r385944, r385978);
        double r385980 = r385979 / r385926;
        double r385981 = r385977 / r385980;
        double r385982 = pow(r385944, r385931);
        double r385983 = r385981 / r385982;
        double r385984 = r385942 * r385983;
        double r385985 = r385931 / r385926;
        double r385986 = r385976 / r385985;
        double r385987 = r385986 / r385982;
        double r385988 = r385984 * r385987;
        double r385989 = r385930 * r385988;
        double r385990 = r385962 ? r385975 : r385989;
        double r385991 = r385929 ? r385960 : r385990;
        return r385991;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (* l l) < 4.670417946003454e-223

   1. Initial program 45.6

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

   2. Simplified 36.2

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

   3. Taylor expanded around inf 16.3

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

   5. Applied sqr-pow 16.3

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

   6. Applied associate-*l* 16.2

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

   8. Applied add-cube-cbrt 16.3

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

   9. Applied unpow-prod-down 16.3

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

   10. Applied associate-/r* 16.0

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

   11. Simplified 11.4

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

   13. Applied add-sqr-sqrt 11.4

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

   14. Applied add-cube-cbrt 11.5

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

   15. Applied add-cube-cbrt 11.5

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

   16. Applied sqr-pow 11.5

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

   17. Applied times-frac 11.0

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

   18. Applied times-frac 11.0

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

   19. Applied *-un-lft-identity 11.0

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

   20. Applied times-frac 10.5

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

   21. Applied times-frac 8.8

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

   22. Applied associate-*r* 6.3

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

   if 4.670417946003454e-223 < (* l l) < 3.087662003733744e+216

   1. Initial program 44.2

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

   2. Simplified 34.3

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

   3. Taylor expanded around inf 9.7

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

   5. Applied sqr-pow 9.7

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

   6. Applied associate-*l* 5.1

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

   8. Applied add-cube-cbrt 5.1

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

   9. Applied times-frac 4.8

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

   10. Applied unpow-prod-down 4.8

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

   11. Applied associate-*l* 2.5

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

   if 3.087662003733744e+216 < (* l l)

   1. Initial program 59.9

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

   2. Simplified 58.0

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

   3. Taylor expanded around inf 53.8

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

   5. Applied sqr-pow 53.8

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

   6. Applied associate-*l* 51.6

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

   8. Applied add-cube-cbrt 51.8

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

   9. Applied unpow-prod-down 51.8

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

   10. Applied associate-/r* 51.8

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

   11. Simplified 51.8

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

   13. Applied sqr-pow 51.8

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

   14. Applied div-inv 51.8

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

   15. Applied add-cube-cbrt 51.8

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

   16. Applied times-frac 51.8

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

   17. Applied times-frac 51.8

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

   18. Applied associate-*r* 34.0

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

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.6704179460034538 \cdot 10^{-223}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}}{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 3.08766200373374408 \cdot 10^{216}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{1}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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))))