Toniolo and Linder, Equation (10-)

Average Error: 48.0 → 13.8

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -7.34197474785895879644805628021741244835 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le 4.162486401097287709225438879584039218636 \cdot 10^{148}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left(\frac{\cos k}{\left|\sin k\right|} \cdot \frac{\ell}{\frac{\left|\sin k\right|}{\ell}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot {\left(\sin k\right)}^{2}}}{{\left(\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{\frac{k}{t}}\right)}^{2}} \cdot \frac{\cos k}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r99859 = 2.0;
        double r99860 = t;
        double r99861 = 3.0;
        double r99862 = pow(r99860, r99861);
        double r99863 = l;
        double r99864 = r99863 * r99863;
        double r99865 = r99862 / r99864;
        double r99866 = k;
        double r99867 = sin(r99866);
        double r99868 = r99865 * r99867;
        double r99869 = tan(r99866);
        double r99870 = r99868 * r99869;
        double r99871 = 1.0;
        double r99872 = r99866 / r99860;
        double r99873 = pow(r99872, r99859);
        double r99874 = r99871 + r99873;
        double r99875 = r99874 - r99871;
        double r99876 = r99870 * r99875;
        double r99877 = r99859 / r99876;
        return r99877;
}

↓

double f(double t, double l, double k) {
        double r99878 = l;
        double r99879 = -7.341974747858959e+153;
        bool r99880 = r99878 <= r99879;
        double r99881 = 2.0;
        double r99882 = t;
        double r99883 = cbrt(r99882);
        double r99884 = r99883 * r99883;
        double r99885 = 3.0;
        double r99886 = pow(r99884, r99885);
        double r99887 = r99886 / r99878;
        double r99888 = pow(r99883, r99885);
        double r99889 = r99888 / r99878;
        double r99890 = k;
        double r99891 = sin(r99890);
        double r99892 = r99889 * r99891;
        double r99893 = r99887 * r99892;
        double r99894 = tan(r99890);
        double r99895 = r99893 * r99894;
        double r99896 = r99881 / r99895;
        double r99897 = r99890 / r99882;
        double r99898 = pow(r99897, r99881);
        double r99899 = r99896 / r99898;
        double r99900 = 4.1624864010972877e+148;
        bool r99901 = r99878 <= r99900;
        double r99902 = 1.0;
        double r99903 = 2.0;
        double r99904 = r99881 / r99903;
        double r99905 = pow(r99890, r99904);
        double r99906 = r99902 / r99905;
        double r99907 = 1.0;
        double r99908 = pow(r99906, r99907);
        double r99909 = cos(r99890);
        double r99910 = fabs(r99891);
        double r99911 = r99909 / r99910;
        double r99912 = r99910 / r99878;
        double r99913 = r99878 / r99912;
        double r99914 = r99911 * r99913;
        double r99915 = pow(r99882, r99907);
        double r99916 = r99905 * r99915;
        double r99917 = r99902 / r99916;
        double r99918 = pow(r99917, r99907);
        double r99919 = r99914 * r99918;
        double r99920 = r99908 * r99919;
        double r99921 = r99881 * r99920;
        double r99922 = pow(r99882, r99885);
        double r99923 = r99922 / r99878;
        double r99924 = r99923 / r99878;
        double r99925 = pow(r99891, r99903);
        double r99926 = r99924 * r99925;
        double r99927 = r99881 / r99926;
        double r99928 = cbrt(r99897);
        double r99929 = r99928 * r99928;
        double r99930 = pow(r99929, r99881);
        double r99931 = r99927 / r99930;
        double r99932 = pow(r99928, r99881);
        double r99933 = r99909 / r99932;
        double r99934 = r99931 * r99933;
        double r99935 = r99901 ? r99921 : r99934;
        double r99936 = r99880 ? r99899 : r99935;
        return r99936;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if l < -7.341974747858959e+153

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   4. Applied add-cube-cbrt 64.0

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

   5. Applied unpow-prod-down 64.0

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

   6. Applied times-frac 50.5

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

   7. Applied associate-*l* 50.5

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

   if -7.341974747858959e+153 < l < 4.1624864010972877e+148

   1. Initial program 45.1

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

   2. Simplified 36.2

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

   3. Taylor expanded around inf 14.0

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

   5. Applied sqr-pow 14.0

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

   6. Applied associate-*l* 11.6

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

   8. Applied *-un-lft-identity 11.6

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

   9. Applied times-frac 11.5

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

   10. Applied unpow-prod-down 11.5

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

   11. Applied associate-*l* 9.7

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

   12. Simplified 9.7

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

   14. Applied add-sqr-sqrt 9.7

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

   15. Applied times-frac 9.8

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

   16. Simplified 9.8

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

   17. Simplified 7.0

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

   if 4.1624864010972877e+148 < l

   1. Initial program 63.6

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

   2. Simplified 63.3

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

   4. Applied add-cube-cbrt 63.3

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

   5. Applied unpow-prod-down 63.3

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

   6. Applied tan-quot 63.3

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

   7. Applied associate-*r/ 63.3

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

   8. Applied associate-/r/ 63.3

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

   9. Applied times-frac 63.3

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

   10. Simplified 51.1

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

4. Final simplification 13.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -7.34197474785895879644805628021741244835 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le 4.162486401097287709225438879584039218636 \cdot 10^{148}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left(\frac{\cos k}{\left|\sin k\right|} \cdot \frac{\ell}{\frac{\left|\sin k\right|}{\ell}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot {\left(\sin k\right)}^{2}}}{{\left(\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{\frac{k}{t}}\right)}^{2}} \cdot \frac{\cos k}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))