Toniolo and Linder, Equation (10-)

Average Error: 48.6 → 8.9

Time: 26.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -4.57723594555655645 \cdot 10^{139} \lor \neg \left(k \le -2.7057282491360021 \cdot 10^{-140} \lor \neg \left(k \le 7.6001933799401753 \cdot 10^{-155} \lor \neg \left(k \le 1.15121578543094186 \cdot 10^{132}\right)\right)\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)}{\frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)\right)}{\frac{\sin k}{\ell}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r88926 = 2.0;
        double r88927 = t;
        double r88928 = 3.0;
        double r88929 = pow(r88927, r88928);
        double r88930 = l;
        double r88931 = r88930 * r88930;
        double r88932 = r88929 / r88931;
        double r88933 = k;
        double r88934 = sin(r88933);
        double r88935 = r88932 * r88934;
        double r88936 = tan(r88933);
        double r88937 = r88935 * r88936;
        double r88938 = 1.0;
        double r88939 = r88933 / r88927;
        double r88940 = pow(r88939, r88926);
        double r88941 = r88938 + r88940;
        double r88942 = r88941 - r88938;
        double r88943 = r88937 * r88942;
        double r88944 = r88926 / r88943;
        return r88944;
}

↓

double f(double t, double l, double k) {
        double r88945 = k;
        double r88946 = -4.5772359455565565e+139;
        bool r88947 = r88945 <= r88946;
        double r88948 = -2.705728249136002e-140;
        bool r88949 = r88945 <= r88948;
        double r88950 = 7.600193379940175e-155;
        bool r88951 = r88945 <= r88950;
        double r88952 = 1.1512157854309419e+132;
        bool r88953 = r88945 <= r88952;
        double r88954 = !r88953;
        bool r88955 = r88951 || r88954;
        double r88956 = !r88955;
        bool r88957 = r88949 || r88956;
        double r88958 = !r88957;
        bool r88959 = r88947 || r88958;
        double r88960 = 2.0;
        double r88961 = 1.0;
        double r88962 = 2.0;
        double r88963 = r88960 / r88962;
        double r88964 = pow(r88945, r88963);
        double r88965 = t;
        double r88966 = 1.0;
        double r88967 = pow(r88965, r88966);
        double r88968 = r88964 * r88967;
        double r88969 = r88964 * r88968;
        double r88970 = r88961 / r88969;
        double r88971 = pow(r88970, r88966);
        double r88972 = cos(r88945);
        double r88973 = sin(r88945);
        double r88974 = r88972 / r88973;
        double r88975 = l;
        double r88976 = r88974 * r88975;
        double r88977 = r88971 * r88976;
        double r88978 = r88973 / r88975;
        double r88979 = r88977 / r88978;
        double r88980 = r88960 * r88979;
        double r88981 = cbrt(r88961);
        double r88982 = r88981 * r88981;
        double r88983 = pow(r88945, r88960);
        double r88984 = r88982 / r88983;
        double r88985 = pow(r88984, r88966);
        double r88986 = r88981 / r88967;
        double r88987 = pow(r88986, r88966);
        double r88988 = r88987 * r88976;
        double r88989 = r88985 * r88988;
        double r88990 = r88989 / r88978;
        double r88991 = r88960 * r88990;
        double r88992 = r88959 ? r88980 : r88991;
        return r88992;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -4.5772359455565565e+139 or -2.705728249136002e-140 < k < 7.600193379940175e-155 or 1.1512157854309419e+132 < k

   1. Initial program 42.0

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

   2. Simplified 36.9

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

   3. Taylor expanded around inf 26.4

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

   5. Applied add-sqr-sqrt 45.4

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

   6. Applied unpow-prod-down 45.4

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

   7. Applied times-frac 45.4

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

   8. Simplified 45.4

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

   9. Simplified 26.2

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

   11. Applied associate-*r/ 25.8

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

   12. Applied associate-*r/ 23.9

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

   14. Applied sqr-pow 23.9

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

   15. Applied associate-*l* 14.6

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

   if -4.5772359455565565e+139 < k < -2.705728249136002e-140 or 7.600193379940175e-155 < k < 1.1512157854309419e+132

   1. Initial program 54.8

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

   2. Simplified 43.8

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

   3. Taylor expanded around inf 18.5

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

   5. Applied add-sqr-sqrt 41.0

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

   6. Applied unpow-prod-down 41.0

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

   7. Applied times-frac 41.0

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

   8. Simplified 41.0

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

   9. Simplified 16.9

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

   11. Applied associate-*r/ 15.5

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

   12. Applied associate-*r/ 8.3

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

   14. Applied add-cube-cbrt 8.3

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

   15. Applied times-frac 8.0

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

   16. Applied unpow-prod-down 8.0

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

   17. Applied associate-*l* 3.4

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -4.57723594555655645 \cdot 10^{139} \lor \neg \left(k \le -2.7057282491360021 \cdot 10^{-140} \lor \neg \left(k \le 7.6001933799401753 \cdot 10^{-155} \lor \neg \left(k \le 1.15121578543094186 \cdot 10^{132}\right)\right)\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)}{\frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)\right)}{\frac{\sin k}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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))))