Toniolo and Linder, Equation (13)

Average Error: 34.2 → 30.0

Time: 43.0s

Precision: 64

Math

\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le 4.739343449525871658159191443467243490565 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\left(n \cdot {\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 7.551608221022998949639242476561684793845 \cdot 10^{-93} \lor \neg \left(t \le 9.260196814501473924371525269420592469666 \cdot 10^{119}\right):\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r84881 = 2.0;
        double r84882 = n;
        double r84883 = r84881 * r84882;
        double r84884 = U;
        double r84885 = r84883 * r84884;
        double r84886 = t;
        double r84887 = l;
        double r84888 = r84887 * r84887;
        double r84889 = Om;
        double r84890 = r84888 / r84889;
        double r84891 = r84881 * r84890;
        double r84892 = r84886 - r84891;
        double r84893 = r84887 / r84889;
        double r84894 = pow(r84893, r84881);
        double r84895 = r84882 * r84894;
        double r84896 = U_;
        double r84897 = r84884 - r84896;
        double r84898 = r84895 * r84897;
        double r84899 = r84892 - r84898;
        double r84900 = r84885 * r84899;
        double r84901 = sqrt(r84900);
        return r84901;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r84902 = t;
        double r84903 = 4.7393434495258717e-151;
        bool r84904 = r84902 <= r84903;
        double r84905 = 2.0;
        double r84906 = l;
        double r84907 = Om;
        double r84908 = r84907 / r84906;
        double r84909 = r84906 / r84908;
        double r84910 = n;
        double r84911 = cbrt(r84906);
        double r84912 = r84911 * r84911;
        double r84913 = 2.0;
        double r84914 = r84905 / r84913;
        double r84915 = pow(r84912, r84914);
        double r84916 = r84910 * r84915;
        double r84917 = r84911 / r84907;
        double r84918 = pow(r84917, r84914);
        double r84919 = r84916 * r84918;
        double r84920 = U;
        double r84921 = U_;
        double r84922 = r84920 - r84921;
        double r84923 = r84906 / r84907;
        double r84924 = pow(r84923, r84914);
        double r84925 = r84922 * r84924;
        double r84926 = r84919 * r84925;
        double r84927 = fma(r84905, r84909, r84926);
        double r84928 = r84902 - r84927;
        double r84929 = r84905 * r84910;
        double r84930 = r84929 * r84920;
        double r84931 = r84928 * r84930;
        double r84932 = sqrt(r84931);
        double r84933 = 7.551608221022999e-93;
        bool r84934 = r84902 <= r84933;
        double r84935 = 9.260196814501474e+119;
        bool r84936 = r84902 <= r84935;
        double r84937 = !r84936;
        bool r84938 = r84934 || r84937;
        double r84939 = r84913 * r84914;
        double r84940 = pow(r84923, r84939);
        double r84941 = r84910 * r84940;
        double r84942 = r84922 * r84941;
        double r84943 = fma(r84905, r84909, r84942);
        double r84944 = r84902 - r84943;
        double r84945 = sqrt(r84944);
        double r84946 = sqrt(r84930);
        double r84947 = r84945 * r84946;
        double r84948 = r84940 * r84910;
        double r84949 = r84948 * r84922;
        double r84950 = fma(r84905, r84909, r84949);
        double r84951 = r84902 - r84950;
        double r84952 = r84951 * r84929;
        double r84953 = r84952 * r84920;
        double r84954 = sqrt(r84953);
        double r84955 = r84938 ? r84947 : r84954;
        double r84956 = r84904 ? r84932 : r84955;
        return r84956;
}

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

1. Split input into 3 regimes

2. if t < 4.7393434495258717e-151

   1. Initial program 34.7

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   2. Simplified 34.7

      \[\leadsto \color{blue}{\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\]
   3. Using strategy rm

   4. Applied associate-/l* 32.0

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
   5. Using strategy rm

   6. Applied sqr-pow 32.0

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   7. Applied associate-*r* 31.2

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
   8. Using strategy rm

   9. Applied associate-*l* 30.9

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   10. Simplified 30.9

       \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
   11. Using strategy rm

   12. Applied *-un-lft-identity 30.9

       \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{\color{blue}{1 \cdot Om}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   13. Applied add-cube-cbrt 31.0

       \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{1 \cdot Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   14. Applied times-frac 31.0

       \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\color{blue}{\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{1} \cdot \frac{\sqrt[3]{\ell}}{Om}\right)}}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   15. Applied unpow-prod-down 31.0

       \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{1}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\sqrt[3]{\ell}}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   16. Applied associate-*r* 31.5

       \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(\left(n \cdot {\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{1}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   17. Simplified 31.5

       \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\color{blue}{\left(n \cdot {\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot {\left(\frac{\sqrt[3]{\ell}}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   if 4.7393434495258717e-151 < t < 7.551608221022999e-93 or 9.260196814501474e+119 < t

   1. Initial program 34.6

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   2. Simplified 34.6

      \[\leadsto \color{blue}{\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\]
   3. Using strategy rm

   4. Applied associate-/l* 32.3

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
   5. Using strategy rm

   6. Applied sqr-pow 32.3

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   7. Applied associate-*r* 32.0

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
   8. Using strategy rm

   9. Applied sqrt-prod 26.1

      \[\leadsto \color{blue}{\sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}}\]

   10. Simplified 26.3

       \[\leadsto \color{blue}{\sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\]

   if 7.551608221022999e-93 < t < 9.260196814501474e+119

   1. Initial program 31.9

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   2. Simplified 31.9

      \[\leadsto \color{blue}{\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\]
   3. Using strategy rm

   4. Applied associate-/l* 29.2

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
   5. Using strategy rm

   6. Applied sqr-pow 29.2

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   7. Applied associate-*r* 28.7

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
   8. Using strategy rm

   9. Applied associate-*l* 28.7

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   10. Simplified 28.7

       \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
   11. Using strategy rm

   12. Applied associate-*r* 28.5

       \[\leadsto \sqrt{\color{blue}{\left(\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}}\]

   13. Simplified 29.0

       \[\leadsto \sqrt{\color{blue}{\left(\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right)} \cdot U}\]
3. Recombined 3 regimes into one program.

4. Final simplification 30.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 4.739343449525871658159191443467243490565 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\left(n \cdot {\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 7.551608221022998949639242476561684793845 \cdot 10^{-93} \lor \neg \left(t \le 9.260196814501473924371525269420592469666 \cdot 10^{119}\right):\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))