Toniolo and Linder, Equation (13)

Average Error: 34.0 → 27.2

Time: 1.0m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.611733182312969328568803254996605738157 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(n \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right), t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 1.30114079115734769467106905376504288527 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{\left(t \cdot U - 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot U}{Om}\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;n \le 1.809873470794950544758355136519339403208 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(n \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right), t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot U} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2286947 = 2.0;
        double r2286948 = n;
        double r2286949 = r2286947 * r2286948;
        double r2286950 = U;
        double r2286951 = r2286949 * r2286950;
        double r2286952 = t;
        double r2286953 = l;
        double r2286954 = r2286953 * r2286953;
        double r2286955 = Om;
        double r2286956 = r2286954 / r2286955;
        double r2286957 = r2286947 * r2286956;
        double r2286958 = r2286952 - r2286957;
        double r2286959 = r2286953 / r2286955;
        double r2286960 = pow(r2286959, r2286947);
        double r2286961 = r2286948 * r2286960;
        double r2286962 = U_;
        double r2286963 = r2286950 - r2286962;
        double r2286964 = r2286961 * r2286963;
        double r2286965 = r2286958 - r2286964;
        double r2286966 = r2286951 * r2286965;
        double r2286967 = sqrt(r2286966);
        return r2286967;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2286968 = n;
        double r2286969 = -1.6117331823129693e-279;
        bool r2286970 = r2286968 <= r2286969;
        double r2286971 = U_;
        double r2286972 = U;
        double r2286973 = r2286971 - r2286972;
        double r2286974 = l;
        double r2286975 = Om;
        double r2286976 = r2286974 / r2286975;
        double r2286977 = 2.0;
        double r2286978 = 2.0;
        double r2286979 = r2286977 / r2286978;
        double r2286980 = pow(r2286976, r2286979);
        double r2286981 = cbrt(r2286974);
        double r2286982 = cbrt(r2286975);
        double r2286983 = r2286981 / r2286982;
        double r2286984 = r2286983 * r2286983;
        double r2286985 = pow(r2286984, r2286979);
        double r2286986 = r2286968 * r2286985;
        double r2286987 = pow(r2286983, r2286979);
        double r2286988 = r2286986 * r2286987;
        double r2286989 = r2286980 * r2286988;
        double r2286990 = t;
        double r2286991 = r2286976 * r2286974;
        double r2286992 = r2286977 * r2286991;
        double r2286993 = r2286990 - r2286992;
        double r2286994 = fma(r2286973, r2286989, r2286993);
        double r2286995 = r2286977 * r2286968;
        double r2286996 = r2286995 * r2286972;
        double r2286997 = r2286994 * r2286996;
        double r2286998 = sqrt(r2286997);
        double r2286999 = 1.3011407911573477e-306;
        bool r2287000 = r2286968 <= r2286999;
        double r2287001 = r2286990 * r2286972;
        double r2287002 = r2286974 * r2286974;
        double r2287003 = r2287002 * r2286972;
        double r2287004 = r2287003 / r2286975;
        double r2287005 = r2286977 * r2287004;
        double r2287006 = r2287001 - r2287005;
        double r2287007 = r2287006 * r2286995;
        double r2287008 = sqrt(r2287007);
        double r2287009 = 1.8098734707949505e-298;
        bool r2287010 = r2286968 <= r2287009;
        double r2287011 = r2286968 * r2286980;
        double r2287012 = r2287011 * r2286980;
        double r2287013 = fma(r2286973, r2287012, r2286993);
        double r2287014 = r2287013 * r2286972;
        double r2287015 = sqrt(r2287014);
        double r2287016 = sqrt(r2286995);
        double r2287017 = r2287015 * r2287016;
        double r2287018 = r2287010 ? r2286998 : r2287017;
        double r2287019 = r2287000 ? r2287008 : r2287018;
        double r2287020 = r2286970 ? r2286998 : r2287019;
        return r2287020;
}

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 n < -1.6117331823129693e-279 or 1.3011407911573477e-306 < n < 1.8098734707949505e-298

   1. Initial program 34.0

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

   2. Simplified 30.9

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

   4. Applied sqr-pow 30.9

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

   5. Applied associate-*r* 30.3

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

   7. Applied add-cube-cbrt 30.3

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

   8. Applied add-cube-cbrt 30.3

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

   9. Applied times-frac 30.3

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

   10. Applied unpow-prod-down 30.3

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

   11. Applied associate-*r* 30.3

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

   12. Simplified 30.3

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

   if -1.6117331823129693e-279 < n < 1.3011407911573477e-306

   1. Initial program 40.0

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

   2. Simplified 36.9

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

   4. Applied sqr-pow 36.9

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

   5. Applied associate-*r* 35.5

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

   7. Applied associate-*l* 37.4

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

   8. Taylor expanded around inf 41.3

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

   9. Simplified 41.3

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

   if 1.8098734707949505e-298 < n

   1. Initial program 33.6

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

   2. Simplified 30.9

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

   4. Applied sqr-pow 30.9

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

   5. Applied associate-*r* 29.8

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

   7. Applied associate-*l* 29.8

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

   9. Applied sqrt-prod 22.9

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

4. Final simplification 27.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.611733182312969328568803254996605738157 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(n \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right), t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 1.30114079115734769467106905376504288527 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{\left(t \cdot U - 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot U}{Om}\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;n \le 1.809873470794950544758355136519339403208 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(n \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right), t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot U} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))