Toniolo and Linder, Equation (13)

Average Error: 34.9 → 28.6

Time: 40.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.00466796771781086 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right) \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\\ \mathbf{elif}\;n \le -9.2277267034582191 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\sqrt[3]{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt[3]{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right) \cdot \sqrt[3]{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right)}\\ \mathbf{elif}\;n \le -4.44897802822854103 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r211972 = 2.0;
        double r211973 = n;
        double r211974 = r211972 * r211973;
        double r211975 = U;
        double r211976 = r211974 * r211975;
        double r211977 = t;
        double r211978 = l;
        double r211979 = r211978 * r211978;
        double r211980 = Om;
        double r211981 = r211979 / r211980;
        double r211982 = r211972 * r211981;
        double r211983 = r211977 - r211982;
        double r211984 = r211978 / r211980;
        double r211985 = pow(r211984, r211972);
        double r211986 = r211973 * r211985;
        double r211987 = U_;
        double r211988 = r211975 - r211987;
        double r211989 = r211986 * r211988;
        double r211990 = r211983 - r211989;
        double r211991 = r211976 * r211990;
        double r211992 = sqrt(r211991);
        return r211992;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r211993 = n;
        double r211994 = -1.0046679677178109e-230;
        bool r211995 = r211993 <= r211994;
        double r211996 = 2.0;
        double r211997 = r211996 * r211993;
        double r211998 = U;
        double r211999 = r211997 * r211998;
        double r212000 = t;
        double r212001 = l;
        double r212002 = Om;
        double r212003 = r212002 / r212001;
        double r212004 = r212001 / r212003;
        double r212005 = r212001 / r212002;
        double r212006 = pow(r212005, r211996);
        double r212007 = r211993 * r212006;
        double r212008 = U_;
        double r212009 = r211998 - r212008;
        double r212010 = r212007 * r212009;
        double r212011 = fma(r211996, r212004, r212010);
        double r212012 = r212000 - r212011;
        double r212013 = r211999 * r212012;
        double r212014 = cbrt(r212013);
        double r212015 = r212014 * r212014;
        double r212016 = r212015 * r212014;
        double r212017 = sqrt(r212016);
        double r212018 = -9.22772670345822e-280;
        bool r212019 = r211993 <= r212018;
        double r212020 = cbrt(r212010);
        double r212021 = r212020 * r212020;
        double r212022 = r212021 * r212020;
        double r212023 = fma(r211996, r212004, r212022);
        double r212024 = r212000 - r212023;
        double r212025 = r211998 * r212024;
        double r212026 = r211997 * r212025;
        double r212027 = sqrt(r212026);
        double r212028 = -4.448978028228541e-305;
        bool r212029 = r211993 <= r212028;
        double r212030 = r211993 * r212001;
        double r212031 = 1.0;
        double r212032 = 1.0;
        double r212033 = pow(r212002, r212032);
        double r212034 = r212031 / r212033;
        double r212035 = pow(r212034, r212032);
        double r212036 = r212030 * r212035;
        double r212037 = 2.0;
        double r212038 = r211996 / r212037;
        double r212039 = pow(r212005, r212038);
        double r212040 = r212036 * r212039;
        double r212041 = r212040 * r212009;
        double r212042 = fma(r211996, r212004, r212041);
        double r212043 = r212000 - r212042;
        double r212044 = r211999 * r212043;
        double r212045 = sqrt(r212044);
        double r212046 = sqrt(r211997);
        double r212047 = r211998 * r212012;
        double r212048 = sqrt(r212047);
        double r212049 = r212046 * r212048;
        double r212050 = r212029 ? r212045 : r212049;
        double r212051 = r212019 ? r212027 : r212050;
        double r212052 = r211995 ? r212017 : r212051;
        return r212052;
}

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 4 regimes

2. if n < -1.0046679677178109e-230

   1. Initial program 34.2

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

   2. Simplified 34.2

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

   4. Applied associate-/l* 31.6

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

   6. Applied associate-*r* 31.5

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

   8. Applied add-cube-cbrt 31.8

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

   if -1.0046679677178109e-230 < n < -9.22772670345822e-280

   1. Initial program 40.1

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

   2. Simplified 39.6

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

   4. Applied associate-/l* 37.7

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

   6. Applied add-cube-cbrt 37.7

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

   if -9.22772670345822e-280 < n < -4.448978028228541e-305

   1. Initial program 41.1

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

   2. Simplified 41.3

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

   4. Applied associate-/l* 37.1

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

   6. Applied associate-*r* 36.7

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

   8. Applied sqr-pow 36.7

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

   9. Applied associate-*r* 36.0

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

   10. Taylor expanded around 0 37.0

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

   if -4.448978028228541e-305 < n

   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 34.4

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

   4. Applied associate-/l* 31.6

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

   6. Applied sqrt-prod 24.6

      \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 28.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.00466796771781086 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right) \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\\ \mathbf{elif}\;n \le -9.2277267034582191 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\sqrt[3]{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt[3]{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right) \cdot \sqrt[3]{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right)}\\ \mathbf{elif}\;n \le -4.44897802822854103 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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*))))))