Toniolo and Linder, Equation (13)

Average Error: 34.6 → 29.0

Time: 57.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -2.484360230131543375186529749532050760457 \cdot 10^{99} \lor \neg \left(U \le 4.221371199983704648470881238153361257851 \cdot 10^{-78}\right):\\ \;\;\;\;\sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, U* - U, \mathsf{fma}\left(\ell, \left(-\frac{\ell}{Om}\right) \cdot 2, t\right)\right)}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, U* - U, \mathsf{fma}\left(\ell, \left(-\frac{\ell}{Om}\right) \cdot 2, t\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot \frac{2}{2}}{2}\right)}, U* - U, \mathsf{fma}\left(\ell, \left(-\frac{\ell}{Om}\right) \cdot 2, t\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r168964 = 2.0;
        double r168965 = n;
        double r168966 = r168964 * r168965;
        double r168967 = U;
        double r168968 = r168966 * r168967;
        double r168969 = t;
        double r168970 = l;
        double r168971 = r168970 * r168970;
        double r168972 = Om;
        double r168973 = r168971 / r168972;
        double r168974 = r168964 * r168973;
        double r168975 = r168969 - r168974;
        double r168976 = r168970 / r168972;
        double r168977 = pow(r168976, r168964);
        double r168978 = r168965 * r168977;
        double r168979 = U_;
        double r168980 = r168967 - r168979;
        double r168981 = r168978 * r168980;
        double r168982 = r168975 - r168981;
        double r168983 = r168968 * r168982;
        double r168984 = sqrt(r168983);
        return r168984;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r168985 = U;
        double r168986 = -2.4843602301315434e+99;
        bool r168987 = r168985 <= r168986;
        double r168988 = 4.2213711999837046e-78;
        bool r168989 = r168985 <= r168988;
        double r168990 = !r168989;
        bool r168991 = r168987 || r168990;
        double r168992 = 2.0;
        double r168993 = n;
        double r168994 = r168992 * r168993;
        double r168995 = r168994 * r168985;
        double r168996 = l;
        double r168997 = Om;
        double r168998 = r168996 / r168997;
        double r168999 = 2.0;
        double r169000 = r168992 / r168999;
        double r169001 = r168999 * r169000;
        double r169002 = pow(r168998, r169001);
        double r169003 = r168993 * r169002;
        double r169004 = U_;
        double r169005 = r169004 - r168985;
        double r169006 = -r168998;
        double r169007 = r169006 * r168992;
        double r169008 = t;
        double r169009 = fma(r168996, r169007, r169008);
        double r169010 = fma(r169003, r169005, r169009);
        double r169011 = r168995 * r169010;
        double r169012 = sqrt(r169011);
        double r169013 = sqrt(r169012);
        double r169014 = r169013 * r169013;
        double r169015 = pow(r168998, r169000);
        double r169016 = r169015 * r168993;
        double r169017 = r169001 / r168999;
        double r169018 = pow(r168998, r169017);
        double r169019 = r169016 * r169018;
        double r169020 = fma(r169019, r169005, r169009);
        double r169021 = r169020 * r168985;
        double r169022 = r169021 * r168994;
        double r169023 = sqrt(r169022);
        double r169024 = r168991 ? r169014 : r169023;
        return r169024;
}

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 2 regimes

2. if U < -2.4843602301315434e+99 or 4.2213711999837046e-78 < U

   1. Initial program 29.6

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

   2. Simplified 26.4

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

   4. Applied sqr-pow 26.4

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

   5. Applied associate-*r* 25.6

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

   7. Applied add-sqr-sqrt 25.8

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

   8. Simplified 26.6

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

   9. Simplified 26.6

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

   if -2.4843602301315434e+99 < U < 4.2213711999837046e-78

   1. Initial program 37.6

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

   2. Simplified 35.2

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

   4. Applied sqr-pow 35.2

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

   5. Applied associate-*r* 34.4

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

   7. Applied associate-*l* 30.5

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

   8. Simplified 31.6

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

   10. Applied sqr-pow 31.6

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

   11. Applied associate-*l* 30.5

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

   12. Simplified 30.5

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

4. Final simplification 29.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -2.484360230131543375186529749532050760457 \cdot 10^{99} \lor \neg \left(U \le 4.221371199983704648470881238153361257851 \cdot 10^{-78}\right):\\ \;\;\;\;\sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, U* - U, \mathsf{fma}\left(\ell, \left(-\frac{\ell}{Om}\right) \cdot 2, t\right)\right)}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, U* - U, \mathsf{fma}\left(\ell, \left(-\frac{\ell}{Om}\right) \cdot 2, t\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot \frac{2}{2}}{2}\right)}, U* - U, \mathsf{fma}\left(\ell, \left(-\frac{\ell}{Om}\right) \cdot 2, t\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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*))))))