Toniolo and Linder, Equation (13)

Average Error: 33.7 → 30.2

Time: 48.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -8.997037644090019 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;n \le -2.81317660591196 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, 2, \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \cdot \sqrt{\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3520944 = 2.0;
        double r3520945 = n;
        double r3520946 = r3520944 * r3520945;
        double r3520947 = U;
        double r3520948 = r3520946 * r3520947;
        double r3520949 = t;
        double r3520950 = l;
        double r3520951 = r3520950 * r3520950;
        double r3520952 = Om;
        double r3520953 = r3520951 / r3520952;
        double r3520954 = r3520944 * r3520953;
        double r3520955 = r3520949 - r3520954;
        double r3520956 = r3520950 / r3520952;
        double r3520957 = pow(r3520956, r3520944);
        double r3520958 = r3520945 * r3520957;
        double r3520959 = U_;
        double r3520960 = r3520947 - r3520959;
        double r3520961 = r3520958 * r3520960;
        double r3520962 = r3520955 - r3520961;
        double r3520963 = r3520948 * r3520962;
        double r3520964 = sqrt(r3520963);
        return r3520964;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3520965 = n;
        double r3520966 = -8.997037644090019e-101;
        bool r3520967 = r3520965 <= r3520966;
        double r3520968 = 2.0;
        double r3520969 = r3520968 * r3520965;
        double r3520970 = U;
        double r3520971 = r3520969 * r3520970;
        double r3520972 = t;
        double r3520973 = l;
        double r3520974 = Om;
        double r3520975 = r3520974 / r3520973;
        double r3520976 = r3520973 / r3520975;
        double r3520977 = r3520968 * r3520976;
        double r3520978 = r3520972 - r3520977;
        double r3520979 = r3520973 / r3520974;
        double r3520980 = U_;
        double r3520981 = r3520970 - r3520980;
        double r3520982 = r3520979 * r3520981;
        double r3520983 = r3520965 * r3520979;
        double r3520984 = r3520982 * r3520983;
        double r3520985 = r3520978 - r3520984;
        double r3520986 = r3520971 * r3520985;
        double r3520987 = sqrt(r3520986);
        double r3520988 = -2.81317660591196e-264;
        bool r3520989 = r3520965 <= r3520988;
        double r3520990 = r3520979 * r3520973;
        double r3520991 = r3520979 * r3520979;
        double r3520992 = r3520965 * r3520991;
        double r3520993 = r3520992 * r3520981;
        double r3520994 = fma(r3520990, r3520968, r3520993);
        double r3520995 = r3520972 - r3520994;
        double r3520996 = r3520970 * r3520995;
        double r3520997 = r3520969 * r3520996;
        double r3520998 = sqrt(r3520997);
        double r3520999 = r3520983 * r3520979;
        double r3521000 = r3520981 * r3520999;
        double r3521001 = r3520978 - r3521000;
        double r3521002 = r3521001 * r3520971;
        double r3521003 = sqrt(r3521002);
        double r3521004 = sqrt(r3521003);
        double r3521005 = r3521004 * r3521004;
        double r3521006 = r3520989 ? r3520998 : r3521005;
        double r3521007 = r3520967 ? r3520987 : r3521006;
        return r3521007;
}

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 < -8.997037644090019e-101

   1. Initial program 30.3

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

   3. Applied associate-/l* 27.9

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

   5. Applied unpow2 27.9

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

   6. Applied associate-*r* 27.1

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

   8. Applied associate-*l* 26.1

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

   if -8.997037644090019e-101 < n < -2.81317660591196e-264

   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. Using strategy rm

   3. Applied associate-*l* 36.3

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

   4. Simplified 34.1

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

   if -2.81317660591196e-264 < n

   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. Using strategy rm

   3. Applied associate-/l* 31.9

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

   5. Applied unpow2 31.9

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

   6. Applied associate-*r* 30.9

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

   8. Applied add-sqr-sqrt 31.1

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

4. Final simplification 30.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -8.997037644090019 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;n \le -2.81317660591196 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, 2, \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \cdot \sqrt{\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))