Toniolo and Linder, Equation (13)

Average Error: 33.3 → 29.1

Time: 46.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 1.8088620139970252 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(\ell, 2 \cdot \frac{\ell}{Om}, \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.5047052485579951 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;t \le 5.16907107140256 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(\ell, 2 \cdot \frac{\ell}{Om}, \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell, 2 \cdot \frac{\ell}{Om}, \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1286955 = 2.0;
        double r1286956 = n;
        double r1286957 = r1286955 * r1286956;
        double r1286958 = U;
        double r1286959 = r1286957 * r1286958;
        double r1286960 = t;
        double r1286961 = l;
        double r1286962 = r1286961 * r1286961;
        double r1286963 = Om;
        double r1286964 = r1286962 / r1286963;
        double r1286965 = r1286955 * r1286964;
        double r1286966 = r1286960 - r1286965;
        double r1286967 = r1286961 / r1286963;
        double r1286968 = pow(r1286967, r1286955);
        double r1286969 = r1286956 * r1286968;
        double r1286970 = U_;
        double r1286971 = r1286958 - r1286970;
        double r1286972 = r1286969 * r1286971;
        double r1286973 = r1286966 - r1286972;
        double r1286974 = r1286959 * r1286973;
        double r1286975 = sqrt(r1286974);
        return r1286975;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1286976 = t;
        double r1286977 = 1.8088620139970252e-254;
        bool r1286978 = r1286976 <= r1286977;
        double r1286979 = 2.0;
        double r1286980 = n;
        double r1286981 = r1286979 * r1286980;
        double r1286982 = U;
        double r1286983 = l;
        double r1286984 = Om;
        double r1286985 = r1286983 / r1286984;
        double r1286986 = r1286979 * r1286985;
        double r1286987 = U_;
        double r1286988 = r1286982 - r1286987;
        double r1286989 = r1286985 * r1286988;
        double r1286990 = r1286980 * r1286989;
        double r1286991 = r1286990 * r1286985;
        double r1286992 = fma(r1286983, r1286986, r1286991);
        double r1286993 = r1286976 - r1286992;
        double r1286994 = r1286982 * r1286993;
        double r1286995 = r1286981 * r1286994;
        double r1286996 = sqrt(r1286995);
        double r1286997 = 1.5047052485579951e-182;
        bool r1286998 = r1286976 <= r1286997;
        double r1286999 = r1286979 * r1286983;
        double r1287000 = r1286980 * r1286985;
        double r1287001 = r1286985 * r1287000;
        double r1287002 = r1287001 * r1286988;
        double r1287003 = fma(r1286999, r1286985, r1287002);
        double r1287004 = r1286976 - r1287003;
        double r1287005 = sqrt(r1287004);
        double r1287006 = r1286981 * r1286982;
        double r1287007 = sqrt(r1287006);
        double r1287008 = r1287005 * r1287007;
        double r1287009 = 5.16907107140256e-46;
        bool r1287010 = r1286976 <= r1287009;
        double r1287011 = sqrt(r1286993);
        double r1287012 = r1287011 * r1287007;
        double r1287013 = r1287010 ? r1286996 : r1287012;
        double r1287014 = r1286998 ? r1287008 : r1287013;
        double r1287015 = r1286978 ? r1286996 : r1287014;
        return r1287015;
}

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 < 1.8088620139970252e-254 or 1.5047052485579951e-182 < t < 5.16907107140256e-46

   1. Initial program 33.2

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

   3. Applied *-un-lft-identity 33.2

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

   4. Applied times-frac 30.5

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

   5. Simplified 30.5

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

   7. Applied unpow2 30.5

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

   8. Applied associate-*r* 29.5

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

   10. Applied associate-*l* 29.3

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

   12. Applied associate-*l* 29.6

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

   13. Simplified 29.8

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

   if 1.8088620139970252e-254 < t < 1.5047052485579951e-182

   1. Initial program 36.8

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

   3. Applied *-un-lft-identity 36.8

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

   4. Applied times-frac 34.6

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

   5. Simplified 34.6

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

   7. Applied unpow2 34.6

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

   8. Applied associate-*r* 32.9

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

   10. Applied sqrt-prod 33.1

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

   11. Simplified 33.1

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

   if 5.16907107140256e-46 < t

   1. Initial program 32.8

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

   3. Applied *-un-lft-identity 32.8

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

   4. Applied times-frac 30.5

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

   5. Simplified 30.5

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

   7. Applied unpow2 30.5

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

   8. Applied associate-*r* 30.1

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

   10. Applied associate-*l* 30.1

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

   12. Applied sqrt-prod 26.5

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

   13. Simplified 26.7

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

4. Final simplification 29.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.8088620139970252 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(\ell, 2 \cdot \frac{\ell}{Om}, \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.5047052485579951 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;t \le 5.16907107140256 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(\ell, 2 \cdot \frac{\ell}{Om}, \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell, 2 \cdot \frac{\ell}{Om}, \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +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*))))))