Toniolo and Linder, Equation (13)

Average Error: 35.1 → 28.0

Time: 2.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le 1.114777108083807533454474777376791311174 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(\sqrt[3]{t + \left(\left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) - \left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell\right)} \cdot \sqrt[3]{t + \left(\left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) - \left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell\right)}\right) \cdot \left(\sqrt[3]{U} \cdot \left(\left(\sqrt[3]{U} \cdot \left(n \cdot 2\right)\right) \cdot \sqrt[3]{\left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2 - t\right)}\right)\right)\right) \cdot \sqrt[3]{U}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(n \cdot 2\right) \cdot \left(\left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot 2}{2}\right)}\right) - \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2 - t\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r351890 = 2.0;
        double r351891 = n;
        double r351892 = r351890 * r351891;
        double r351893 = U;
        double r351894 = r351892 * r351893;
        double r351895 = t;
        double r351896 = l;
        double r351897 = r351896 * r351896;
        double r351898 = Om;
        double r351899 = r351897 / r351898;
        double r351900 = r351890 * r351899;
        double r351901 = r351895 - r351900;
        double r351902 = r351896 / r351898;
        double r351903 = pow(r351902, r351890);
        double r351904 = r351891 * r351903;
        double r351905 = U_;
        double r351906 = r351893 - r351905;
        double r351907 = r351904 * r351906;
        double r351908 = r351901 - r351907;
        double r351909 = r351894 * r351908;
        double r351910 = sqrt(r351909);
        return r351910;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r351911 = U;
        double r351912 = 1.1147771080838e-310;
        bool r351913 = r351911 <= r351912;
        double r351914 = t;
        double r351915 = U_;
        double r351916 = r351915 - r351911;
        double r351917 = l;
        double r351918 = Om;
        double r351919 = r351917 / r351918;
        double r351920 = 2.0;
        double r351921 = 2.0;
        double r351922 = r351920 / r351921;
        double r351923 = pow(r351919, r351922);
        double r351924 = r351916 * r351923;
        double r351925 = n;
        double r351926 = r351925 * r351917;
        double r351927 = 1.0;
        double r351928 = 1.0;
        double r351929 = pow(r351918, r351928);
        double r351930 = r351927 / r351929;
        double r351931 = pow(r351930, r351928);
        double r351932 = r351926 * r351931;
        double r351933 = r351924 * r351932;
        double r351934 = r351920 * r351919;
        double r351935 = r351934 * r351917;
        double r351936 = r351933 - r351935;
        double r351937 = r351914 + r351936;
        double r351938 = cbrt(r351937);
        double r351939 = r351938 * r351938;
        double r351940 = cbrt(r351911);
        double r351941 = r351925 * r351920;
        double r351942 = r351940 * r351941;
        double r351943 = r351932 * r351923;
        double r351944 = r351943 * r351916;
        double r351945 = r351919 * r351917;
        double r351946 = r351945 * r351920;
        double r351947 = r351946 - r351914;
        double r351948 = r351944 - r351947;
        double r351949 = cbrt(r351948);
        double r351950 = r351942 * r351949;
        double r351951 = r351940 * r351950;
        double r351952 = r351939 * r351951;
        double r351953 = r351952 * r351940;
        double r351954 = sqrt(r351953);
        double r351955 = sqrt(r351911);
        double r351956 = r351920 * r351921;
        double r351957 = r351956 / r351921;
        double r351958 = pow(r351919, r351957);
        double r351959 = r351925 * r351958;
        double r351960 = r351916 * r351959;
        double r351961 = r351960 - r351947;
        double r351962 = r351941 * r351961;
        double r351963 = sqrt(r351962);
        double r351964 = r351955 * r351963;
        double r351965 = r351913 ? r351954 : r351964;
        return r351965;
}

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 < 1.1147771080838e-310

   1. Initial program 35.6

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

   2. Simplified 33.2

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

   4. Applied sqr-pow 33.2

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

   5. Applied associate-*r* 32.0

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

   6. Simplified 32.0

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

   7. Taylor expanded around 0 32.6

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

   8. Simplified 32.1

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

   10. Applied add-cube-cbrt 32.4

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \left(\left(\left(n \cdot \left(\ell \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \sqrt[3]{U}\right)}}\]

   11. Applied associate-*r* 32.4

       \[\leadsto \sqrt{\color{blue}{\left(\left(\left(2 \cdot n\right) \cdot \left(t + \left(\left(\left(n \cdot \left(\ell \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)\right) \cdot \sqrt[3]{U}}}\]

   12. Simplified 32.2

       \[\leadsto \sqrt{\color{blue}{\left(\left(t + \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U* - U\right)\right) - \left(\frac{\ell}{Om} \cdot 2\right) \cdot \ell\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)\right)\right)} \cdot \sqrt[3]{U}}\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 32.3

       \[\leadsto \sqrt{\left(\color{blue}{\left(\left(\sqrt[3]{t + \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U* - U\right)\right) - \left(\frac{\ell}{Om} \cdot 2\right) \cdot \ell\right)} \cdot \sqrt[3]{t + \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U* - U\right)\right) - \left(\frac{\ell}{Om} \cdot 2\right) \cdot \ell\right)}\right) \cdot \sqrt[3]{t + \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U* - U\right)\right) - \left(\frac{\ell}{Om} \cdot 2\right) \cdot \ell\right)}\right)} \cdot \left(\left(2 \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)\right)\right) \cdot \sqrt[3]{U}}\]

   15. Applied associate-*l* 32.3

       \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt[3]{t + \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U* - U\right)\right) - \left(\frac{\ell}{Om} \cdot 2\right) \cdot \ell\right)} \cdot \sqrt[3]{t + \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U* - U\right)\right) - \left(\frac{\ell}{Om} \cdot 2\right) \cdot \ell\right)}\right) \cdot \left(\sqrt[3]{t + \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U* - U\right)\right) - \left(\frac{\ell}{Om} \cdot 2\right) \cdot \ell\right)} \cdot \left(\left(2 \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)\right)\right)\right)} \cdot \sqrt[3]{U}}\]

   16. Simplified 31.4

       \[\leadsto \sqrt{\left(\left(\sqrt[3]{t + \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U* - U\right)\right) - \left(\frac{\ell}{Om} \cdot 2\right) \cdot \ell\right)} \cdot \sqrt[3]{t + \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U* - U\right)\right) - \left(\frac{\ell}{Om} \cdot 2\right) \cdot \ell\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\ell \cdot n\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right)\right) - \left(2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) - t\right)} \cdot \left(\left(n \cdot 2\right) \cdot \sqrt[3]{U}\right)\right) \cdot \sqrt[3]{U}\right)}\right) \cdot \sqrt[3]{U}}\]

   if 1.1147771080838e-310 < U

   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 32.0

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

   4. Applied sqr-pow 32.0

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

   5. Applied associate-*r* 31.0

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

   6. Simplified 31.0

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

   8. Applied sqrt-prod 23.3

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

   9. Simplified 24.5

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

4. Final simplification 28.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le 1.114777108083807533454474777376791311174 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(\sqrt[3]{t + \left(\left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) - \left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell\right)} \cdot \sqrt[3]{t + \left(\left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) - \left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell\right)}\right) \cdot \left(\sqrt[3]{U} \cdot \left(\left(\sqrt[3]{U} \cdot \left(n \cdot 2\right)\right) \cdot \sqrt[3]{\left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2 - t\right)}\right)\right)\right) \cdot \sqrt[3]{U}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(n \cdot 2\right) \cdot \left(\left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot 2}{2}\right)}\right) - \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2 - t\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(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*))))))