Toniolo and Linder, Equation (13)

Average Error: 34.4 → 32.0

Time: 47.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -7.011733841081500777334264329602193954177 \cdot 10^{54}:\\ \;\;\;\;\sqrt{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}}\\ \mathbf{elif}\;U \le 1.714350869156095708205362740940600427368 \cdot 10^{62}:\\ \;\;\;\;\sqrt{\left(\left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)}^{2}\right) \cdot \left(\left(U* - U\right) \cdot n\right) + \left(\mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r115891 = 2.0;
        double r115892 = n;
        double r115893 = r115891 * r115892;
        double r115894 = U;
        double r115895 = r115893 * r115894;
        double r115896 = t;
        double r115897 = l;
        double r115898 = r115897 * r115897;
        double r115899 = Om;
        double r115900 = r115898 / r115899;
        double r115901 = r115891 * r115900;
        double r115902 = r115896 - r115901;
        double r115903 = r115897 / r115899;
        double r115904 = pow(r115903, r115891);
        double r115905 = r115892 * r115904;
        double r115906 = U_;
        double r115907 = r115894 - r115906;
        double r115908 = r115905 * r115907;
        double r115909 = r115902 - r115908;
        double r115910 = r115895 * r115909;
        double r115911 = sqrt(r115910);
        return r115911;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r115912 = U;
        double r115913 = -7.011733841081501e+54;
        bool r115914 = r115912 <= r115913;
        double r115915 = U_;
        double r115916 = r115915 - r115912;
        double r115917 = n;
        double r115918 = r115916 * r115917;
        double r115919 = l;
        double r115920 = Om;
        double r115921 = r115919 / r115920;
        double r115922 = 2.0;
        double r115923 = pow(r115921, r115922);
        double r115924 = -r115922;
        double r115925 = r115924 * r115919;
        double r115926 = t;
        double r115927 = fma(r115921, r115925, r115926);
        double r115928 = fma(r115918, r115923, r115927);
        double r115929 = r115917 * r115912;
        double r115930 = r115922 * r115929;
        double r115931 = r115928 * r115930;
        double r115932 = sqrt(r115931);
        double r115933 = sqrt(r115932);
        double r115934 = r115933 * r115933;
        double r115935 = 1.7143508691560957e+62;
        bool r115936 = r115912 <= r115935;
        double r115937 = cbrt(r115921);
        double r115938 = pow(r115937, r115922);
        double r115939 = r115912 * r115922;
        double r115940 = r115917 * r115939;
        double r115941 = r115938 * r115940;
        double r115942 = r115937 * r115937;
        double r115943 = pow(r115942, r115922);
        double r115944 = r115941 * r115943;
        double r115945 = r115944 * r115918;
        double r115946 = r115927 * r115912;
        double r115947 = r115917 * r115922;
        double r115948 = r115946 * r115947;
        double r115949 = r115945 + r115948;
        double r115950 = sqrt(r115949);
        double r115951 = sqrt(r115930);
        double r115952 = sqrt(r115928);
        double r115953 = r115951 * r115952;
        double r115954 = r115936 ? r115950 : r115953;
        double r115955 = r115914 ? r115934 : r115954;
        return r115955;
}

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 U < -7.011733841081501e+54

   1. Initial program 29.1

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

   2. Simplified 26.1

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

   4. Applied add-sqr-sqrt 26.3

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

   5. Simplified 27.2

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

   6. Simplified 27.1

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

   if -7.011733841081501e+54 < U < 1.7143508691560957e+62

   1. Initial program 36.1

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

   2. Simplified 33.2

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

   4. Applied fma-udef 33.2

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

   5. Applied distribute-lft-in 33.2

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

   6. Simplified 34.9

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

   7. Simplified 32.1

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

   9. Applied add-cube-cbrt 32.2

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

   10. Applied unpow-prod-down 32.1

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

   11. Applied associate-*l* 31.0

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

   12. Simplified 31.0

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

   if 1.7143508691560957e+62 < U

   1. Initial program 31.3

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

   2. Simplified 28.0

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

   4. Applied sqrt-prod 42.1

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

   5. Simplified 42.1

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

   6. Simplified 42.6

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

4. Final simplification 32.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -7.011733841081500777334264329602193954177 \cdot 10^{54}:\\ \;\;\;\;\sqrt{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}}\\ \mathbf{elif}\;U \le 1.714350869156095708205362740940600427368 \cdot 10^{62}:\\ \;\;\;\;\sqrt{\left(\left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right) \cdot {\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)}^{2}\right) \cdot \left(\left(U* - U\right) \cdot n\right) + \left(\mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, {\left(\frac{\ell}{Om}\right)}^{2}, \mathsf{fma}\left(\frac{\ell}{Om}, \left(-2\right) \cdot \ell, t\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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*))))))