Toniolo and Linder, Equation (13)

Average Error: 33.1 → 20.9

Time: 49.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le 7.586071649758195 \cdot 10^{-274}:\\ \;\;\;\;\left|\sqrt[3]{U}\right| \cdot \sqrt{\left(n \cdot \left(t - \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, 2, n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right)\right)\right) \cdot \left(2 \cdot \sqrt[3]{U}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(2 \cdot \sqrt[3]{U}\right) \cdot \left(t - \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, 2, n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right)\right)} \cdot \sqrt{n}\right) \cdot \left|\sqrt[3]{U}\right|\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1520980 = 2.0;
        double r1520981 = n;
        double r1520982 = r1520980 * r1520981;
        double r1520983 = U;
        double r1520984 = r1520982 * r1520983;
        double r1520985 = t;
        double r1520986 = l;
        double r1520987 = r1520986 * r1520986;
        double r1520988 = Om;
        double r1520989 = r1520987 / r1520988;
        double r1520990 = r1520980 * r1520989;
        double r1520991 = r1520985 - r1520990;
        double r1520992 = r1520986 / r1520988;
        double r1520993 = pow(r1520992, r1520980);
        double r1520994 = r1520981 * r1520993;
        double r1520995 = U_;
        double r1520996 = r1520983 - r1520995;
        double r1520997 = r1520994 * r1520996;
        double r1520998 = r1520991 - r1520997;
        double r1520999 = r1520984 * r1520998;
        double r1521000 = sqrt(r1520999);
        return r1521000;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1521001 = n;
        double r1521002 = 7.586071649758195e-274;
        bool r1521003 = r1521001 <= r1521002;
        double r1521004 = U;
        double r1521005 = cbrt(r1521004);
        double r1521006 = fabs(r1521005);
        double r1521007 = t;
        double r1521008 = l;
        double r1521009 = Om;
        double r1521010 = r1521008 / r1521009;
        double r1521011 = 2.0;
        double r1521012 = U_;
        double r1521013 = r1521004 - r1521012;
        double r1521014 = r1521013 * r1521010;
        double r1521015 = r1521001 * r1521014;
        double r1521016 = fma(r1521008, r1521011, r1521015);
        double r1521017 = r1521010 * r1521016;
        double r1521018 = r1521007 - r1521017;
        double r1521019 = r1521001 * r1521018;
        double r1521020 = r1521011 * r1521005;
        double r1521021 = r1521019 * r1521020;
        double r1521022 = sqrt(r1521021);
        double r1521023 = r1521006 * r1521022;
        double r1521024 = r1521020 * r1521018;
        double r1521025 = sqrt(r1521024);
        double r1521026 = sqrt(r1521001);
        double r1521027 = r1521025 * r1521026;
        double r1521028 = r1521027 * r1521006;
        double r1521029 = r1521003 ? r1521023 : r1521028;
        return r1521029;
}

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 n < 7.586071649758195e-274

   1. Initial program 33.7

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

   2. Simplified 30.8

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

   4. Applied add-cube-cbrt 31.1

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

   5. Applied associate-*l* 31.1

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

   6. Simplified 29.2

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

   8. Applied sqrt-prod 24.3

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

   9. Simplified 24.3

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

   10. Simplified 25.0

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

   12. Applied associate-*l* 25.1

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

   if 7.586071649758195e-274 < n

   1. Initial program 32.2

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

   2. Simplified 30.8

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

   4. Applied add-cube-cbrt 31.1

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

   5. Applied associate-*l* 31.1

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

   6. Simplified 29.1

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

   8. Applied sqrt-prod 23.9

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

   9. Simplified 23.9

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

   10. Simplified 23.8

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

   12. Applied sqrt-prod 15.8

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

4. Final simplification 20.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le 7.586071649758195 \cdot 10^{-274}:\\ \;\;\;\;\left|\sqrt[3]{U}\right| \cdot \sqrt{\left(n \cdot \left(t - \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, 2, n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right)\right)\right) \cdot \left(2 \cdot \sqrt[3]{U}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(2 \cdot \sqrt[3]{U}\right) \cdot \left(t - \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, 2, n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right)\right)} \cdot \sqrt{n}\right) \cdot \left|\sqrt[3]{U}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +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*))))))