Toniolo and Linder, Equation (13)

Average Error: 34.6 → 27.9

Time: 1.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -4.797165734624695899226646467454361223135 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot 2}{2}\right)}\right) \cdot \left(U* - U\right) - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)} \cdot \sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot 2}{2}\right)}\right) \cdot \left(U* - U\right) - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)}}\\ \mathbf{elif}\;U \le -7.292518127255309422901006800230409602213 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot 2}{2}\right)}\right) \cdot \left(U* - U\right) - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right)}\\ \mathbf{elif}\;U \le -2.544301569058814338086007489613547011313 \cdot 10^{-310} \lor \neg \left(U \le 4.913407422060663127687960956485861242372 \cdot 10^{-171}\right) \land U \le 8.219354400775285904861674568102655539055 \cdot 10^{94}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t + \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right)\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right) \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot 2}{2}\right)}\right) \cdot \left(U* - U\right) - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right) \cdot \left(n \cdot 2\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r261805 = 2.0;
        double r261806 = n;
        double r261807 = r261805 * r261806;
        double r261808 = U;
        double r261809 = r261807 * r261808;
        double r261810 = t;
        double r261811 = l;
        double r261812 = r261811 * r261811;
        double r261813 = Om;
        double r261814 = r261812 / r261813;
        double r261815 = r261805 * r261814;
        double r261816 = r261810 - r261815;
        double r261817 = r261811 / r261813;
        double r261818 = pow(r261817, r261805);
        double r261819 = r261806 * r261818;
        double r261820 = U_;
        double r261821 = r261808 - r261820;
        double r261822 = r261819 * r261821;
        double r261823 = r261816 - r261822;
        double r261824 = r261809 * r261823;
        double r261825 = sqrt(r261824);
        return r261825;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r261826 = U;
        double r261827 = -4.797165734624696e-88;
        bool r261828 = r261826 <= r261827;
        double r261829 = n;
        double r261830 = 2.0;
        double r261831 = r261829 * r261830;
        double r261832 = r261831 * r261826;
        double r261833 = t;
        double r261834 = l;
        double r261835 = Om;
        double r261836 = r261834 / r261835;
        double r261837 = 2.0;
        double r261838 = r261837 * r261830;
        double r261839 = r261838 / r261837;
        double r261840 = pow(r261836, r261839);
        double r261841 = r261829 * r261840;
        double r261842 = U_;
        double r261843 = r261842 - r261826;
        double r261844 = r261841 * r261843;
        double r261845 = r261834 * r261836;
        double r261846 = r261845 * r261830;
        double r261847 = r261844 - r261846;
        double r261848 = r261833 + r261847;
        double r261849 = r261832 * r261848;
        double r261850 = sqrt(r261849);
        double r261851 = r261850 * r261850;
        double r261852 = sqrt(r261851);
        double r261853 = -7.292518127255309e-277;
        bool r261854 = r261826 <= r261853;
        double r261855 = r261826 * r261848;
        double r261856 = r261831 * r261855;
        double r261857 = sqrt(r261856);
        double r261858 = -2.5443015690588e-310;
        bool r261859 = r261826 <= r261858;
        double r261860 = 4.913407422060663e-171;
        bool r261861 = r261826 <= r261860;
        double r261862 = !r261861;
        double r261863 = 8.219354400775286e+94;
        bool r261864 = r261826 <= r261863;
        bool r261865 = r261862 && r261864;
        bool r261866 = r261859 || r261865;
        double r261867 = r261830 / r261837;
        double r261868 = pow(r261836, r261867);
        double r261869 = r261868 * r261829;
        double r261870 = r261868 * r261869;
        double r261871 = r261870 * r261843;
        double r261872 = r261830 * r261834;
        double r261873 = r261836 * r261872;
        double r261874 = r261871 - r261873;
        double r261875 = r261833 + r261874;
        double r261876 = r261875 * r261831;
        double r261877 = r261826 * r261876;
        double r261878 = sqrt(r261877);
        double r261879 = sqrt(r261826);
        double r261880 = r261848 * r261831;
        double r261881 = sqrt(r261880);
        double r261882 = r261879 * r261881;
        double r261883 = r261866 ? r261878 : r261882;
        double r261884 = r261854 ? r261857 : r261883;
        double r261885 = r261828 ? r261852 : r261884;
        return r261885;
}

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 4 regimes

2. if U < -4.797165734624696e-88

   1. Initial program 30.0

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

   2. Simplified 28.3

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

   4. Applied sqr-pow 28.3

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

   5. Applied associate-*r* 27.5

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

   6. Simplified 27.5

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

   8. Applied add-sqr-sqrt 27.5

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

   9. Simplified 29.7

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

   10. Simplified 26.7

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

   if -4.797165734624696e-88 < U < -7.292518127255309e-277

   1. Initial program 38.6

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

   2. Simplified 35.1

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

   4. Applied sqr-pow 35.1

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

   5. Applied associate-*r* 33.7

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

   6. Simplified 33.7

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

   8. Applied associate-*l* 29.8

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

   9. Simplified 31.2

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

   if -7.292518127255309e-277 < U < -2.5443015690588e-310 or 4.913407422060663e-171 < U < 8.219354400775286e+94

   1. Initial program 33.6

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

   2. Simplified 31.0

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

   4. Applied sqr-pow 31.0

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

   5. Applied associate-*r* 29.6

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

   6. Simplified 29.6

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

   if -2.5443015690588e-310 < U < 4.913407422060663e-171 or 8.219354400775286e+94 < U

   1. Initial program 37.4

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

   2. Simplified 35.5

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

   4. Applied sqr-pow 35.5

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

   5. Applied associate-*r* 34.7

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

   6. Simplified 34.7

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

   8. Applied sqrt-prod 24.0

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

   9. Simplified 25.0

      \[\leadsto \color{blue}{\sqrt{\left(\left(\left(U* - U\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot 2}{2}\right)}\right) - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right) + t\right) \cdot \left(n \cdot 2\right)}} \cdot \sqrt{U}\]
3. Recombined 4 regimes into one program.

4. Final simplification 27.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -4.797165734624695899226646467454361223135 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot 2}{2}\right)}\right) \cdot \left(U* - U\right) - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)} \cdot \sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot 2}{2}\right)}\right) \cdot \left(U* - U\right) - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)}}\\ \mathbf{elif}\;U \le -7.292518127255309422901006800230409602213 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot 2}{2}\right)}\right) \cdot \left(U* - U\right) - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right)}\\ \mathbf{elif}\;U \le -2.544301569058814338086007489613547011313 \cdot 10^{-310} \lor \neg \left(U \le 4.913407422060663127687960956485861242372 \cdot 10^{-171}\right) \land U \le 8.219354400775285904861674568102655539055 \cdot 10^{94}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t + \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right)\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right) \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot 2}{2}\right)}\right) \cdot \left(U* - U\right) - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right) \cdot \left(n \cdot 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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*))))))