Toniolo and Linder, Equation (13)

Average Error: 33.3 → 26.0

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -2.164770592264949 \cdot 10^{+29}:\\ \;\;\;\;{\left(t \cdot \left(n \cdot \left(2 \cdot U\right)\right) + \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right) \cdot \left(-2 \cdot \ell - \frac{U - U*}{\frac{Om}{\ell \cdot n}}\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;U \le 1.2507119813575732 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot t + \left(\left(-2 \cdot \ell - \left(U - U*\right) \cdot \frac{\ell}{\frac{Om}{n}}\right) \cdot \left(U \cdot \left(\left(\frac{1}{Om} \cdot n\right) \cdot \ell\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot t + 2 \cdot \left(\left(U \cdot \left(\left(n \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)\right)\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \frac{\ell}{\frac{Om}{n}}\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r6954044 = 2.0;
        double r6954045 = n;
        double r6954046 = r6954044 * r6954045;
        double r6954047 = U;
        double r6954048 = r6954046 * r6954047;
        double r6954049 = t;
        double r6954050 = l;
        double r6954051 = r6954050 * r6954050;
        double r6954052 = Om;
        double r6954053 = r6954051 / r6954052;
        double r6954054 = r6954044 * r6954053;
        double r6954055 = r6954049 - r6954054;
        double r6954056 = r6954050 / r6954052;
        double r6954057 = pow(r6954056, r6954044);
        double r6954058 = r6954045 * r6954057;
        double r6954059 = U_;
        double r6954060 = r6954047 - r6954059;
        double r6954061 = r6954058 * r6954060;
        double r6954062 = r6954055 - r6954061;
        double r6954063 = r6954048 * r6954062;
        double r6954064 = sqrt(r6954063);
        return r6954064;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r6954065 = U;
        double r6954066 = -2.164770592264949e+29;
        bool r6954067 = r6954065 <= r6954066;
        double r6954068 = t;
        double r6954069 = n;
        double r6954070 = 2.0;
        double r6954071 = r6954070 * r6954065;
        double r6954072 = r6954069 * r6954071;
        double r6954073 = r6954068 * r6954072;
        double r6954074 = l;
        double r6954075 = Om;
        double r6954076 = r6954074 / r6954075;
        double r6954077 = r6954076 * r6954072;
        double r6954078 = -2.0;
        double r6954079 = r6954078 * r6954074;
        double r6954080 = U_;
        double r6954081 = r6954065 - r6954080;
        double r6954082 = r6954074 * r6954069;
        double r6954083 = r6954075 / r6954082;
        double r6954084 = r6954081 / r6954083;
        double r6954085 = r6954079 - r6954084;
        double r6954086 = r6954077 * r6954085;
        double r6954087 = r6954073 + r6954086;
        double r6954088 = 0.5;
        double r6954089 = pow(r6954087, r6954088);
        double r6954090 = 1.2507119813575732e-34;
        bool r6954091 = r6954065 <= r6954090;
        double r6954092 = r6954069 * r6954070;
        double r6954093 = r6954065 * r6954092;
        double r6954094 = r6954093 * r6954068;
        double r6954095 = r6954075 / r6954069;
        double r6954096 = r6954074 / r6954095;
        double r6954097 = r6954081 * r6954096;
        double r6954098 = r6954079 - r6954097;
        double r6954099 = 1.0;
        double r6954100 = r6954099 / r6954075;
        double r6954101 = r6954100 * r6954069;
        double r6954102 = r6954101 * r6954074;
        double r6954103 = r6954065 * r6954102;
        double r6954104 = r6954098 * r6954103;
        double r6954105 = r6954104 * r6954070;
        double r6954106 = r6954094 + r6954105;
        double r6954107 = sqrt(r6954106);
        double r6954108 = cbrt(r6954074);
        double r6954109 = cbrt(r6954075);
        double r6954110 = r6954108 / r6954109;
        double r6954111 = r6954069 * r6954110;
        double r6954112 = r6954110 * r6954110;
        double r6954113 = r6954111 * r6954112;
        double r6954114 = r6954065 * r6954113;
        double r6954115 = r6954114 * r6954098;
        double r6954116 = r6954070 * r6954115;
        double r6954117 = r6954094 + r6954116;
        double r6954118 = sqrt(r6954117);
        double r6954119 = r6954091 ? r6954107 : r6954118;
        double r6954120 = r6954067 ? r6954089 : r6954119;
        return r6954120;
}

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 < -2.164770592264949e+29

   1. Initial program 28.2

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

   3. Applied *-un-lft-identity 28.2

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

   4. Applied associate-*r* 28.2

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

   5. Simplified 24.0

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

   7. Applied sub-neg 24.0

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

   8. Applied distribute-rgt-in 24.0

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

   9. Simplified 23.6

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

   11. Applied pow1 23.6

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

   12. Applied sqrt-pow1 23.6

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

   13. Simplified 23.4

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

   if -2.164770592264949e+29 < U < 1.2507119813575732e-34

   1. Initial program 36.7

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

   3. Applied *-un-lft-identity 36.7

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

   4. Applied associate-*r* 36.7

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

   5. Simplified 32.9

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

   7. Applied sub-neg 32.9

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

   8. Applied distribute-rgt-in 32.9

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

   9. Simplified 27.2

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

   10. Taylor expanded around inf 31.4

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

   11. Simplified 28.4

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

   13. Applied div-inv 28.4

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

   14. Applied associate-*l* 27.8

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

   if 1.2507119813575732e-34 < U

   1. Initial program 27.8

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

   3. Applied *-un-lft-identity 27.8

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

   4. Applied associate-*r* 27.8

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

   5. Simplified 23.2

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

   7. Applied sub-neg 23.2

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

   8. Applied distribute-rgt-in 23.2

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

   9. Simplified 22.6

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

   10. Taylor expanded around inf 25.2

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

   11. Simplified 22.6

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

   13. Applied add-cube-cbrt 22.7

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

   14. Applied add-cube-cbrt 22.8

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

   15. Applied times-frac 22.8

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

   16. Applied associate-*l* 22.6

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

   17. Simplified 22.6

       \[\leadsto \sqrt{t \cdot \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) + \left(\left(\left(\color{blue}{\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)} \cdot \left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot n\right)\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \frac{\ell}{\frac{Om}{n}} \cdot \left(U - U*\right)\right)\right) \cdot 2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 26.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -2.164770592264949 \cdot 10^{+29}:\\ \;\;\;\;{\left(t \cdot \left(n \cdot \left(2 \cdot U\right)\right) + \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right) \cdot \left(-2 \cdot \ell - \frac{U - U*}{\frac{Om}{\ell \cdot n}}\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;U \le 1.2507119813575732 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot t + \left(\left(-2 \cdot \ell - \left(U - U*\right) \cdot \frac{\ell}{\frac{Om}{n}}\right) \cdot \left(U \cdot \left(\left(\frac{1}{Om} \cdot n\right) \cdot \ell\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot t + 2 \cdot \left(\left(U \cdot \left(\left(n \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)\right)\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \frac{\ell}{\frac{Om}{n}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(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*))))))