Toniolo and Linder, Equation (13)

Average Error: 33.3 → 26.0

Time: 1.5m

Precision: 64

Math

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

↓

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

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r7643172 = 2.0;
        double r7643173 = n;
        double r7643174 = r7643172 * r7643173;
        double r7643175 = U;
        double r7643176 = r7643174 * r7643175;
        double r7643177 = t;
        double r7643178 = l;
        double r7643179 = r7643178 * r7643178;
        double r7643180 = Om;
        double r7643181 = r7643179 / r7643180;
        double r7643182 = r7643172 * r7643181;
        double r7643183 = r7643177 - r7643182;
        double r7643184 = r7643178 / r7643180;
        double r7643185 = pow(r7643184, r7643172);
        double r7643186 = r7643173 * r7643185;
        double r7643187 = U_;
        double r7643188 = r7643175 - r7643187;
        double r7643189 = r7643186 * r7643188;
        double r7643190 = r7643183 - r7643189;
        double r7643191 = r7643176 * r7643190;
        double r7643192 = sqrt(r7643191);
        return r7643192;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r7643193 = U;
        double r7643194 = -2.164770592264949e+29;
        bool r7643195 = r7643193 <= r7643194;
        double r7643196 = t;
        double r7643197 = n;
        double r7643198 = 2.0;
        double r7643199 = r7643198 * r7643193;
        double r7643200 = r7643197 * r7643199;
        double r7643201 = r7643196 * r7643200;
        double r7643202 = l;
        double r7643203 = Om;
        double r7643204 = r7643202 / r7643203;
        double r7643205 = r7643204 * r7643200;
        double r7643206 = -2.0;
        double r7643207 = r7643206 * r7643202;
        double r7643208 = U_;
        double r7643209 = r7643193 - r7643208;
        double r7643210 = r7643202 * r7643197;
        double r7643211 = r7643203 / r7643210;
        double r7643212 = r7643209 / r7643211;
        double r7643213 = r7643207 - r7643212;
        double r7643214 = r7643205 * r7643213;
        double r7643215 = r7643201 + r7643214;
        double r7643216 = 0.5;
        double r7643217 = pow(r7643215, r7643216);
        double r7643218 = 1.2507119813575732e-34;
        bool r7643219 = r7643193 <= r7643218;
        double r7643220 = r7643197 * r7643198;
        double r7643221 = r7643193 * r7643220;
        double r7643222 = r7643221 * r7643196;
        double r7643223 = r7643203 / r7643197;
        double r7643224 = r7643202 / r7643223;
        double r7643225 = r7643209 * r7643224;
        double r7643226 = r7643207 - r7643225;
        double r7643227 = 1.0;
        double r7643228 = r7643227 / r7643203;
        double r7643229 = r7643228 * r7643197;
        double r7643230 = r7643229 * r7643202;
        double r7643231 = r7643193 * r7643230;
        double r7643232 = r7643226 * r7643231;
        double r7643233 = r7643232 * r7643198;
        double r7643234 = r7643222 + r7643233;
        double r7643235 = sqrt(r7643234);
        double r7643236 = cbrt(r7643202);
        double r7643237 = cbrt(r7643203);
        double r7643238 = r7643236 / r7643237;
        double r7643239 = r7643197 * r7643238;
        double r7643240 = r7643238 * r7643238;
        double r7643241 = r7643239 * r7643240;
        double r7643242 = r7643193 * r7643241;
        double r7643243 = r7643242 * r7643226;
        double r7643244 = r7643198 * r7643243;
        double r7643245 = r7643222 + r7643244;
        double r7643246 = sqrt(r7643245);
        double r7643247 = r7643219 ? r7643235 : r7643246;
        double r7643248 = r7643195 ? r7643217 : r7643247;
        return r7643248;
}

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*))))))