Toniolo and Linder, Equation (13)

Average Error: 34.6 → 27.9

Time: 1.3m

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 r247146 = 2.0;
        double r247147 = n;
        double r247148 = r247146 * r247147;
        double r247149 = U;
        double r247150 = r247148 * r247149;
        double r247151 = t;
        double r247152 = l;
        double r247153 = r247152 * r247152;
        double r247154 = Om;
        double r247155 = r247153 / r247154;
        double r247156 = r247146 * r247155;
        double r247157 = r247151 - r247156;
        double r247158 = r247152 / r247154;
        double r247159 = pow(r247158, r247146);
        double r247160 = r247147 * r247159;
        double r247161 = U_;
        double r247162 = r247149 - r247161;
        double r247163 = r247160 * r247162;
        double r247164 = r247157 - r247163;
        double r247165 = r247150 * r247164;
        double r247166 = sqrt(r247165);
        return r247166;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r247167 = U;
        double r247168 = -4.797165734624696e-88;
        bool r247169 = r247167 <= r247168;
        double r247170 = n;
        double r247171 = 2.0;
        double r247172 = r247170 * r247171;
        double r247173 = r247172 * r247167;
        double r247174 = t;
        double r247175 = l;
        double r247176 = Om;
        double r247177 = r247175 / r247176;
        double r247178 = 2.0;
        double r247179 = r247178 * r247171;
        double r247180 = r247179 / r247178;
        double r247181 = pow(r247177, r247180);
        double r247182 = r247170 * r247181;
        double r247183 = U_;
        double r247184 = r247183 - r247167;
        double r247185 = r247182 * r247184;
        double r247186 = r247175 * r247177;
        double r247187 = r247186 * r247171;
        double r247188 = r247185 - r247187;
        double r247189 = r247174 + r247188;
        double r247190 = r247173 * r247189;
        double r247191 = sqrt(r247190);
        double r247192 = r247191 * r247191;
        double r247193 = sqrt(r247192);
        double r247194 = -7.292518127255309e-277;
        bool r247195 = r247167 <= r247194;
        double r247196 = r247167 * r247189;
        double r247197 = r247172 * r247196;
        double r247198 = sqrt(r247197);
        double r247199 = -2.5443015690588e-310;
        bool r247200 = r247167 <= r247199;
        double r247201 = 4.913407422060663e-171;
        bool r247202 = r247167 <= r247201;
        double r247203 = !r247202;
        double r247204 = 8.219354400775286e+94;
        bool r247205 = r247167 <= r247204;
        bool r247206 = r247203 && r247205;
        bool r247207 = r247200 || r247206;
        double r247208 = r247171 / r247178;
        double r247209 = pow(r247177, r247208);
        double r247210 = r247209 * r247170;
        double r247211 = r247209 * r247210;
        double r247212 = r247211 * r247184;
        double r247213 = r247171 * r247175;
        double r247214 = r247177 * r247213;
        double r247215 = r247212 - r247214;
        double r247216 = r247174 + r247215;
        double r247217 = r247216 * r247172;
        double r247218 = r247167 * r247217;
        double r247219 = sqrt(r247218);
        double r247220 = sqrt(r247167);
        double r247221 = r247189 * r247172;
        double r247222 = sqrt(r247221);
        double r247223 = r247220 * r247222;
        double r247224 = r247207 ? r247219 : r247223;
        double r247225 = r247195 ? r247198 : r247224;
        double r247226 = r247169 ? r247193 : r247225;
        return r247226;
}

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*))))))