Toniolo and Linder, Equation (13)

Average Error: 33.1 → 27.4

Time: 40.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -6.580462540237744 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le 1.340924181894456 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 7.197261728987005 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;n \le 7.547628596937392 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left(\left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 2.4979801865310678 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r983157 = 2.0;
        double r983158 = n;
        double r983159 = r983157 * r983158;
        double r983160 = U;
        double r983161 = r983159 * r983160;
        double r983162 = t;
        double r983163 = l;
        double r983164 = r983163 * r983163;
        double r983165 = Om;
        double r983166 = r983164 / r983165;
        double r983167 = r983157 * r983166;
        double r983168 = r983162 - r983167;
        double r983169 = r983163 / r983165;
        double r983170 = pow(r983169, r983157);
        double r983171 = r983158 * r983170;
        double r983172 = U_;
        double r983173 = r983160 - r983172;
        double r983174 = r983171 * r983173;
        double r983175 = r983168 - r983174;
        double r983176 = r983161 * r983175;
        double r983177 = sqrt(r983176);
        return r983177;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r983178 = n;
        double r983179 = -6.580462540237744e-211;
        bool r983180 = r983178 <= r983179;
        double r983181 = 2.0;
        double r983182 = r983181 * r983178;
        double r983183 = U;
        double r983184 = t;
        double r983185 = l;
        double r983186 = Om;
        double r983187 = r983185 / r983186;
        double r983188 = r983187 * r983185;
        double r983189 = r983188 * r983181;
        double r983190 = r983184 - r983189;
        double r983191 = r983187 * r983178;
        double r983192 = U_;
        double r983193 = r983183 - r983192;
        double r983194 = r983187 * r983193;
        double r983195 = r983191 * r983194;
        double r983196 = r983190 - r983195;
        double r983197 = r983183 * r983196;
        double r983198 = r983182 * r983197;
        double r983199 = sqrt(r983198);
        double r983200 = 1.340924181894456e-308;
        bool r983201 = r983178 <= r983200;
        double r983202 = r983186 / r983185;
        double r983203 = r983185 / r983202;
        double r983204 = r983187 * r983187;
        double r983205 = r983204 * r983178;
        double r983206 = r983193 * r983205;
        double r983207 = fma(r983181, r983203, r983206);
        double r983208 = r983184 - r983207;
        double r983209 = r983182 * r983183;
        double r983210 = r983208 * r983209;
        double r983211 = sqrt(r983210);
        double r983212 = 7.197261728987005e-257;
        bool r983213 = r983178 <= r983212;
        double r983214 = sqrt(r983182);
        double r983215 = r983185 * r983185;
        double r983216 = r983215 / r983186;
        double r983217 = r983216 * r983181;
        double r983218 = r983184 - r983217;
        double r983219 = r983218 - r983195;
        double r983220 = r983183 * r983219;
        double r983221 = sqrt(r983220);
        double r983222 = r983214 * r983221;
        double r983223 = 7.547628596937392e-254;
        bool r983224 = r983178 <= r983223;
        double r983225 = /* ERROR: no posit support in C */;
        double r983226 = /* ERROR: no posit support in C */;
        double r983227 = r983193 * r983226;
        double r983228 = r983218 - r983227;
        double r983229 = r983228 * r983209;
        double r983230 = sqrt(r983229);
        double r983231 = 2.4979801865310678e+51;
        bool r983232 = r983178 <= r983231;
        double r983233 = r983232 ? r983199 : r983222;
        double r983234 = r983224 ? r983230 : r983233;
        double r983235 = r983213 ? r983222 : r983234;
        double r983236 = r983201 ? r983211 : r983235;
        double r983237 = r983180 ? r983199 : r983236;
        return r983237;
}

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 n < -6.580462540237744e-211 or 7.547628596937392e-254 < n < 2.4979801865310678e+51

   1. Initial program 31.6

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

   3. Applied unpow2 31.6

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

   4. Applied associate-*r* 30.9

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

   6. Applied associate-*l* 30.7

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

   8. Applied associate-*l* 30.2

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

   10. Applied *-un-lft-identity 30.2

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

   11. Applied times-frac 27.3

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

   12. Simplified 27.3

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

   if -6.580462540237744e-211 < n < 1.340924181894456e-308

   1. Initial program 40.2

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

   3. Applied unpow2 40.2

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

   4. Applied associate-*r* 39.1

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

   6. Applied pow1 39.1

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

   7. Applied pow1 39.1

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

   8. Applied pow1 39.1

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

   9. Applied pow1 39.1

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

   10. Applied pow-prod-down 39.1

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

   11. Applied pow-prod-down 39.1

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

   12. Applied pow-prod-down 39.1

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

   13. Simplified 38.0

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

   if 1.340924181894456e-308 < n < 7.197261728987005e-257 or 2.4979801865310678e+51 < n

   1. Initial program 34.4

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

   3. Applied unpow2 34.4

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

   4. Applied associate-*r* 32.7

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

   6. Applied associate-*l* 32.2

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

   8. Applied associate-*l* 32.4

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

   10. Applied sqrt-prod 22.4

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

   if 7.197261728987005e-257 < n < 7.547628596937392e-254

   1. Initial program 36.3

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

   3. Applied insert-posit16 38.9

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

   4. Simplified 38.9

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

4. Final simplification 27.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -6.580462540237744 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le 1.340924181894456 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 7.197261728987005 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;n \le 7.547628596937392 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left(\left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 2.4979801865310678 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +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*))))))