Toniolo and Linder, Equation (13)

Average Error: 33.2 → 26.6

Time: 5.4m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -9.183046660180342 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{n \cdot \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;U \le 1.0657230548376832 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r26890183 = 2.0;
        double r26890184 = n;
        double r26890185 = r26890183 * r26890184;
        double r26890186 = U;
        double r26890187 = r26890185 * r26890186;
        double r26890188 = t;
        double r26890189 = l;
        double r26890190 = r26890189 * r26890189;
        double r26890191 = Om;
        double r26890192 = r26890190 / r26890191;
        double r26890193 = r26890183 * r26890192;
        double r26890194 = r26890188 - r26890193;
        double r26890195 = r26890189 / r26890191;
        double r26890196 = pow(r26890195, r26890183);
        double r26890197 = r26890184 * r26890196;
        double r26890198 = U_;
        double r26890199 = r26890186 - r26890198;
        double r26890200 = r26890197 * r26890199;
        double r26890201 = r26890194 - r26890200;
        double r26890202 = r26890187 * r26890201;
        double r26890203 = sqrt(r26890202);
        return r26890203;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r26890204 = U;
        double r26890205 = -9.183046660180342e-109;
        bool r26890206 = r26890204 <= r26890205;
        double r26890207 = 2.0;
        double r26890208 = n;
        double r26890209 = r26890207 * r26890208;
        double r26890210 = r26890209 * r26890204;
        double r26890211 = t;
        double r26890212 = l;
        double r26890213 = Om;
        double r26890214 = r26890213 / r26890212;
        double r26890215 = r26890212 / r26890214;
        double r26890216 = r26890207 * r26890215;
        double r26890217 = r26890211 - r26890216;
        double r26890218 = 1.0;
        double r26890219 = r26890218 / r26890214;
        double r26890220 = U_;
        double r26890221 = r26890204 - r26890220;
        double r26890222 = r26890219 * r26890221;
        double r26890223 = r26890208 * r26890222;
        double r26890224 = r26890223 / r26890214;
        double r26890225 = r26890217 - r26890224;
        double r26890226 = r26890210 * r26890225;
        double r26890227 = sqrt(r26890226);
        double r26890228 = 1.0657230548376832e+24;
        bool r26890229 = r26890204 <= r26890228;
        double r26890230 = r26890208 / r26890214;
        double r26890231 = r26890221 * r26890230;
        double r26890232 = r26890231 / r26890214;
        double r26890233 = r26890217 - r26890232;
        double r26890234 = r26890233 * r26890204;
        double r26890235 = r26890209 * r26890234;
        double r26890236 = sqrt(r26890235);
        double r26890237 = r26890210 * r26890233;
        double r26890238 = sqrt(r26890237);
        double r26890239 = sqrt(r26890238);
        double r26890240 = r26890239 * r26890239;
        double r26890241 = r26890229 ? r26890236 : r26890240;
        double r26890242 = r26890206 ? r26890227 : r26890241;
        return r26890242;
}

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 < -9.183046660180342e-109

   1. Initial program 28.3

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

   3. Applied associate-/l* 24.9

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

   4. Taylor expanded around inf 31.1

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

   5. Simplified 24.1

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

   7. Applied associate-*l/ 23.6

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

   9. Applied div-inv 23.6

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

   10. Applied associate-*l* 24.0

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

   if -9.183046660180342e-109 < U < 1.0657230548376832e+24

   1. Initial program 37.3

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

   3. Applied associate-/l* 34.9

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

   4. Taylor expanded around inf 41.1

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

   5. Simplified 33.9

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

   7. Applied associate-*l/ 33.4

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

   9. Applied associate-*l* 28.6

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

   if 1.0657230548376832e+24 < U

   1. Initial program 28.5

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

   3. Applied associate-/l* 25.9

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

   4. Taylor expanded around inf 30.9

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

   5. Simplified 25.2

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

   7. Applied associate-*l/ 24.6

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

   9. Applied add-sqr-sqrt 24.8

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

4. Final simplification 26.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -9.183046660180342 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{n \cdot \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}}\right)}\\ \mathbf{elif}\;U \le 1.0657230548376832 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +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*))))))