Toniolo and Linder, Equation (13)

Average Error: 34.5 → 28.7

Time: 52.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -8.419437228522636679217683308706909883767 \cdot 10^{-4} \lor n \le -3.88758576516516698363399032912355760136 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right)\right) \cdot \sqrt[3]{U - U*}\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r192190 = 2.0;
        double r192191 = n;
        double r192192 = r192190 * r192191;
        double r192193 = U;
        double r192194 = r192192 * r192193;
        double r192195 = t;
        double r192196 = l;
        double r192197 = r192196 * r192196;
        double r192198 = Om;
        double r192199 = r192197 / r192198;
        double r192200 = r192190 * r192199;
        double r192201 = r192195 - r192200;
        double r192202 = r192196 / r192198;
        double r192203 = pow(r192202, r192190);
        double r192204 = r192191 * r192203;
        double r192205 = U_;
        double r192206 = r192193 - r192205;
        double r192207 = r192204 * r192206;
        double r192208 = r192201 - r192207;
        double r192209 = r192194 * r192208;
        double r192210 = sqrt(r192209);
        return r192210;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r192211 = n;
        double r192212 = -0.0008419437228522637;
        bool r192213 = r192211 <= r192212;
        double r192214 = -3.887585765165167e-308;
        bool r192215 = r192211 <= r192214;
        bool r192216 = r192213 || r192215;
        double r192217 = 2.0;
        double r192218 = r192217 * r192211;
        double r192219 = U;
        double r192220 = t;
        double r192221 = l;
        double r192222 = Om;
        double r192223 = r192222 / r192221;
        double r192224 = r192221 / r192223;
        double r192225 = r192217 * r192224;
        double r192226 = r192220 - r192225;
        double r192227 = r192221 / r192222;
        double r192228 = pow(r192227, r192217);
        double r192229 = U_;
        double r192230 = r192219 - r192229;
        double r192231 = r192228 * r192230;
        double r192232 = r192211 * r192231;
        double r192233 = r192226 - r192232;
        double r192234 = r192219 * r192233;
        double r192235 = r192218 * r192234;
        double r192236 = sqrt(r192235);
        double r192237 = sqrt(r192218);
        double r192238 = r192211 * r192228;
        double r192239 = cbrt(r192230);
        double r192240 = r192239 * r192239;
        double r192241 = r192238 * r192240;
        double r192242 = r192241 * r192239;
        double r192243 = r192226 - r192242;
        double r192244 = r192219 * r192243;
        double r192245 = sqrt(r192244);
        double r192246 = r192237 * r192245;
        double r192247 = r192216 ? r192236 : r192246;
        return r192247;
}

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 n < -0.0008419437228522637

   1. Initial program 32.5

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

   3. Applied associate-/l* 30.5

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

   5. Applied associate-*l* 31.6

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

   7. Applied associate-*l* 30.9

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

   if -0.0008419437228522637 < n < -3.887585765165167e-308

   1. Initial program 35.6

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

   3. Applied associate-/l* 32.5

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

   5. Applied associate-*l* 32.5

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

   7. Applied add-cube-cbrt 32.5

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

   8. Applied associate-*r* 32.5

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right)\right) \cdot \sqrt[3]{U - U*}}\right)\right)}\]
   9. Using strategy rm

   10. Applied pow1 32.5

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

   11. Applied pow1 32.5

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

   12. Applied pow-prod-down 32.5

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

   13. Applied pow1 32.5

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

   14. Applied pow1 32.5

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

   15. Applied pow-prod-down 32.5

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

   16. Applied pow-prod-down 32.5

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

   17. Simplified 29.6

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

   if -3.887585765165167e-308 < n

   1. Initial program 34.7

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

   3. Applied associate-/l* 31.8

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

   5. Applied associate-*l* 32.5

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

   7. Applied add-cube-cbrt 32.5

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

   8. Applied associate-*r* 32.5

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right)\right) \cdot \sqrt[3]{U - U*}}\right)\right)}\]
   9. Using strategy rm

   10. Applied sqrt-prod 24.8

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

4. Final simplification 28.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -8.419437228522636679217683308706909883767 \cdot 10^{-4} \lor n \le -3.88758576516516698363399032912355760136 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right)\right) \cdot \sqrt[3]{U - U*}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))