Toniolo and Linder, Equation (13)

Average Error: 32.8 → 28.1

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 3.546451389182172 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{\frac{\sqrt[3]{n}}{\frac{\sqrt[3]{Om}}{\ell}}}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;t \le 5.193191025670422 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \sqrt{\left(U \cdot 2\right) \cdot n}\\ \mathbf{elif}\;t \le 1.2796693870036841 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot \mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{\frac{\sqrt[3]{n}}{\frac{\sqrt[3]{Om}}{\ell}}}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \sqrt{\left(U \cdot 2\right) \cdot n}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2457167 = 2.0;
        double r2457168 = n;
        double r2457169 = r2457167 * r2457168;
        double r2457170 = U;
        double r2457171 = r2457169 * r2457170;
        double r2457172 = t;
        double r2457173 = l;
        double r2457174 = r2457173 * r2457173;
        double r2457175 = Om;
        double r2457176 = r2457174 / r2457175;
        double r2457177 = r2457167 * r2457176;
        double r2457178 = r2457172 - r2457177;
        double r2457179 = r2457173 / r2457175;
        double r2457180 = pow(r2457179, r2457167);
        double r2457181 = r2457168 * r2457180;
        double r2457182 = U_;
        double r2457183 = r2457170 - r2457182;
        double r2457184 = r2457181 * r2457183;
        double r2457185 = r2457178 - r2457184;
        double r2457186 = r2457171 * r2457185;
        double r2457187 = sqrt(r2457186);
        return r2457187;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2457188 = t;
        double r2457189 = 3.546451389182172e-302;
        bool r2457190 = r2457188 <= r2457189;
        double r2457191 = U;
        double r2457192 = 2.0;
        double r2457193 = r2457191 * r2457192;
        double r2457194 = n;
        double r2457195 = U_;
        double r2457196 = r2457195 - r2457191;
        double r2457197 = cbrt(r2457194);
        double r2457198 = r2457197 * r2457197;
        double r2457199 = Om;
        double r2457200 = cbrt(r2457199);
        double r2457201 = r2457200 * r2457200;
        double r2457202 = l;
        double r2457203 = r2457200 / r2457202;
        double r2457204 = r2457197 / r2457203;
        double r2457205 = r2457201 / r2457204;
        double r2457206 = r2457198 / r2457205;
        double r2457207 = r2457199 / r2457202;
        double r2457208 = r2457206 / r2457207;
        double r2457209 = r2457202 / r2457207;
        double r2457210 = -2.0;
        double r2457211 = fma(r2457209, r2457210, r2457188);
        double r2457212 = fma(r2457196, r2457208, r2457211);
        double r2457213 = r2457194 * r2457212;
        double r2457214 = r2457193 * r2457213;
        double r2457215 = sqrt(r2457214);
        double r2457216 = 5.193191025670422e-187;
        bool r2457217 = r2457188 <= r2457216;
        double r2457218 = r2457194 / r2457207;
        double r2457219 = r2457218 / r2457207;
        double r2457220 = fma(r2457196, r2457219, r2457211);
        double r2457221 = sqrt(r2457220);
        double r2457222 = r2457193 * r2457194;
        double r2457223 = sqrt(r2457222);
        double r2457224 = r2457221 * r2457223;
        double r2457225 = 1.2796693870036841e+137;
        bool r2457226 = r2457188 <= r2457225;
        double r2457227 = r2457222 * r2457212;
        double r2457228 = sqrt(r2457227);
        double r2457229 = r2457226 ? r2457228 : r2457224;
        double r2457230 = r2457217 ? r2457224 : r2457229;
        double r2457231 = r2457190 ? r2457215 : r2457230;
        return r2457231;
}

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 t < 3.546451389182172e-302

   1. Initial program 32.3

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

   2. Simplified 28.9

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

   4. Applied associate-*r* 28.8

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

   6. Applied add-cube-cbrt 28.9

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

   7. Applied associate-/l* 28.9

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

   9. Applied *-un-lft-identity 28.9

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

   10. Applied add-cube-cbrt 28.9

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

   11. Applied times-frac 28.9

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

   12. Applied associate-/l* 28.9

       \[\leadsto \sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot \mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{\color{blue}{\frac{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{1}}{\frac{\sqrt[3]{n}}{\frac{\sqrt[3]{Om}}{\ell}}}}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\]
   13. Using strategy rm

   14. Applied associate-*l* 29.0

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

   15. Simplified 29.0

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

   if 3.546451389182172e-302 < t < 5.193191025670422e-187 or 1.2796693870036841e+137 < t

   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. Simplified 33.5

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

   4. Applied associate-*r* 33.0

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

   6. Applied sqrt-prod 27.0

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

   if 5.193191025670422e-187 < t < 1.2796693870036841e+137

   1. Initial program 30.5

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

   2. Simplified 26.8

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

   4. Applied associate-*r* 27.2

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

   6. Applied add-cube-cbrt 27.3

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

   7. Applied associate-/l* 27.3

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

   9. Applied *-un-lft-identity 27.3

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

   10. Applied add-cube-cbrt 27.3

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

   11. Applied times-frac 27.3

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

   12. Applied associate-/l* 27.3

       \[\leadsto \sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot \mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{\color{blue}{\frac{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{1}}{\frac{\sqrt[3]{n}}{\frac{\sqrt[3]{Om}}{\ell}}}}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 28.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 3.546451389182172 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot \mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{\frac{\sqrt[3]{n}}{\frac{\sqrt[3]{Om}}{\ell}}}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)\right)}\\ \mathbf{elif}\;t \le 5.193191025670422 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \sqrt{\left(U \cdot 2\right) \cdot n}\\ \mathbf{elif}\;t \le 1.2796693870036841 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot \mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{\frac{\sqrt[3]{n}}{\frac{\sqrt[3]{Om}}{\ell}}}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \sqrt{\left(U \cdot 2\right) \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +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*))))))