Toniolo and Linder, Equation (13)

Average Error: 32.8 → 28.1

Time: 2.1m

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 r3045185 = 2.0;
        double r3045186 = n;
        double r3045187 = r3045185 * r3045186;
        double r3045188 = U;
        double r3045189 = r3045187 * r3045188;
        double r3045190 = t;
        double r3045191 = l;
        double r3045192 = r3045191 * r3045191;
        double r3045193 = Om;
        double r3045194 = r3045192 / r3045193;
        double r3045195 = r3045185 * r3045194;
        double r3045196 = r3045190 - r3045195;
        double r3045197 = r3045191 / r3045193;
        double r3045198 = pow(r3045197, r3045185);
        double r3045199 = r3045186 * r3045198;
        double r3045200 = U_;
        double r3045201 = r3045188 - r3045200;
        double r3045202 = r3045199 * r3045201;
        double r3045203 = r3045196 - r3045202;
        double r3045204 = r3045189 * r3045203;
        double r3045205 = sqrt(r3045204);
        return r3045205;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3045206 = t;
        double r3045207 = 3.546451389182172e-302;
        bool r3045208 = r3045206 <= r3045207;
        double r3045209 = U;
        double r3045210 = 2.0;
        double r3045211 = r3045209 * r3045210;
        double r3045212 = n;
        double r3045213 = U_;
        double r3045214 = r3045213 - r3045209;
        double r3045215 = cbrt(r3045212);
        double r3045216 = r3045215 * r3045215;
        double r3045217 = Om;
        double r3045218 = cbrt(r3045217);
        double r3045219 = r3045218 * r3045218;
        double r3045220 = l;
        double r3045221 = r3045218 / r3045220;
        double r3045222 = r3045215 / r3045221;
        double r3045223 = r3045219 / r3045222;
        double r3045224 = r3045216 / r3045223;
        double r3045225 = r3045217 / r3045220;
        double r3045226 = r3045224 / r3045225;
        double r3045227 = r3045220 / r3045225;
        double r3045228 = -2.0;
        double r3045229 = fma(r3045227, r3045228, r3045206);
        double r3045230 = fma(r3045214, r3045226, r3045229);
        double r3045231 = r3045212 * r3045230;
        double r3045232 = r3045211 * r3045231;
        double r3045233 = sqrt(r3045232);
        double r3045234 = 5.193191025670422e-187;
        bool r3045235 = r3045206 <= r3045234;
        double r3045236 = r3045212 / r3045225;
        double r3045237 = r3045236 / r3045225;
        double r3045238 = fma(r3045214, r3045237, r3045229);
        double r3045239 = sqrt(r3045238);
        double r3045240 = r3045211 * r3045212;
        double r3045241 = sqrt(r3045240);
        double r3045242 = r3045239 * r3045241;
        double r3045243 = 1.2796693870036841e+137;
        bool r3045244 = r3045206 <= r3045243;
        double r3045245 = r3045240 * r3045230;
        double r3045246 = sqrt(r3045245);
        double r3045247 = r3045244 ? r3045246 : r3045242;
        double r3045248 = r3045235 ? r3045242 : r3045247;
        double r3045249 = r3045208 ? r3045233 : r3045248;
        return r3045249;
}

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*))))))