Toniolo and Linder, Equation (13)

Average Error: 34.7 → 30.3

Time: 42.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 4.69377638721043389061730150231150526438 \cdot 10^{-278} \lor \neg \left(t \le 1.201900539256645067229719544001952040476 \cdot 10^{-58}\right) \land t \le 5.865833813155722386854733087312992545187 \cdot 10^{144}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(\left(n \cdot {\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r104293 = 2.0;
        double r104294 = n;
        double r104295 = r104293 * r104294;
        double r104296 = U;
        double r104297 = r104295 * r104296;
        double r104298 = t;
        double r104299 = l;
        double r104300 = r104299 * r104299;
        double r104301 = Om;
        double r104302 = r104300 / r104301;
        double r104303 = r104293 * r104302;
        double r104304 = r104298 - r104303;
        double r104305 = r104299 / r104301;
        double r104306 = pow(r104305, r104293);
        double r104307 = r104294 * r104306;
        double r104308 = U_;
        double r104309 = r104296 - r104308;
        double r104310 = r104307 * r104309;
        double r104311 = r104304 - r104310;
        double r104312 = r104297 * r104311;
        double r104313 = sqrt(r104312);
        return r104313;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r104314 = t;
        double r104315 = 4.693776387210434e-278;
        bool r104316 = r104314 <= r104315;
        double r104317 = 1.201900539256645e-58;
        bool r104318 = r104314 <= r104317;
        double r104319 = !r104318;
        double r104320 = 5.8658338131557224e+144;
        bool r104321 = r104314 <= r104320;
        bool r104322 = r104319 && r104321;
        bool r104323 = r104316 || r104322;
        double r104324 = 2.0;
        double r104325 = l;
        double r104326 = Om;
        double r104327 = r104325 / r104326;
        double r104328 = r104325 * r104327;
        double r104329 = n;
        double r104330 = cbrt(r104325);
        double r104331 = r104330 * r104330;
        double r104332 = 2.0;
        double r104333 = r104324 / r104332;
        double r104334 = pow(r104331, r104333);
        double r104335 = r104329 * r104334;
        double r104336 = r104330 / r104326;
        double r104337 = pow(r104336, r104333);
        double r104338 = r104335 * r104337;
        double r104339 = pow(r104327, r104333);
        double r104340 = U;
        double r104341 = U_;
        double r104342 = r104340 - r104341;
        double r104343 = r104339 * r104342;
        double r104344 = r104338 * r104343;
        double r104345 = fma(r104324, r104328, r104344);
        double r104346 = r104314 - r104345;
        double r104347 = r104324 * r104329;
        double r104348 = r104347 * r104340;
        double r104349 = r104346 * r104348;
        double r104350 = sqrt(r104349);
        double r104351 = r104332 * r104333;
        double r104352 = pow(r104327, r104351);
        double r104353 = r104329 * r104352;
        double r104354 = r104342 * r104353;
        double r104355 = fma(r104324, r104328, r104354);
        double r104356 = r104314 - r104355;
        double r104357 = sqrt(r104356);
        double r104358 = sqrt(r104348);
        double r104359 = r104357 * r104358;
        double r104360 = r104323 ? r104350 : r104359;
        return r104360;
}

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 2 regimes

2. if t < 4.693776387210434e-278 or 1.201900539256645e-58 < t < 5.8658338131557224e+144

   1. Initial program 33.7

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

   2. Simplified 33.7

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

   4. Applied *-un-lft-identity 33.7

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

   5. Applied times-frac 31.0

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

   6. Simplified 31.0

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

   8. Applied sqr-pow 31.0

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

   9. Applied associate-*r* 30.2

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

   11. Applied associate-*l* 29.9

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

   13. Applied *-un-lft-identity 29.9

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

   14. Applied add-cube-cbrt 29.9

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

   15. Applied times-frac 29.9

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

   16. Applied unpow-prod-down 29.9

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

   17. Applied associate-*r* 30.3

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

   18. Simplified 30.3

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

   if 4.693776387210434e-278 < t < 1.201900539256645e-58 or 5.8658338131557224e+144 < t

   1. Initial program 37.1

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

   2. Simplified 37.1

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

   4. Applied *-un-lft-identity 37.1

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

   5. Applied times-frac 34.7

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

   6. Simplified 34.7

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

   8. Applied sqr-pow 34.7

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

   9. Applied associate-*r* 33.9

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

   11. Applied sqrt-prod 29.6

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

   12. Simplified 30.1

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

4. Final simplification 30.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 4.69377638721043389061730150231150526438 \cdot 10^{-278} \lor \neg \left(t \le 1.201900539256645067229719544001952040476 \cdot 10^{-58}\right) \land t \le 5.865833813155722386854733087312992545187 \cdot 10^{144}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(\left(n \cdot {\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(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*))))))