Toniolo and Linder, Equation (13)

Average Error: 34.3 → 29.4

Time: 1.1m

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 -9.379719815826530589603530820932364731187 \cdot 10^{241}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right)}\\ \mathbf{elif}\;t \le -1.966200757559735377429830381935693182299 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(U* - U, \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)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 2336909186103.619140625:\\ \;\;\;\;\sqrt{\sqrt{\sqrt[3]{\mathsf{fma}\left(U* - U, \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)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \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)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \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)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(U* - U, \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)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\mathsf{fma}\left(U* - U, \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)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2806370 = 2.0;
        double r2806371 = n;
        double r2806372 = r2806370 * r2806371;
        double r2806373 = U;
        double r2806374 = r2806372 * r2806373;
        double r2806375 = t;
        double r2806376 = l;
        double r2806377 = r2806376 * r2806376;
        double r2806378 = Om;
        double r2806379 = r2806377 / r2806378;
        double r2806380 = r2806370 * r2806379;
        double r2806381 = r2806375 - r2806380;
        double r2806382 = r2806376 / r2806378;
        double r2806383 = pow(r2806382, r2806370);
        double r2806384 = r2806371 * r2806383;
        double r2806385 = U_;
        double r2806386 = r2806373 - r2806385;
        double r2806387 = r2806384 * r2806386;
        double r2806388 = r2806381 - r2806387;
        double r2806389 = r2806374 * r2806388;
        double r2806390 = sqrt(r2806389);
        return r2806390;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2806391 = t;
        double r2806392 = -9.37971981582653e+241;
        bool r2806393 = r2806391 <= r2806392;
        double r2806394 = 2.0;
        double r2806395 = n;
        double r2806396 = r2806394 * r2806395;
        double r2806397 = U;
        double r2806398 = r2806396 * r2806397;
        double r2806399 = l;
        double r2806400 = Om;
        double r2806401 = r2806399 / r2806400;
        double r2806402 = r2806399 * r2806394;
        double r2806403 = r2806401 * r2806402;
        double r2806404 = r2806391 - r2806403;
        double r2806405 = r2806398 * r2806404;
        double r2806406 = sqrt(r2806405);
        double r2806407 = -1.9662007575597354e-176;
        bool r2806408 = r2806391 <= r2806407;
        double r2806409 = U_;
        double r2806410 = r2806409 - r2806397;
        double r2806411 = 2.0;
        double r2806412 = r2806394 / r2806411;
        double r2806413 = pow(r2806401, r2806412);
        double r2806414 = r2806395 * r2806413;
        double r2806415 = r2806414 * r2806413;
        double r2806416 = r2806401 * r2806399;
        double r2806417 = r2806394 * r2806416;
        double r2806418 = r2806391 - r2806417;
        double r2806419 = fma(r2806410, r2806415, r2806418);
        double r2806420 = r2806419 * r2806397;
        double r2806421 = r2806396 * r2806420;
        double r2806422 = sqrt(r2806421);
        double r2806423 = 2336909186103.619;
        bool r2806424 = r2806391 <= r2806423;
        double r2806425 = r2806419 * r2806398;
        double r2806426 = cbrt(r2806425);
        double r2806427 = r2806426 * r2806426;
        double r2806428 = r2806426 * r2806427;
        double r2806429 = sqrt(r2806428);
        double r2806430 = sqrt(r2806429);
        double r2806431 = sqrt(r2806425);
        double r2806432 = sqrt(r2806431);
        double r2806433 = r2806430 * r2806432;
        double r2806434 = sqrt(r2806398);
        double r2806435 = sqrt(r2806419);
        double r2806436 = r2806434 * r2806435;
        double r2806437 = r2806424 ? r2806433 : r2806436;
        double r2806438 = r2806408 ? r2806422 : r2806437;
        double r2806439 = r2806393 ? r2806406 : r2806438;
        return r2806439;
}

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 4 regimes

2. if t < -9.37971981582653e+241

   1. Initial program 41.0

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

   2. Simplified 38.7

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

   4. Applied sqr-pow 38.7

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

   5. Applied associate-*r* 38.5

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

   6. Taylor expanded around 0 40.3

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

   7. Simplified 37.9

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

   if -9.37971981582653e+241 < t < -1.9662007575597354e-176

   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. Simplified 29.7

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

   4. Applied sqr-pow 29.7

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

   5. Applied associate-*r* 29.0

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

   7. Applied associate-*l* 28.6

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

   if -1.9662007575597354e-176 < t < 2336909186103.619

   1. Initial program 34.8

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

   2. Simplified 31.6

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

   4. Applied sqr-pow 31.6

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

   5. Applied associate-*r* 30.5

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

   7. Applied add-sqr-sqrt 30.6

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

   9. Applied add-cube-cbrt 30.7

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

   if 2336909186103.619 < t

   1. Initial program 34.8

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

   2. Simplified 32.4

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

   4. Applied sqr-pow 32.4

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

   5. Applied associate-*r* 31.9

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

   7. Applied sqrt-prod 26.8

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 29.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.379719815826530589603530820932364731187 \cdot 10^{241}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right)}\\ \mathbf{elif}\;t \le -1.966200757559735377429830381935693182299 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(U* - U, \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)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 2336909186103.619140625:\\ \;\;\;\;\sqrt{\sqrt{\sqrt[3]{\mathsf{fma}\left(U* - U, \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)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \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)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \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)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(U* - U, \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)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\mathsf{fma}\left(U* - U, \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)}, t - 2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))