Toniolo and Linder, Equation (13)

Average Error: 33.2 → 26.4

Time: 1.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -6.903174177282536 \cdot 10^{+63}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(\left(-2 \cdot \ell - \frac{U - U*}{\frac{\frac{Om}{n}}{\ell}}\right) \cdot \frac{U}{\frac{Om}{n \cdot \ell}} + t \cdot \left(U \cdot n\right)\right)}\right|\\ \mathbf{elif}\;\ell \le -5.425968795100289 \cdot 10^{-221}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t + \left(\left(U \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t + \left(\left(U \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot 2}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r7750408 = 2.0;
        double r7750409 = n;
        double r7750410 = r7750408 * r7750409;
        double r7750411 = U;
        double r7750412 = r7750410 * r7750411;
        double r7750413 = t;
        double r7750414 = l;
        double r7750415 = r7750414 * r7750414;
        double r7750416 = Om;
        double r7750417 = r7750415 / r7750416;
        double r7750418 = r7750408 * r7750417;
        double r7750419 = r7750413 - r7750418;
        double r7750420 = r7750414 / r7750416;
        double r7750421 = pow(r7750420, r7750408);
        double r7750422 = r7750409 * r7750421;
        double r7750423 = U_;
        double r7750424 = r7750411 - r7750423;
        double r7750425 = r7750422 * r7750424;
        double r7750426 = r7750419 - r7750425;
        double r7750427 = r7750412 * r7750426;
        double r7750428 = sqrt(r7750427);
        return r7750428;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r7750429 = l;
        double r7750430 = -6.903174177282536e+63;
        bool r7750431 = r7750429 <= r7750430;
        double r7750432 = 2.0;
        double r7750433 = -2.0;
        double r7750434 = r7750433 * r7750429;
        double r7750435 = U;
        double r7750436 = U_;
        double r7750437 = r7750435 - r7750436;
        double r7750438 = Om;
        double r7750439 = n;
        double r7750440 = r7750438 / r7750439;
        double r7750441 = r7750440 / r7750429;
        double r7750442 = r7750437 / r7750441;
        double r7750443 = r7750434 - r7750442;
        double r7750444 = r7750439 * r7750429;
        double r7750445 = r7750438 / r7750444;
        double r7750446 = r7750435 / r7750445;
        double r7750447 = r7750443 * r7750446;
        double r7750448 = t;
        double r7750449 = r7750435 * r7750439;
        double r7750450 = r7750448 * r7750449;
        double r7750451 = r7750447 + r7750450;
        double r7750452 = r7750432 * r7750451;
        double r7750453 = sqrt(r7750452);
        double r7750454 = fabs(r7750453);
        double r7750455 = -5.425968795100289e-221;
        bool r7750456 = r7750429 <= r7750455;
        double r7750457 = r7750429 * r7750429;
        double r7750458 = r7750457 / r7750438;
        double r7750459 = r7750458 * r7750432;
        double r7750460 = r7750448 - r7750459;
        double r7750461 = r7750429 / r7750438;
        double r7750462 = pow(r7750461, r7750432);
        double r7750463 = r7750462 * r7750439;
        double r7750464 = r7750437 * r7750463;
        double r7750465 = r7750460 - r7750464;
        double r7750466 = r7750435 * r7750465;
        double r7750467 = r7750432 * r7750439;
        double r7750468 = r7750466 * r7750467;
        double r7750469 = sqrt(r7750468);
        double r7750470 = r7750467 * r7750435;
        double r7750471 = r7750470 * r7750448;
        double r7750472 = r7750461 * r7750439;
        double r7750473 = r7750435 * r7750472;
        double r7750474 = r7750437 * r7750472;
        double r7750475 = r7750434 - r7750474;
        double r7750476 = r7750473 * r7750475;
        double r7750477 = r7750476 * r7750432;
        double r7750478 = r7750471 + r7750477;
        double r7750479 = sqrt(r7750478);
        double r7750480 = sqrt(r7750479);
        double r7750481 = r7750480 * r7750480;
        double r7750482 = r7750456 ? r7750469 : r7750481;
        double r7750483 = r7750431 ? r7750454 : r7750482;
        return r7750483;
}

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 l < -6.903174177282536e+63

   1. Initial program 49.0

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

   3. Applied *-un-lft-identity 49.0

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

   4. Applied associate-*r* 49.0

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

   5. Simplified 39.5

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

   7. Applied sub-neg 39.5

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

   8. Applied distribute-rgt-in 39.5

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

   9. Simplified 31.2

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

   11. Applied add-sqr-sqrt 31.2

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

   12. Applied rem-sqrt-square 31.2

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

   13. Simplified 30.3

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

   15. Applied associate-/r* 30.3

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

   if -6.903174177282536e+63 < l < -5.425968795100289e-221

   1. Initial program 27.0

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

   3. Applied associate-*l* 27.0

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

   if -5.425968795100289e-221 < l

   1. Initial program 31.7

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

   3. Applied *-un-lft-identity 31.7

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

   4. Applied associate-*r* 31.7

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

   5. Simplified 28.2

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

   7. Applied sub-neg 28.2

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

   8. Applied distribute-rgt-in 28.2

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

   9. Simplified 24.9

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

   11. Applied add-sqr-sqrt 24.9

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

   12. Applied sqrt-prod 25.1

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

4. Final simplification 26.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -6.903174177282536 \cdot 10^{+63}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(\left(-2 \cdot \ell - \frac{U - U*}{\frac{\frac{Om}{n}}{\ell}}\right) \cdot \frac{U}{\frac{Om}{n \cdot \ell}} + t \cdot \left(U \cdot n\right)\right)}\right|\\ \mathbf{elif}\;\ell \le -5.425968795100289 \cdot 10^{-221}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t + \left(\left(U \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t + \left(\left(U \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(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*))))))