Toniolo and Linder, Equation (13)

Average Error: 33.2 → 26.4

Time: 1.0m

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 r7185335 = 2.0;
        double r7185336 = n;
        double r7185337 = r7185335 * r7185336;
        double r7185338 = U;
        double r7185339 = r7185337 * r7185338;
        double r7185340 = t;
        double r7185341 = l;
        double r7185342 = r7185341 * r7185341;
        double r7185343 = Om;
        double r7185344 = r7185342 / r7185343;
        double r7185345 = r7185335 * r7185344;
        double r7185346 = r7185340 - r7185345;
        double r7185347 = r7185341 / r7185343;
        double r7185348 = pow(r7185347, r7185335);
        double r7185349 = r7185336 * r7185348;
        double r7185350 = U_;
        double r7185351 = r7185338 - r7185350;
        double r7185352 = r7185349 * r7185351;
        double r7185353 = r7185346 - r7185352;
        double r7185354 = r7185339 * r7185353;
        double r7185355 = sqrt(r7185354);
        return r7185355;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r7185356 = l;
        double r7185357 = -6.903174177282536e+63;
        bool r7185358 = r7185356 <= r7185357;
        double r7185359 = 2.0;
        double r7185360 = -2.0;
        double r7185361 = r7185360 * r7185356;
        double r7185362 = U;
        double r7185363 = U_;
        double r7185364 = r7185362 - r7185363;
        double r7185365 = Om;
        double r7185366 = n;
        double r7185367 = r7185365 / r7185366;
        double r7185368 = r7185367 / r7185356;
        double r7185369 = r7185364 / r7185368;
        double r7185370 = r7185361 - r7185369;
        double r7185371 = r7185366 * r7185356;
        double r7185372 = r7185365 / r7185371;
        double r7185373 = r7185362 / r7185372;
        double r7185374 = r7185370 * r7185373;
        double r7185375 = t;
        double r7185376 = r7185362 * r7185366;
        double r7185377 = r7185375 * r7185376;
        double r7185378 = r7185374 + r7185377;
        double r7185379 = r7185359 * r7185378;
        double r7185380 = sqrt(r7185379);
        double r7185381 = fabs(r7185380);
        double r7185382 = -5.425968795100289e-221;
        bool r7185383 = r7185356 <= r7185382;
        double r7185384 = r7185356 * r7185356;
        double r7185385 = r7185384 / r7185365;
        double r7185386 = r7185385 * r7185359;
        double r7185387 = r7185375 - r7185386;
        double r7185388 = r7185356 / r7185365;
        double r7185389 = pow(r7185388, r7185359);
        double r7185390 = r7185389 * r7185366;
        double r7185391 = r7185364 * r7185390;
        double r7185392 = r7185387 - r7185391;
        double r7185393 = r7185362 * r7185392;
        double r7185394 = r7185359 * r7185366;
        double r7185395 = r7185393 * r7185394;
        double r7185396 = sqrt(r7185395);
        double r7185397 = r7185394 * r7185362;
        double r7185398 = r7185397 * r7185375;
        double r7185399 = r7185388 * r7185366;
        double r7185400 = r7185362 * r7185399;
        double r7185401 = r7185364 * r7185399;
        double r7185402 = r7185361 - r7185401;
        double r7185403 = r7185400 * r7185402;
        double r7185404 = r7185403 * r7185359;
        double r7185405 = r7185398 + r7185404;
        double r7185406 = sqrt(r7185405);
        double r7185407 = sqrt(r7185406);
        double r7185408 = r7185407 * r7185407;
        double r7185409 = r7185383 ? r7185396 : r7185408;
        double r7185410 = r7185358 ? r7185381 : r7185409;
        return r7185410;
}

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 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*))))))