Toniolo and Linder, Equation (13)

Average Error: 33.4 → 28.4

Time: 45.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 9.930109553502569 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 7.97946278428516 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;t \le 8.755812112715264 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2167367 = 2.0;
        double r2167368 = n;
        double r2167369 = r2167367 * r2167368;
        double r2167370 = U;
        double r2167371 = r2167369 * r2167370;
        double r2167372 = t;
        double r2167373 = l;
        double r2167374 = r2167373 * r2167373;
        double r2167375 = Om;
        double r2167376 = r2167374 / r2167375;
        double r2167377 = r2167367 * r2167376;
        double r2167378 = r2167372 - r2167377;
        double r2167379 = r2167373 / r2167375;
        double r2167380 = pow(r2167379, r2167367);
        double r2167381 = r2167368 * r2167380;
        double r2167382 = U_;
        double r2167383 = r2167370 - r2167382;
        double r2167384 = r2167381 * r2167383;
        double r2167385 = r2167378 - r2167384;
        double r2167386 = r2167371 * r2167385;
        double r2167387 = sqrt(r2167386);
        return r2167387;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2167388 = t;
        double r2167389 = 9.930109553502569e-303;
        bool r2167390 = r2167388 <= r2167389;
        double r2167391 = 2.0;
        double r2167392 = l;
        double r2167393 = Om;
        double r2167394 = r2167392 / r2167393;
        double r2167395 = r2167392 * r2167394;
        double r2167396 = r2167391 * r2167395;
        double r2167397 = r2167388 - r2167396;
        double r2167398 = n;
        double r2167399 = r2167398 * r2167394;
        double r2167400 = U;
        double r2167401 = U_;
        double r2167402 = r2167400 - r2167401;
        double r2167403 = r2167402 * r2167394;
        double r2167404 = r2167399 * r2167403;
        double r2167405 = r2167397 - r2167404;
        double r2167406 = r2167391 * r2167398;
        double r2167407 = r2167406 * r2167400;
        double r2167408 = r2167405 * r2167407;
        double r2167409 = sqrt(r2167408);
        double r2167410 = 7.97946278428516e-27;
        bool r2167411 = r2167388 <= r2167410;
        double r2167412 = fma(r2167395, r2167391, r2167404);
        double r2167413 = r2167388 - r2167412;
        double r2167414 = sqrt(r2167413);
        double r2167415 = sqrt(r2167407);
        double r2167416 = r2167414 * r2167415;
        double r2167417 = 8.755812112715264e+71;
        bool r2167418 = r2167388 <= r2167417;
        double r2167419 = r2167392 * r2167391;
        double r2167420 = r2167394 * r2167394;
        double r2167421 = r2167420 * r2167398;
        double r2167422 = r2167421 * r2167402;
        double r2167423 = fma(r2167419, r2167394, r2167422);
        double r2167424 = r2167388 - r2167423;
        double r2167425 = r2167424 * r2167400;
        double r2167426 = r2167406 * r2167425;
        double r2167427 = sqrt(r2167426);
        double r2167428 = r2167418 ? r2167427 : r2167416;
        double r2167429 = r2167411 ? r2167416 : r2167428;
        double r2167430 = r2167390 ? r2167409 : r2167429;
        return r2167430;
}

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 < 9.930109553502569e-303

   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. Using strategy rm

   3. Applied *-un-lft-identity 33.7

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

   4. Applied times-frac 31.5

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

   5. Simplified 31.5

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

   7. Applied unpow2 31.5

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

   8. Applied associate-*r* 30.5

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

   10. Applied *-commutative 30.5

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

   12. Applied associate-*l* 30.3

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

   if 9.930109553502569e-303 < t < 7.97946278428516e-27 or 8.755812112715264e+71 < t

   1. Initial program 34.3

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

   3. Applied *-un-lft-identity 34.3

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

   4. Applied times-frac 31.6

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

   5. Simplified 31.6

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

   7. Applied unpow2 31.6

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

   8. Applied associate-*r* 30.7

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

   10. Applied *-commutative 30.7

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

   12. Applied sqrt-prod 27.2

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

   13. Simplified 27.0

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

   if 7.97946278428516e-27 < t < 8.755812112715264e+71

   1. Initial program 26.7

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

   3. Applied *-un-lft-identity 26.7

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

   4. Applied times-frac 24.3

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

   5. Simplified 24.3

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

   7. Applied unpow2 24.3

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

   8. Applied associate-*r* 23.7

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

   10. Applied associate-*l* 22.5

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

   11. Simplified 23.4

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

4. Final simplification 28.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 9.930109553502569 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 7.97946278428516 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;t \le 8.755812112715264 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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*))))))