Toniolo and Linder, Equation (13)

Average Error: 33.4 → 28.7

Time: 42.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 1.065763946703352 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 7.97946278428516 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;t \le 8.755812112715264 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3018358 = 2.0;
        double r3018359 = n;
        double r3018360 = r3018358 * r3018359;
        double r3018361 = U;
        double r3018362 = r3018360 * r3018361;
        double r3018363 = t;
        double r3018364 = l;
        double r3018365 = r3018364 * r3018364;
        double r3018366 = Om;
        double r3018367 = r3018365 / r3018366;
        double r3018368 = r3018358 * r3018367;
        double r3018369 = r3018363 - r3018368;
        double r3018370 = r3018364 / r3018366;
        double r3018371 = pow(r3018370, r3018358);
        double r3018372 = r3018359 * r3018371;
        double r3018373 = U_;
        double r3018374 = r3018361 - r3018373;
        double r3018375 = r3018372 * r3018374;
        double r3018376 = r3018369 - r3018375;
        double r3018377 = r3018362 * r3018376;
        double r3018378 = sqrt(r3018377);
        return r3018378;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3018379 = t;
        double r3018380 = 1.065763946703352e-302;
        bool r3018381 = r3018379 <= r3018380;
        double r3018382 = 2.0;
        double r3018383 = l;
        double r3018384 = Om;
        double r3018385 = r3018383 / r3018384;
        double r3018386 = r3018383 * r3018385;
        double r3018387 = r3018382 * r3018386;
        double r3018388 = r3018379 - r3018387;
        double r3018389 = n;
        double r3018390 = r3018389 * r3018385;
        double r3018391 = U;
        double r3018392 = U_;
        double r3018393 = r3018391 - r3018392;
        double r3018394 = r3018393 * r3018385;
        double r3018395 = r3018390 * r3018394;
        double r3018396 = r3018388 - r3018395;
        double r3018397 = r3018382 * r3018389;
        double r3018398 = r3018397 * r3018391;
        double r3018399 = r3018396 * r3018398;
        double r3018400 = sqrt(r3018399);
        double r3018401 = 7.97946278428516e-27;
        bool r3018402 = r3018379 <= r3018401;
        double r3018403 = r3018383 * r3018382;
        double r3018404 = r3018385 * r3018385;
        double r3018405 = r3018389 * r3018404;
        double r3018406 = r3018393 * r3018405;
        double r3018407 = fma(r3018403, r3018385, r3018406);
        double r3018408 = r3018379 - r3018407;
        double r3018409 = sqrt(r3018408);
        double r3018410 = sqrt(r3018398);
        double r3018411 = r3018409 * r3018410;
        double r3018412 = 8.755812112715264e+71;
        bool r3018413 = r3018379 <= r3018412;
        double r3018414 = r3018391 * r3018408;
        double r3018415 = r3018414 * r3018397;
        double r3018416 = sqrt(r3018415);
        double r3018417 = r3018413 ? r3018416 : r3018411;
        double r3018418 = r3018402 ? r3018411 : r3018417;
        double r3018419 = r3018381 ? r3018400 : r3018418;
        return r3018419;
}

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 < 1.065763946703352e-302

   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 associate-*l* 30.3

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

   if 1.065763946703352e-302 < 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 sqrt-prod 27.2

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

   11. Simplified 27.8

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

   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 28.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.065763946703352 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 7.97946278428516 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;t \le 8.755812112715264 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

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*))))))