Toniolo and Linder, Equation (13)

Average Error: 34.7 → 30.1

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 4.69377638721043389061730150231150526438 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(\left(\sqrt[3]{n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt[3]{n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{elif}\;t \le 2.559404422323993207029950230040666238883 \cdot 10^{-58} \lor \neg \left(t \le 5.263875707831979755454541422910156991261 \cdot 10^{143}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r180324 = 2.0;
        double r180325 = n;
        double r180326 = r180324 * r180325;
        double r180327 = U;
        double r180328 = r180326 * r180327;
        double r180329 = t;
        double r180330 = l;
        double r180331 = r180330 * r180330;
        double r180332 = Om;
        double r180333 = r180331 / r180332;
        double r180334 = r180324 * r180333;
        double r180335 = r180329 - r180334;
        double r180336 = r180330 / r180332;
        double r180337 = pow(r180336, r180324);
        double r180338 = r180325 * r180337;
        double r180339 = U_;
        double r180340 = r180327 - r180339;
        double r180341 = r180338 * r180340;
        double r180342 = r180335 - r180341;
        double r180343 = r180328 * r180342;
        double r180344 = sqrt(r180343);
        return r180344;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r180345 = t;
        double r180346 = 4.693776387210434e-278;
        bool r180347 = r180345 <= r180346;
        double r180348 = 2.0;
        double r180349 = n;
        double r180350 = r180348 * r180349;
        double r180351 = U;
        double r180352 = r180350 * r180351;
        double r180353 = l;
        double r180354 = Om;
        double r180355 = r180353 / r180354;
        double r180356 = r180353 * r180355;
        double r180357 = r180348 * r180356;
        double r180358 = r180345 - r180357;
        double r180359 = 2.0;
        double r180360 = r180348 / r180359;
        double r180361 = pow(r180355, r180360);
        double r180362 = r180349 * r180361;
        double r180363 = cbrt(r180362);
        double r180364 = r180363 * r180363;
        double r180365 = r180364 * r180363;
        double r180366 = U_;
        double r180367 = r180351 - r180366;
        double r180368 = r180367 * r180361;
        double r180369 = r180365 * r180368;
        double r180370 = r180358 - r180369;
        double r180371 = r180352 * r180370;
        double r180372 = sqrt(r180371);
        double r180373 = 2.5594044223239932e-58;
        bool r180374 = r180345 <= r180373;
        double r180375 = 5.26387570783198e+143;
        bool r180376 = r180345 <= r180375;
        double r180377 = !r180376;
        bool r180378 = r180374 || r180377;
        double r180379 = sqrt(r180352);
        double r180380 = r180348 * r180353;
        double r180381 = r180359 * r180360;
        double r180382 = pow(r180355, r180381);
        double r180383 = r180349 * r180382;
        double r180384 = r180367 * r180383;
        double r180385 = fma(r180380, r180355, r180384);
        double r180386 = r180345 - r180385;
        double r180387 = sqrt(r180386);
        double r180388 = r180379 * r180387;
        double r180389 = r180349 * r180353;
        double r180390 = 1.0;
        double r180391 = 1.0;
        double r180392 = pow(r180354, r180391);
        double r180393 = r180390 / r180392;
        double r180394 = pow(r180393, r180391);
        double r180395 = r180389 * r180394;
        double r180396 = r180395 * r180368;
        double r180397 = r180358 - r180396;
        double r180398 = r180352 * r180397;
        double r180399 = sqrt(r180398);
        double r180400 = r180378 ? r180388 : r180399;
        double r180401 = r180347 ? r180372 : r180400;
        return r180401;
}

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 < 4.693776387210434e-278

   1. Initial program 34.6

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

   3. Applied *-un-lft-identity 34.6

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

   4. Applied times-frac 31.7

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

   5. Simplified 31.7

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

   7. Applied sqr-pow 31.7

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

   8. Applied associate-*r* 30.8

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

   10. Applied associate-*l* 30.4

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

   11. Simplified 30.4

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

   13. Applied add-cube-cbrt 30.5

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

   if 4.693776387210434e-278 < t < 2.5594044223239932e-58 or 5.26387570783198e+143 < t

   1. Initial program 37.1

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

   3. Applied *-un-lft-identity 37.1

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

   4. Applied times-frac 34.7

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

   5. Simplified 34.7

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

   7. Applied sqr-pow 34.7

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

   8. Applied associate-*r* 33.9

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

   10. Applied sqrt-prod 29.6

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

   11. Simplified 30.1

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

   if 2.5594044223239932e-58 < t < 5.26387570783198e+143

   1. Initial program 30.7

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

   3. Applied *-un-lft-identity 30.7

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

   4. Applied times-frac 28.5

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

   5. Simplified 28.5

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

   7. Applied sqr-pow 28.5

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

   8. Applied associate-*r* 28.2

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

   10. Applied associate-*l* 28.2

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

   11. Simplified 28.2

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

   12. Taylor expanded around 0 28.7

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

4. Final simplification 30.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 4.69377638721043389061730150231150526438 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(\left(\sqrt[3]{n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt[3]{n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{elif}\;t \le 2.559404422323993207029950230040666238883 \cdot 10^{-58} \lor \neg \left(t \le 5.263875707831979755454541422910156991261 \cdot 10^{143}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))