Toniolo and Linder, Equation (13)

Average Error: 34.7 → 31.4

Time: 34.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le 1.1077086227437005 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r217402 = 2.0;
        double r217403 = n;
        double r217404 = r217402 * r217403;
        double r217405 = U;
        double r217406 = r217404 * r217405;
        double r217407 = t;
        double r217408 = l;
        double r217409 = r217408 * r217408;
        double r217410 = Om;
        double r217411 = r217409 / r217410;
        double r217412 = r217402 * r217411;
        double r217413 = r217407 - r217412;
        double r217414 = r217408 / r217410;
        double r217415 = pow(r217414, r217402);
        double r217416 = r217403 * r217415;
        double r217417 = U_;
        double r217418 = r217405 - r217417;
        double r217419 = r217416 * r217418;
        double r217420 = r217413 - r217419;
        double r217421 = r217406 * r217420;
        double r217422 = sqrt(r217421);
        return r217422;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r217423 = n;
        double r217424 = 1.1077086227437005e-233;
        bool r217425 = r217423 <= r217424;
        double r217426 = 2.0;
        double r217427 = r217426 * r217423;
        double r217428 = U;
        double r217429 = t;
        double r217430 = l;
        double r217431 = Om;
        double r217432 = r217430 / r217431;
        double r217433 = r217430 * r217432;
        double r217434 = r217426 * r217433;
        double r217435 = r217429 - r217434;
        double r217436 = 2.0;
        double r217437 = r217426 / r217436;
        double r217438 = pow(r217432, r217437);
        double r217439 = r217423 * r217438;
        double r217440 = U_;
        double r217441 = r217428 - r217440;
        double r217442 = r217438 * r217441;
        double r217443 = r217439 * r217442;
        double r217444 = r217435 - r217443;
        double r217445 = r217428 * r217444;
        double r217446 = r217427 * r217445;
        double r217447 = sqrt(r217446);
        double r217448 = r217427 * r217428;
        double r217449 = 1.0;
        double r217450 = cbrt(r217431);
        double r217451 = r217450 * r217450;
        double r217452 = r217449 / r217451;
        double r217453 = pow(r217452, r217437);
        double r217454 = r217430 / r217450;
        double r217455 = pow(r217454, r217437);
        double r217456 = r217455 * r217441;
        double r217457 = r217453 * r217456;
        double r217458 = r217439 * r217457;
        double r217459 = r217435 - r217458;
        double r217460 = r217448 * r217459;
        double r217461 = sqrt(r217460);
        double r217462 = r217425 ? r217447 : r217461;
        return r217462;
}

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 2 regimes

2. if n < 1.1077086227437005e-233

   1. Initial program 35.8

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

   3. Applied *-un-lft-identity 35.8

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

   4. Applied times-frac 33.4

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

   5. Simplified 33.4

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

   7. Applied sqr-pow 33.4

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

   8. Applied associate-*r* 32.6

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

   10. Applied associate-*l* 32.8

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

   12. Applied associate-*l* 32.3

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

   if 1.1077086227437005e-233 < n

   1. Initial program 33.2

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

   3. Applied *-un-lft-identity 33.2

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

   4. Applied times-frac 30.5

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

   5. Simplified 30.5

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

   7. Applied sqr-pow 30.5

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

   8. Applied associate-*r* 29.7

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

   10. Applied associate-*l* 29.3

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

   12. Applied add-cube-cbrt 29.3

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

   13. Applied *-un-lft-identity 29.3

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

   14. Applied times-frac 29.3

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

   15. Applied unpow-prod-down 29.3

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

   16. Applied associate-*l* 30.1

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

4. Final simplification 31.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le 1.1077086227437005 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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*))))))