Toniolo and Linder, Equation (13)

Average Error: 34.6 → 27.1

Time: 1.8m

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}\;U \le -1.240246637385966641601913454268540267419 \cdot 10^{99}:\\ \;\;\;\;\sqrt{\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\left(\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) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right)\right)}} \cdot \sqrt{\sqrt{\left(\sqrt[3]{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\left(\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) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right)\right)} \cdot \sqrt[3]{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\left(\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) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right)\right)}\right) \cdot \sqrt[3]{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\left(\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) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right)\right)}}}\\ \mathbf{elif}\;U \le 1.023131932272581970653900996532248181447 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(\left(\left(\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) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right) \cdot \left(2 \cdot n\right)} \cdot \sqrt{U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3968506 = 2.0;
        double r3968507 = n;
        double r3968508 = r3968506 * r3968507;
        double r3968509 = U;
        double r3968510 = r3968508 * r3968509;
        double r3968511 = t;
        double r3968512 = l;
        double r3968513 = r3968512 * r3968512;
        double r3968514 = Om;
        double r3968515 = r3968513 / r3968514;
        double r3968516 = r3968506 * r3968515;
        double r3968517 = r3968511 - r3968516;
        double r3968518 = r3968512 / r3968514;
        double r3968519 = pow(r3968518, r3968506);
        double r3968520 = r3968507 * r3968519;
        double r3968521 = U_;
        double r3968522 = r3968509 - r3968521;
        double r3968523 = r3968520 * r3968522;
        double r3968524 = r3968517 - r3968523;
        double r3968525 = r3968510 * r3968524;
        double r3968526 = sqrt(r3968525);
        return r3968526;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3968527 = U;
        double r3968528 = -1.2402466373859666e+99;
        bool r3968529 = r3968527 <= r3968528;
        double r3968530 = 2.0;
        double r3968531 = n;
        double r3968532 = r3968530 * r3968531;
        double r3968533 = l;
        double r3968534 = Om;
        double r3968535 = r3968533 / r3968534;
        double r3968536 = 2.0;
        double r3968537 = r3968530 / r3968536;
        double r3968538 = pow(r3968535, r3968537);
        double r3968539 = r3968531 * r3968538;
        double r3968540 = r3968539 * r3968538;
        double r3968541 = U_;
        double r3968542 = r3968541 - r3968527;
        double r3968543 = r3968540 * r3968542;
        double r3968544 = r3968530 * r3968533;
        double r3968545 = r3968535 * r3968544;
        double r3968546 = r3968543 - r3968545;
        double r3968547 = t;
        double r3968548 = r3968546 + r3968547;
        double r3968549 = r3968532 * r3968548;
        double r3968550 = r3968527 * r3968549;
        double r3968551 = sqrt(r3968550);
        double r3968552 = sqrt(r3968551);
        double r3968553 = cbrt(r3968550);
        double r3968554 = r3968553 * r3968553;
        double r3968555 = r3968554 * r3968553;
        double r3968556 = sqrt(r3968555);
        double r3968557 = sqrt(r3968556);
        double r3968558 = r3968552 * r3968557;
        double r3968559 = 1.023131932272582e-302;
        bool r3968560 = r3968527 <= r3968559;
        double r3968561 = r3968548 * r3968527;
        double r3968562 = r3968561 * r3968532;
        double r3968563 = sqrt(r3968562);
        double r3968564 = pow(r3968535, r3968530);
        double r3968565 = r3968531 * r3968564;
        double r3968566 = r3968565 * r3968542;
        double r3968567 = r3968566 - r3968545;
        double r3968568 = r3968567 + r3968547;
        double r3968569 = r3968568 * r3968532;
        double r3968570 = sqrt(r3968569);
        double r3968571 = sqrt(r3968527);
        double r3968572 = r3968570 * r3968571;
        double r3968573 = r3968560 ? r3968563 : r3968572;
        double r3968574 = r3968529 ? r3968558 : r3968573;
        return r3968574;
}

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 U < -1.2402466373859666e+99

   1. Initial program 30.2

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

   2. Simplified 29.5

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

   4. Applied sqr-pow 29.5

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

   5. Applied associate-*r* 29.2

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

   7. Applied add-sqr-sqrt 29.4

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

   9. Applied add-cube-cbrt 29.5

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

   if -1.2402466373859666e+99 < U < 1.023131932272582e-302

   1. Initial program 36.5

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

   2. Simplified 33.3

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

   4. Applied sqr-pow 33.3

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

   5. Applied associate-*r* 32.1

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

   7. Applied associate-*l* 30.1

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

   if 1.023131932272582e-302 < U

   1. Initial program 34.1

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

   2. Simplified 31.2

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

   4. Applied sqrt-prod 24.2

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

4. Final simplification 27.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -1.240246637385966641601913454268540267419 \cdot 10^{99}:\\ \;\;\;\;\sqrt{\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\left(\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) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right)\right)}} \cdot \sqrt{\sqrt{\left(\sqrt[3]{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\left(\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) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right)\right)} \cdot \sqrt[3]{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\left(\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) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right)\right)}\right) \cdot \sqrt[3]{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\left(\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) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right)\right)}}}\\ \mathbf{elif}\;U \le 1.023131932272581970653900996532248181447 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(\left(\left(\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) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right) \cdot \left(2 \cdot n\right)} \cdot \sqrt{U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))