Toniolo and Linder, Equation (13)

Average Error: 33.3 → 26.7

Time: 52.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -7.524653843956402 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\sqrt{\left(\sqrt[3]{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)} \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)}}} \cdot \sqrt{\sqrt{\left(\sqrt[3]{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)} \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)}}}\\ \mathbf{elif}\;U \le 1.09532300335197 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(2 \cdot U\right) \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1771561 = 2.0;
        double r1771562 = n;
        double r1771563 = r1771561 * r1771562;
        double r1771564 = U;
        double r1771565 = r1771563 * r1771564;
        double r1771566 = t;
        double r1771567 = l;
        double r1771568 = r1771567 * r1771567;
        double r1771569 = Om;
        double r1771570 = r1771568 / r1771569;
        double r1771571 = r1771561 * r1771570;
        double r1771572 = r1771566 - r1771571;
        double r1771573 = r1771567 / r1771569;
        double r1771574 = pow(r1771573, r1771561);
        double r1771575 = r1771562 * r1771574;
        double r1771576 = U_;
        double r1771577 = r1771564 - r1771576;
        double r1771578 = r1771575 * r1771577;
        double r1771579 = r1771572 - r1771578;
        double r1771580 = r1771565 * r1771579;
        double r1771581 = sqrt(r1771580);
        return r1771581;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1771582 = U;
        double r1771583 = -7.524653843956402e+68;
        bool r1771584 = r1771582 <= r1771583;
        double r1771585 = 2.0;
        double r1771586 = r1771585 * r1771582;
        double r1771587 = n;
        double r1771588 = t;
        double r1771589 = Om;
        double r1771590 = l;
        double r1771591 = r1771589 / r1771590;
        double r1771592 = r1771591 * r1771591;
        double r1771593 = r1771587 / r1771592;
        double r1771594 = U_;
        double r1771595 = r1771582 - r1771594;
        double r1771596 = r1771593 * r1771595;
        double r1771597 = r1771588 - r1771596;
        double r1771598 = -2.0;
        double r1771599 = r1771598 / r1771591;
        double r1771600 = r1771590 * r1771599;
        double r1771601 = r1771597 + r1771600;
        double r1771602 = r1771587 * r1771601;
        double r1771603 = r1771586 * r1771602;
        double r1771604 = cbrt(r1771603);
        double r1771605 = r1771604 * r1771604;
        double r1771606 = r1771605 * r1771604;
        double r1771607 = sqrt(r1771606);
        double r1771608 = sqrt(r1771607);
        double r1771609 = r1771608 * r1771608;
        double r1771610 = 1.09532300335197e-309;
        bool r1771611 = r1771582 <= r1771610;
        double r1771612 = r1771586 * r1771601;
        double r1771613 = r1771587 * r1771612;
        double r1771614 = sqrt(r1771613);
        double r1771615 = sqrt(r1771582);
        double r1771616 = r1771585 * r1771587;
        double r1771617 = r1771616 * r1771601;
        double r1771618 = sqrt(r1771617);
        double r1771619 = r1771615 * r1771618;
        double r1771620 = r1771611 ? r1771614 : r1771619;
        double r1771621 = r1771584 ? r1771609 : r1771620;
        return r1771621;
}

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 < -7.524653843956402e+68

   1. Initial program 27.8

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

   2. Simplified 30.6

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

   4. Applied *-un-lft-identity 30.6

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

   5. Applied associate-*l* 30.6

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

   6. Simplified 28.1

      \[\leadsto \sqrt{1 \cdot \color{blue}{\left(\left(U \cdot 2\right) \cdot \left(\left(\frac{-2}{\frac{Om}{\ell}} \cdot \ell + \left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot n\right)\right)}}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 28.4

      \[\leadsto \sqrt{1 \cdot \color{blue}{\left(\left(\sqrt[3]{\left(U \cdot 2\right) \cdot \left(\left(\frac{-2}{\frac{Om}{\ell}} \cdot \ell + \left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot n\right)} \cdot \sqrt[3]{\left(U \cdot 2\right) \cdot \left(\left(\frac{-2}{\frac{Om}{\ell}} \cdot \ell + \left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot n\right)}\right) \cdot \sqrt[3]{\left(U \cdot 2\right) \cdot \left(\left(\frac{-2}{\frac{Om}{\ell}} \cdot \ell + \left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot n\right)}\right)}}\]
   9. Using strategy rm

   10. Applied add-sqr-sqrt 28.4

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

   if -7.524653843956402e+68 < U < 1.09532300335197e-309

   1. Initial program 35.5

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

   2. Simplified 32.9

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

   4. Applied *-un-lft-identity 32.9

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

   5. Applied associate-*l* 32.9

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

   6. Simplified 31.5

      \[\leadsto \sqrt{1 \cdot \color{blue}{\left(\left(U \cdot 2\right) \cdot \left(\left(\frac{-2}{\frac{Om}{\ell}} \cdot \ell + \left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot n\right)\right)}}\]
   7. Using strategy rm

   8. Applied associate-*r* 29.8

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

   if 1.09532300335197e-309 < U

   1. Initial program 33.1

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

   2. Simplified 32.4

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

   4. Applied sqrt-prod 25.6

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

   5. Simplified 23.9

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

4. Final simplification 26.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -7.524653843956402 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\sqrt{\left(\sqrt[3]{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)} \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)}}} \cdot \sqrt{\sqrt{\left(\sqrt[3]{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)} \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)}}}\\ \mathbf{elif}\;U \le 1.09532300335197 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(2 \cdot U\right) \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \ell \cdot \frac{-2}{\frac{Om}{\ell}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(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*))))))