Toniolo and Linder, Equation (13)

Average Error: 33.2 → 25.9

Time: 47.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le 2.9259008121443 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(\left(t - \left(\ell \cdot 2 - \sqrt[3]{\left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{Om} \cdot n\right)\right)} \cdot \left(\sqrt[3]{\left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{Om} \cdot n\right)\right)} \cdot \sqrt[3]{\left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{Om} \cdot n\right)\right)}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot n\right) \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{Om}\right)\right)\right) \cdot 2\right)} \cdot \sqrt{U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1908632 = 2.0;
        double r1908633 = n;
        double r1908634 = r1908632 * r1908633;
        double r1908635 = U;
        double r1908636 = r1908634 * r1908635;
        double r1908637 = t;
        double r1908638 = l;
        double r1908639 = r1908638 * r1908638;
        double r1908640 = Om;
        double r1908641 = r1908639 / r1908640;
        double r1908642 = r1908632 * r1908641;
        double r1908643 = r1908637 - r1908642;
        double r1908644 = r1908638 / r1908640;
        double r1908645 = pow(r1908644, r1908632);
        double r1908646 = r1908633 * r1908645;
        double r1908647 = U_;
        double r1908648 = r1908635 - r1908647;
        double r1908649 = r1908646 * r1908648;
        double r1908650 = r1908643 - r1908649;
        double r1908651 = r1908636 * r1908650;
        double r1908652 = sqrt(r1908651);
        return r1908652;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1908653 = U;
        double r1908654 = 2.9259008121443e-311;
        bool r1908655 = r1908653 <= r1908654;
        double r1908656 = t;
        double r1908657 = l;
        double r1908658 = 2.0;
        double r1908659 = r1908657 * r1908658;
        double r1908660 = U_;
        double r1908661 = r1908660 - r1908653;
        double r1908662 = cbrt(r1908657);
        double r1908663 = r1908662 * r1908662;
        double r1908664 = Om;
        double r1908665 = r1908662 / r1908664;
        double r1908666 = n;
        double r1908667 = r1908665 * r1908666;
        double r1908668 = r1908663 * r1908667;
        double r1908669 = r1908661 * r1908668;
        double r1908670 = cbrt(r1908669);
        double r1908671 = r1908670 * r1908670;
        double r1908672 = r1908670 * r1908671;
        double r1908673 = r1908659 - r1908672;
        double r1908674 = r1908657 / r1908664;
        double r1908675 = r1908673 * r1908674;
        double r1908676 = r1908656 - r1908675;
        double r1908677 = r1908676 * r1908658;
        double r1908678 = r1908677 * r1908666;
        double r1908679 = r1908653 * r1908678;
        double r1908680 = sqrt(r1908679);
        double r1908681 = r1908662 * r1908666;
        double r1908682 = r1908663 / r1908664;
        double r1908683 = r1908681 * r1908682;
        double r1908684 = r1908661 * r1908683;
        double r1908685 = r1908659 - r1908684;
        double r1908686 = r1908674 * r1908685;
        double r1908687 = r1908656 - r1908686;
        double r1908688 = r1908687 * r1908658;
        double r1908689 = r1908666 * r1908688;
        double r1908690 = sqrt(r1908689);
        double r1908691 = sqrt(r1908653);
        double r1908692 = r1908690 * r1908691;
        double r1908693 = r1908655 ? r1908680 : r1908692;
        return r1908693;
}

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 U < 2.9259008121443e-311

   1. Initial program 33.5

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

   2. Simplified 30.0

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

   4. Applied div-inv 30.0

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

   5. Applied add-cube-cbrt 30.0

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

   6. Applied times-frac 29.5

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

   7. Simplified 29.5

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

   9. Applied associate-*l* 29.6

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

   11. Applied div-inv 29.6

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

   12. Applied associate-*l* 29.7

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

   13. Simplified 29.6

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

   15. Applied add-cube-cbrt 29.6

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

   if 2.9259008121443e-311 < U

   1. Initial program 33.0

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

   2. Simplified 30.1

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

   4. Applied div-inv 30.1

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

   5. Applied add-cube-cbrt 30.1

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

   6. Applied times-frac 29.7

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

   7. Simplified 29.7

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

   9. Applied associate-*l* 29.1

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

   11. Applied sqrt-prod 22.2

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

4. Final simplification 25.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le 2.9259008121443 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(\left(t - \left(\ell \cdot 2 - \sqrt[3]{\left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{Om} \cdot n\right)\right)} \cdot \left(\sqrt[3]{\left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{Om} \cdot n\right)\right)} \cdot \sqrt[3]{\left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{Om} \cdot n\right)\right)}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot n\right) \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{Om}\right)\right)\right) \cdot 2\right)} \cdot \sqrt{U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(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*))))))