Toniolo and Linder, Equation (13)

Average Error: 34.4 → 27.3

Time: 3.0m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -5.889660218948970277661379651605054643937 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\left(t - \mathsf{fma}\left(\frac{\ell}{Om} \cdot 2, \ell, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(\left(U - U*\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3502596 = 2.0;
        double r3502597 = n;
        double r3502598 = r3502596 * r3502597;
        double r3502599 = U;
        double r3502600 = r3502598 * r3502599;
        double r3502601 = t;
        double r3502602 = l;
        double r3502603 = r3502602 * r3502602;
        double r3502604 = Om;
        double r3502605 = r3502603 / r3502604;
        double r3502606 = r3502596 * r3502605;
        double r3502607 = r3502601 - r3502606;
        double r3502608 = r3502602 / r3502604;
        double r3502609 = pow(r3502608, r3502596);
        double r3502610 = r3502597 * r3502609;
        double r3502611 = U_;
        double r3502612 = r3502599 - r3502611;
        double r3502613 = r3502610 * r3502612;
        double r3502614 = r3502607 - r3502613;
        double r3502615 = r3502600 * r3502614;
        double r3502616 = sqrt(r3502615);
        return r3502616;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3502617 = U;
        double r3502618 = -5.889660218949e-311;
        bool r3502619 = r3502617 <= r3502618;
        double r3502620 = n;
        double r3502621 = r3502620 * r3502617;
        double r3502622 = t;
        double r3502623 = l;
        double r3502624 = Om;
        double r3502625 = r3502623 / r3502624;
        double r3502626 = 2.0;
        double r3502627 = r3502625 * r3502626;
        double r3502628 = 2.0;
        double r3502629 = r3502626 / r3502628;
        double r3502630 = pow(r3502625, r3502629);
        double r3502631 = r3502620 * r3502630;
        double r3502632 = U_;
        double r3502633 = r3502617 - r3502632;
        double r3502634 = cbrt(r3502623);
        double r3502635 = cbrt(r3502624);
        double r3502636 = r3502634 / r3502635;
        double r3502637 = pow(r3502636, r3502629);
        double r3502638 = r3502633 * r3502637;
        double r3502639 = r3502636 * r3502636;
        double r3502640 = pow(r3502639, r3502629);
        double r3502641 = r3502638 * r3502640;
        double r3502642 = r3502631 * r3502641;
        double r3502643 = fma(r3502627, r3502623, r3502642);
        double r3502644 = r3502622 - r3502643;
        double r3502645 = r3502644 * r3502626;
        double r3502646 = r3502621 * r3502645;
        double r3502647 = sqrt(r3502646);
        double r3502648 = r3502626 * r3502623;
        double r3502649 = r3502630 * r3502633;
        double r3502650 = r3502649 * r3502631;
        double r3502651 = fma(r3502625, r3502648, r3502650);
        double r3502652 = r3502622 - r3502651;
        double r3502653 = r3502652 * r3502620;
        double r3502654 = r3502653 * r3502626;
        double r3502655 = sqrt(r3502654);
        double r3502656 = sqrt(r3502617);
        double r3502657 = r3502655 * r3502656;
        double r3502658 = r3502619 ? r3502647 : r3502657;
        return r3502658;
}

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 < -5.889660218949e-311

   1. Initial program 34.8

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

   2. Simplified 31.9

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

   4. Applied sqr-pow 31.9

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

   5. Applied associate-*r* 30.8

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

   7. Applied associate-*l* 31.1

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

   9. Applied add-cube-cbrt 31.2

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

   10. Applied add-cube-cbrt 31.2

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

   11. Applied times-frac 31.2

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

   12. Applied unpow-prod-down 31.2

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

   13. Applied associate-*l* 31.2

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

   15. Applied *-un-lft-identity 31.2

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

   16. Applied associate-*l* 31.2

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

   17. Simplified 31.0

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

   if -5.889660218949e-311 < U

   1. Initial program 34.0

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

   2. Simplified 31.6

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

   4. Applied sqr-pow 31.6

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

   5. Applied associate-*r* 30.7

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

   7. Applied associate-*l* 30.8

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

   9. Applied sqrt-prod 23.6

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

4. Final simplification 27.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -5.889660218948970277661379651605054643937 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\left(t - \mathsf{fma}\left(\frac{\ell}{Om} \cdot 2, \ell, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(\left(U - U*\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(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*))))))