Toniolo and Linder, Equation (13)

Average Error: 34.3 → 28.2

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -5.147845724912201976044071050075956463843 \cdot 10^{-5} \lor \neg \left(n \le 12422347331609260556550144\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, 2, \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U + U \cdot \left(\left(\left(\left(-\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right) + \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot n\right) \cdot 2\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r181096 = 2.0;
        double r181097 = n;
        double r181098 = r181096 * r181097;
        double r181099 = U;
        double r181100 = r181098 * r181099;
        double r181101 = t;
        double r181102 = l;
        double r181103 = r181102 * r181102;
        double r181104 = Om;
        double r181105 = r181103 / r181104;
        double r181106 = r181096 * r181105;
        double r181107 = r181101 - r181106;
        double r181108 = r181102 / r181104;
        double r181109 = pow(r181108, r181096);
        double r181110 = r181097 * r181109;
        double r181111 = U_;
        double r181112 = r181099 - r181111;
        double r181113 = r181110 * r181112;
        double r181114 = r181107 - r181113;
        double r181115 = r181100 * r181114;
        double r181116 = sqrt(r181115);
        return r181116;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r181117 = n;
        double r181118 = -5.147845724912202e-05;
        bool r181119 = r181117 <= r181118;
        double r181120 = 1.242234733160926e+25;
        bool r181121 = r181117 <= r181120;
        double r181122 = !r181121;
        bool r181123 = r181119 || r181122;
        double r181124 = 2.0;
        double r181125 = r181124 * r181117;
        double r181126 = U;
        double r181127 = r181125 * r181126;
        double r181128 = t;
        double r181129 = l;
        double r181130 = Om;
        double r181131 = r181130 / r181129;
        double r181132 = r181129 / r181131;
        double r181133 = r181124 * r181132;
        double r181134 = r181128 - r181133;
        double r181135 = r181129 / r181130;
        double r181136 = 2.0;
        double r181137 = r181124 / r181136;
        double r181138 = pow(r181135, r181137);
        double r181139 = r181117 * r181138;
        double r181140 = U_;
        double r181141 = r181126 - r181140;
        double r181142 = r181141 * r181138;
        double r181143 = r181139 * r181142;
        double r181144 = r181134 - r181143;
        double r181145 = r181127 * r181144;
        double r181146 = sqrt(r181145);
        double r181147 = r181136 * r181137;
        double r181148 = pow(r181135, r181147);
        double r181149 = r181148 * r181117;
        double r181150 = r181149 * r181141;
        double r181151 = fma(r181132, r181124, r181150);
        double r181152 = r181128 - r181151;
        double r181153 = r181152 * r181125;
        double r181154 = r181153 * r181126;
        double r181155 = -r181150;
        double r181156 = r181155 + r181150;
        double r181157 = r181156 * r181117;
        double r181158 = r181157 * r181124;
        double r181159 = r181126 * r181158;
        double r181160 = r181154 + r181159;
        double r181161 = sqrt(r181160);
        double r181162 = r181123 ? r181146 : r181161;
        return r181162;
}

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 < -5.147845724912202e-05 or 1.242234733160926e+25 < 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 associate-/l* 31.0

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

   5. Applied sqr-pow 31.0

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

   6. Applied associate-*r* 30.0

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

   8. Applied associate-*l* 29.1

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

   9. Simplified 29.1

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

   if -5.147845724912202e-05 < n < 1.242234733160926e+25

   1. Initial program 35.0

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

   3. Applied associate-/l* 31.5

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

   5. Applied sqr-pow 31.5

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

   6. Applied associate-*r* 30.8

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

   8. Applied associate-*l* 31.3

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

   9. Simplified 31.3

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

   11. Applied add-sqr-sqrt 47.6

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

   12. Applied prod-diff 47.6

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

   13. Applied distribute-lft-in 47.6

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

   14. Simplified 28.7

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

   15. Simplified 27.7

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

4. Final simplification 28.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5.147845724912201976044071050075956463843 \cdot 10^{-5} \lor \neg \left(n \le 12422347331609260556550144\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, 2, \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U + U \cdot \left(\left(\left(\left(-\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right) + \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot n\right) \cdot 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(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*))))))