Toniolo and Linder, Equation (13)

Average Error: 34.3 → 31.5

Time: 1.7m

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 -115.2035769670583960078147356398403644562:\\ \;\;\;\;\sqrt{e^{\log \left(\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right) \cdot 2\right)}}\\ \mathbf{elif}\;n \le -1.928586768576104907029448118743871413968 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(U - U*\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le -1.991223724932492166185779454526004556741 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{\left(n \cdot n\right) \cdot U}{Om}, {\left(\frac{1}{{-1}^{2}}\right)}^{1}, \left(U \cdot n\right) \cdot t\right) \cdot 2 - 4 \cdot \left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)\right)}\\ \mathbf{elif}\;n \le 1.190270096378231867728639520214394049373 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(U - U*\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le 3.245686032965869293542566860037617070177 \cdot 10^{109}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{\left(n \cdot n\right) \cdot U}{Om}, {\left(\frac{1}{{-1}^{2}}\right)}^{1}, \left(U \cdot n\right) \cdot t\right) \cdot 2 - 4 \cdot \left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right) \cdot 2\right) \cdot \sqrt{\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right) \cdot 2}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3915033 = 2.0;
        double r3915034 = n;
        double r3915035 = r3915033 * r3915034;
        double r3915036 = U;
        double r3915037 = r3915035 * r3915036;
        double r3915038 = t;
        double r3915039 = l;
        double r3915040 = r3915039 * r3915039;
        double r3915041 = Om;
        double r3915042 = r3915040 / r3915041;
        double r3915043 = r3915033 * r3915042;
        double r3915044 = r3915038 - r3915043;
        double r3915045 = r3915039 / r3915041;
        double r3915046 = pow(r3915045, r3915033);
        double r3915047 = r3915034 * r3915046;
        double r3915048 = U_;
        double r3915049 = r3915036 - r3915048;
        double r3915050 = r3915047 * r3915049;
        double r3915051 = r3915044 - r3915050;
        double r3915052 = r3915037 * r3915051;
        double r3915053 = sqrt(r3915052);
        return r3915053;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3915054 = n;
        double r3915055 = -115.2035769670584;
        bool r3915056 = r3915054 <= r3915055;
        double r3915057 = U;
        double r3915058 = r3915057 * r3915054;
        double r3915059 = t;
        double r3915060 = l;
        double r3915061 = Om;
        double r3915062 = r3915060 / r3915061;
        double r3915063 = 2.0;
        double r3915064 = 2.0;
        double r3915065 = r3915063 / r3915064;
        double r3915066 = pow(r3915062, r3915065);
        double r3915067 = U_;
        double r3915068 = r3915057 - r3915067;
        double r3915069 = r3915066 * r3915068;
        double r3915070 = r3915066 * r3915054;
        double r3915071 = r3915060 * r3915062;
        double r3915072 = r3915071 * r3915063;
        double r3915073 = fma(r3915069, r3915070, r3915072);
        double r3915074 = r3915059 - r3915073;
        double r3915075 = r3915058 * r3915074;
        double r3915076 = r3915075 * r3915063;
        double r3915077 = log(r3915076);
        double r3915078 = exp(r3915077);
        double r3915079 = sqrt(r3915078);
        double r3915080 = -1.928586768576105e-95;
        bool r3915081 = r3915054 <= r3915080;
        double r3915082 = sqrt(r3915057);
        double r3915083 = r3915063 * r3915060;
        double r3915084 = r3915070 * r3915066;
        double r3915085 = r3915068 * r3915084;
        double r3915086 = fma(r3915062, r3915083, r3915085);
        double r3915087 = r3915059 - r3915086;
        double r3915088 = r3915054 * r3915087;
        double r3915089 = r3915063 * r3915088;
        double r3915090 = sqrt(r3915089);
        double r3915091 = r3915082 * r3915090;
        double r3915092 = -1.9912237249324922e-147;
        bool r3915093 = r3915054 <= r3915092;
        double r3915094 = r3915060 * r3915060;
        double r3915095 = r3915067 * r3915094;
        double r3915096 = r3915095 / r3915061;
        double r3915097 = r3915054 * r3915054;
        double r3915098 = r3915097 * r3915057;
        double r3915099 = r3915098 / r3915061;
        double r3915100 = r3915096 * r3915099;
        double r3915101 = 1.0;
        double r3915102 = -1.0;
        double r3915103 = pow(r3915102, r3915063);
        double r3915104 = r3915101 / r3915103;
        double r3915105 = 1.0;
        double r3915106 = pow(r3915104, r3915105);
        double r3915107 = r3915058 * r3915059;
        double r3915108 = fma(r3915100, r3915106, r3915107);
        double r3915109 = r3915108 * r3915063;
        double r3915110 = 4.0;
        double r3915111 = r3915057 / r3915061;
        double r3915112 = r3915054 * r3915060;
        double r3915113 = r3915112 * r3915060;
        double r3915114 = r3915111 * r3915113;
        double r3915115 = r3915110 * r3915114;
        double r3915116 = r3915109 - r3915115;
        double r3915117 = sqrt(r3915116);
        double r3915118 = 1.1902700963782319e-56;
        bool r3915119 = r3915054 <= r3915118;
        double r3915120 = r3915057 * r3915089;
        double r3915121 = sqrt(r3915120);
        double r3915122 = 3.2456860329658693e+109;
        bool r3915123 = r3915054 <= r3915122;
        double r3915124 = sqrt(r3915076);
        double r3915125 = r3915076 * r3915124;
        double r3915126 = cbrt(r3915125);
        double r3915127 = r3915123 ? r3915117 : r3915126;
        double r3915128 = r3915119 ? r3915121 : r3915127;
        double r3915129 = r3915093 ? r3915117 : r3915128;
        double r3915130 = r3915081 ? r3915091 : r3915129;
        double r3915131 = r3915056 ? r3915079 : r3915130;
        return r3915131;
}

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 5 regimes

2. if n < -115.2035769670584

   1. Initial program 32.2

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

   2. Simplified 36.2

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

   4. Applied sqr-pow 36.2

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

   5. Applied associate-*r* 35.1

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

   7. Applied add-exp-log 35.1

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

   8. Applied add-exp-log 50.8

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

   9. Applied add-exp-log 64.0

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

   10. Applied prod-exp 64.0

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

   11. Applied prod-exp 64.0

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

   12. Applied add-exp-log 64.0

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

   13. Applied prod-exp 64.0

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

   14. Simplified 30.1

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

   if -115.2035769670584 < n < -1.928586768576105e-95

   1. Initial program 32.3

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

   2. Simplified 26.7

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

   4. Applied sqr-pow 26.7

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

   5. Applied associate-*r* 26.5

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

   7. Applied sqrt-prod 38.5

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

   if -1.928586768576105e-95 < n < -1.9912237249324922e-147 or 1.1902700963782319e-56 < n < 3.2456860329658693e+109

   1. Initial program 29.9

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

   2. Simplified 27.0

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

   4. Applied sqr-pow 27.0

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

   5. Applied associate-*r* 26.5

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

   7. Applied *-un-lft-identity 26.5

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

   8. Applied add-sqr-sqrt 45.0

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

   9. Applied prod-diff 45.0

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

   10. Applied distribute-lft-in 45.0

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

   11. Simplified 26.1

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

   12. Taylor expanded around -inf 35.5

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(t \cdot \left(U \cdot n\right)\right) + 2 \cdot \left(\frac{U \cdot \left({n}^{2} \cdot \left(U* \cdot {\ell}^{2}\right)\right)}{{Om}^{2}} \cdot {\left(\frac{1}{{-1}^{2}}\right)}^{1}\right)\right) - 4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}}\]

   13. Simplified 32.4

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot \left(n \cdot n\right)}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot U*}{Om}, {\left(\frac{1}{{-1}^{2}}\right)}^{1}, t \cdot \left(U \cdot n\right)\right) \cdot 2 - 4 \cdot \left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)\right)}}\]

   if -1.9912237249324922e-147 < n < 1.1902700963782319e-56

   1. Initial program 37.3

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

   2. Simplified 29.8

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

   4. Applied sqr-pow 29.8

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

   5. Applied associate-*r* 28.3

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

   if 3.2456860329658693e+109 < n

   1. Initial program 35.2

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

   2. Simplified 40.6

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

   4. Applied sqr-pow 40.6

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

   5. Applied associate-*r* 39.8

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

   7. Applied add-cbrt-cube 44.9

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

   8. Simplified 40.1

      \[\leadsto \sqrt[3]{\color{blue}{\left(\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)\right)\right) \cdot 2\right) \cdot \sqrt{\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)\right)\right) \cdot 2}}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 31.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -115.2035769670583960078147356398403644562:\\ \;\;\;\;\sqrt{e^{\log \left(\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right) \cdot 2\right)}}\\ \mathbf{elif}\;n \le -1.928586768576104907029448118743871413968 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(U - U*\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le -1.991223724932492166185779454526004556741 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{\left(n \cdot n\right) \cdot U}{Om}, {\left(\frac{1}{{-1}^{2}}\right)}^{1}, \left(U \cdot n\right) \cdot t\right) \cdot 2 - 4 \cdot \left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)\right)}\\ \mathbf{elif}\;n \le 1.190270096378231867728639520214394049373 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(U - U*\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le 3.245686032965869293542566860037617070177 \cdot 10^{109}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{\left(n \cdot n\right) \cdot U}{Om}, {\left(\frac{1}{{-1}^{2}}\right)}^{1}, \left(U \cdot n\right) \cdot t\right) \cdot 2 - 4 \cdot \left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right) \cdot 2\right) \cdot \sqrt{\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right) \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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*))))))