Toniolo and Linder, Equation (13)

Average Error: 33.2 → 28.7

Time: 44.1s

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}\;t \le -9.628340007082016 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\right)} \cdot \left(\left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}\\ \mathbf{elif}\;t \le -3.2377717899062056 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{\left(\left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n}\right) \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n}\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{elif}\;t \le 7.3820940872889895 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\right)} \cdot \left(\left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}\\ \mathbf{elif}\;t \le 1.412213387362571 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot 2\right) \cdot \left(\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n\right)}} \cdot \sqrt{\sqrt{\left(U \cdot 2\right) \cdot \left(\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1556076 = 2.0;
        double r1556077 = n;
        double r1556078 = r1556076 * r1556077;
        double r1556079 = U;
        double r1556080 = r1556078 * r1556079;
        double r1556081 = t;
        double r1556082 = l;
        double r1556083 = r1556082 * r1556082;
        double r1556084 = Om;
        double r1556085 = r1556083 / r1556084;
        double r1556086 = r1556076 * r1556085;
        double r1556087 = r1556081 - r1556086;
        double r1556088 = r1556082 / r1556084;
        double r1556089 = pow(r1556088, r1556076);
        double r1556090 = r1556077 * r1556089;
        double r1556091 = U_;
        double r1556092 = r1556079 - r1556091;
        double r1556093 = r1556090 * r1556092;
        double r1556094 = r1556087 - r1556093;
        double r1556095 = r1556080 * r1556094;
        double r1556096 = sqrt(r1556095);
        return r1556096;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1556097 = t;
        double r1556098 = -9.628340007082016e+98;
        bool r1556099 = r1556097 <= r1556098;
        double r1556100 = U_;
        double r1556101 = U;
        double r1556102 = r1556100 - r1556101;
        double r1556103 = n;
        double r1556104 = Om;
        double r1556105 = l;
        double r1556106 = r1556104 / r1556105;
        double r1556107 = r1556103 / r1556106;
        double r1556108 = r1556107 / r1556106;
        double r1556109 = r1556105 / r1556106;
        double r1556110 = -2.0;
        double r1556111 = fma(r1556109, r1556110, r1556097);
        double r1556112 = fma(r1556102, r1556108, r1556111);
        double r1556113 = cbrt(r1556112);
        double r1556114 = r1556113 * r1556113;
        double r1556115 = r1556113 * r1556114;
        double r1556116 = cbrt(r1556115);
        double r1556117 = 2.0;
        double r1556118 = r1556101 * r1556117;
        double r1556119 = r1556118 * r1556103;
        double r1556120 = r1556114 * r1556119;
        double r1556121 = r1556116 * r1556120;
        double r1556122 = sqrt(r1556121);
        double r1556123 = -3.2377717899062056e-141;
        bool r1556124 = r1556097 <= r1556123;
        double r1556125 = r1556112 * r1556103;
        double r1556126 = cbrt(r1556125);
        double r1556127 = r1556126 * r1556126;
        double r1556128 = r1556127 * r1556126;
        double r1556129 = r1556128 * r1556118;
        double r1556130 = sqrt(r1556129);
        double r1556131 = 7.3820940872889895e-298;
        bool r1556132 = r1556097 <= r1556131;
        double r1556133 = 1.412213387362571e+29;
        bool r1556134 = r1556097 <= r1556133;
        double r1556135 = r1556118 * r1556125;
        double r1556136 = sqrt(r1556135);
        double r1556137 = sqrt(r1556136);
        double r1556138 = r1556137 * r1556137;
        double r1556139 = sqrt(r1556119);
        double r1556140 = sqrt(r1556112);
        double r1556141 = r1556139 * r1556140;
        double r1556142 = r1556134 ? r1556138 : r1556141;
        double r1556143 = r1556132 ? r1556122 : r1556142;
        double r1556144 = r1556124 ? r1556130 : r1556143;
        double r1556145 = r1556099 ? r1556122 : r1556144;
        return r1556145;
}

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 4 regimes

2. if t < -9.628340007082016e+98 or -3.2377717899062056e-141 < t < 7.3820940872889895e-298

   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. Simplified 31.4

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

   4. Applied associate-*r* 31.5

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

   6. Applied add-cube-cbrt 31.7

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

   7. Applied associate-*r* 31.7

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

   9. Applied add-cbrt-cube 31.8

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

   if -9.628340007082016e+98 < t < -3.2377717899062056e-141

   1. Initial program 30.8

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

   2. Simplified 27.1

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

   4. Applied add-cube-cbrt 27.5

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

   if 7.3820940872889895e-298 < t < 1.412213387362571e+29

   1. Initial program 32.8

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

   2. Simplified 29.5

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

   4. Applied add-sqr-sqrt 29.7

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

   if 1.412213387362571e+29 < t

   1. Initial program 33.4

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

   2. Simplified 29.0

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

   4. Applied associate-*r* 30.4

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

   6. Applied sqrt-prod 24.3

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 28.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.628340007082016 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\right)} \cdot \left(\left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}\\ \mathbf{elif}\;t \le -3.2377717899062056 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{\left(\left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n}\right) \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n}\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{elif}\;t \le 7.3820940872889895 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\right)} \cdot \left(\left(\sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}\\ \mathbf{elif}\;t \le 1.412213387362571 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot 2\right) \cdot \left(\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n\right)}} \cdot \sqrt{\sqrt{\left(U \cdot 2\right) \cdot \left(\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(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*))))))