Toniolo and Linder, Equation (13)

Percentage Accurate: 48.8% → 62.7%

Time: 20.8s

Alternatives: 17

Speedup: 0.4×

Specification

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 48.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 62.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(l\_m, -2 \cdot \frac{l\_m}{Om}, t\right)\\ t_2 := 2 \cdot \left(n \cdot U\right)\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(t\_1 \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+307}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(\frac{U* \cdot l\_m}{Om}, \frac{l\_m}{Om} \cdot n, t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma l_m (* -2.0 (/ l_m Om)) t))
        (t_2 (* 2.0 (* n U)))
        (t_3
         (*
          t_2
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_3 0.0)
     (sqrt (* n (* t_1 (* 2.0 U))))
     (if (<= t_3 1e+307)
       (sqrt (* t_2 (fma (/ (* U* l_m) Om) (* (/ l_m Om) n) t_1)))
       (*
        (sqrt (* (* n U) (- (/ (* n (- U* U)) (* Om Om)) (/ 2.0 Om))))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma(l_m, (-2.0 * (l_m / Om)), t);
	double t_2 = 2.0 * (n * U);
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((n * (t_1 * (2.0 * U))));
	} else if (t_3 <= 1e+307) {
		tmp = sqrt((t_2 * fma(((U_42_ * l_m) / Om), ((l_m / Om) * n), t_1)));
	} else {
		tmp = sqrt(((n * U) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om)))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(l_m, Float64(-2.0 * Float64(l_m / Om)), t)
	t_2 = Float64(2.0 * Float64(n * U))
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(n * Float64(t_1 * Float64(2.0 * U))));
	elseif (t_3 <= 1e+307)
		tmp = sqrt(Float64(t_2 * fma(Float64(Float64(U_42_ * l_m) / Om), Float64(Float64(l_m / Om) * n), t_1)));
	else
		tmp = Float64(sqrt(Float64(Float64(n * U) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) - Float64(2.0 / Om)))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(l$95$m * N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(n * N[(t$95$1 * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+307], N[Sqrt[N[(t$95$2 * N[(N[(N[(U$42$ * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 12.2%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 18.1

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

   5. Simplified 18.1%

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

   7. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. sqrt-unprod N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   9. Applied egg-rr 48.8%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 98.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied egg-rr 99.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 97.7

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

   7. Simplified 97.7%

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

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 20.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied egg-rr 26.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   7. Simplified 21.0%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(\frac{U* \cdot \ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_3 := n \cdot \left(U* - U\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(l\_m, -2 \cdot \frac{l\_m}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\frac{l\_m}{Om}, t\_3, l\_m \cdot -2\right), t\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{t\_3}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U)))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        (t_3 (* n (- U* U))))
   (if (<= t_2 0.0)
     (sqrt (* n (* (fma l_m (* -2.0 (/ l_m Om)) t) (* 2.0 U))))
     (if (<= t_2 2e-94)
       (sqrt (* t_1 (fma (/ l_m Om) (fma (/ l_m Om) t_3 (* l_m -2.0)) t)))
       (if (<= t_2 1e+307)
         (sqrt
          (* t_1 (- t (/ (* (* l_m l_m) (fma n (/ (- U U*) Om) 2.0)) Om))))
         (*
          (sqrt (* (* n U) (- (/ t_3 (* Om Om)) (/ 2.0 Om))))
          (* l_m (sqrt 2.0))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double t_3 = n * (U_42_ - U);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n * (fma(l_m, (-2.0 * (l_m / Om)), t) * (2.0 * U))));
	} else if (t_2 <= 2e-94) {
		tmp = sqrt((t_1 * fma((l_m / Om), fma((l_m / Om), t_3, (l_m * -2.0)), t)));
	} else if (t_2 <= 1e+307) {
		tmp = sqrt((t_1 * (t - (((l_m * l_m) * fma(n, ((U - U_42_) / Om), 2.0)) / Om))));
	} else {
		tmp = sqrt(((n * U) * ((t_3 / (Om * Om)) - (2.0 / Om)))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	t_3 = Float64(n * Float64(U_42_ - U))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(n * Float64(fma(l_m, Float64(-2.0 * Float64(l_m / Om)), t) * Float64(2.0 * U))));
	elseif (t_2 <= 2e-94)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m / Om), fma(Float64(l_m / Om), t_3, Float64(l_m * -2.0)), t)));
	elseif (t_2 <= 1e+307)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Om), 2.0)) / Om))));
	else
		tmp = Float64(sqrt(Float64(Float64(n * U) * Float64(Float64(t_3 / Float64(Om * Om)) - Float64(2.0 / Om)))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(n * N[(N[(l$95$m * N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e-94], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * t$95$3 + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(t$95$3 / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 12.2%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 18.1

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

   5. Simplified 18.1%

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

   7. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. sqrt-unprod N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   9. Applied egg-rr 48.8%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.9999999999999999e-94

   1. Initial program 98.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied egg-rr 98.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   6. Applied egg-rr 44.3%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(U* + \left(0 - U\right), \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \color{blue}{\left(0 - U\right)}\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(U* + \left(0 - U\right)\right)} \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \color{blue}{\left(\ell \cdot n\right)}}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\color{blue}{\ell \cdot \left(\ell \cdot n\right)}}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{\color{blue}{Om \cdot Om}} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \color{blue}{\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \left(\color{blue}{\frac{\ell}{Om}} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\ell \cdot -2\right)} + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      9. lift-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)}\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

   8. Applied egg-rr 47.6%

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

   10. Applied egg-rr 98.4%

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

   if 1.9999999999999999e-94 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 98.1%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. lower-/.f64 N/A

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

   5. Simplified 91.9%

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

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 20.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied egg-rr 26.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   7. Simplified 21.0%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\frac{\ell}{Om}, n \cdot \left(U* - U\right), \ell \cdot -2\right), t\right)}\\ \mathbf{elif}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(l\_m, -2 \cdot \frac{l\_m}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\frac{l\_m}{Om}, n \cdot \left(U* - U\right), l\_m \cdot -2\right), t\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(U* \cdot \frac{n}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U)))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (sqrt (* n (* (fma l_m (* -2.0 (/ l_m Om)) t) (* 2.0 U))))
     (if (<= t_2 2e-94)
       (sqrt
        (*
         t_1
         (fma (/ l_m Om) (fma (/ l_m Om) (* n (- U* U)) (* l_m -2.0)) t)))
       (if (<= t_2 1e+307)
         (sqrt
          (* t_1 (- t (/ (* (* l_m l_m) (fma n (/ (- U U*) Om) 2.0)) Om))))
         (*
          (sqrt (* (* n U) (- (* U* (/ n (* Om Om))) (/ 2.0 Om))))
          (* l_m (sqrt 2.0))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n * (fma(l_m, (-2.0 * (l_m / Om)), t) * (2.0 * U))));
	} else if (t_2 <= 2e-94) {
		tmp = sqrt((t_1 * fma((l_m / Om), fma((l_m / Om), (n * (U_42_ - U)), (l_m * -2.0)), t)));
	} else if (t_2 <= 1e+307) {
		tmp = sqrt((t_1 * (t - (((l_m * l_m) * fma(n, ((U - U_42_) / Om), 2.0)) / Om))));
	} else {
		tmp = sqrt(((n * U) * ((U_42_ * (n / (Om * Om))) - (2.0 / Om)))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(n * Float64(fma(l_m, Float64(-2.0 * Float64(l_m / Om)), t) * Float64(2.0 * U))));
	elseif (t_2 <= 2e-94)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m / Om), fma(Float64(l_m / Om), Float64(n * Float64(U_42_ - U)), Float64(l_m * -2.0)), t)));
	elseif (t_2 <= 1e+307)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Om), 2.0)) / Om))));
	else
		tmp = Float64(sqrt(Float64(Float64(n * U) * Float64(Float64(U_42_ * Float64(n / Float64(Om * Om))) - Float64(2.0 / Om)))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(n * N[(N[(l$95$m * N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e-94], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(U$42$ * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 12.2%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 18.1

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

   5. Simplified 18.1%

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

   7. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. sqrt-unprod N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   9. Applied egg-rr 48.8%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.9999999999999999e-94

   1. Initial program 98.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied egg-rr 98.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   6. Applied egg-rr 44.3%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(U* + \left(0 - U\right), \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \color{blue}{\left(0 - U\right)}\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(U* + \left(0 - U\right)\right)} \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \color{blue}{\left(\ell \cdot n\right)}}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\color{blue}{\ell \cdot \left(\ell \cdot n\right)}}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{\color{blue}{Om \cdot Om}} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \color{blue}{\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \left(\color{blue}{\frac{\ell}{Om}} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\ell \cdot -2\right)} + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      9. lift-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)}\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

   8. Applied egg-rr 47.6%

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

   10. Applied egg-rr 98.4%

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

   if 1.9999999999999999e-94 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 98.1%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. lower-/.f64 N/A

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

   5. Simplified 91.9%

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

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 20.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied egg-rr 26.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 25.1

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

   7. Simplified 25.1%

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

   8. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 20.0

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

   10. Simplified 20.0%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\frac{\ell}{Om}, n \cdot \left(U* - U\right), \ell \cdot -2\right), t\right)}\\ \mathbf{elif}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(U* \cdot \frac{n}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(l\_m, -2 \cdot \frac{l\_m}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\frac{l\_m}{Om}, n \cdot \left(U* - U\right), l\_m \cdot -2\right), t\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{\left(n \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U)))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (sqrt (* n (* (fma l_m (* -2.0 (/ l_m Om)) t) (* 2.0 U))))
     (if (<= t_2 2e-94)
       (sqrt
        (*
         t_1
         (fma (/ l_m Om) (fma (/ l_m Om) (* n (- U* U)) (* l_m -2.0)) t)))
       (if (<= t_2 1e+307)
         (sqrt
          (* t_1 (- t (/ (* (* l_m l_m) (fma n (/ (- U U*) Om) 2.0)) Om))))
         (*
          l_m
          (sqrt
           (*
            (* n (fma n (/ (- U* U) (* Om Om)) (/ -2.0 Om)))
            (* 2.0 U)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n * (fma(l_m, (-2.0 * (l_m / Om)), t) * (2.0 * U))));
	} else if (t_2 <= 2e-94) {
		tmp = sqrt((t_1 * fma((l_m / Om), fma((l_m / Om), (n * (U_42_ - U)), (l_m * -2.0)), t)));
	} else if (t_2 <= 1e+307) {
		tmp = sqrt((t_1 * (t - (((l_m * l_m) * fma(n, ((U - U_42_) / Om), 2.0)) / Om))));
	} else {
		tmp = l_m * sqrt(((n * fma(n, ((U_42_ - U) / (Om * Om)), (-2.0 / Om))) * (2.0 * U)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(n * Float64(fma(l_m, Float64(-2.0 * Float64(l_m / Om)), t) * Float64(2.0 * U))));
	elseif (t_2 <= 2e-94)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m / Om), fma(Float64(l_m / Om), Float64(n * Float64(U_42_ - U)), Float64(l_m * -2.0)), t)));
	elseif (t_2 <= 1e+307)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Om), 2.0)) / Om))));
	else
		tmp = Float64(l_m * sqrt(Float64(Float64(n * fma(n, Float64(Float64(U_42_ - U) / Float64(Om * Om)), Float64(-2.0 / Om))) * Float64(2.0 * U))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(n * N[(N[(l$95$m * N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e-94], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(N[(n * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 12.2%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 18.1

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

   5. Simplified 18.1%

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

   7. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. sqrt-unprod N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   9. Applied egg-rr 48.8%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.9999999999999999e-94

   1. Initial program 98.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied egg-rr 98.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   6. Applied egg-rr 44.3%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(U* + \left(0 - U\right), \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \color{blue}{\left(0 - U\right)}\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(U* + \left(0 - U\right)\right)} \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \color{blue}{\left(\ell \cdot n\right)}}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\color{blue}{\ell \cdot \left(\ell \cdot n\right)}}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{\color{blue}{Om \cdot Om}} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \color{blue}{\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}} + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \left(\color{blue}{\frac{\ell}{Om}} \cdot \left(\ell \cdot -2\right) + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\ell \cdot -2\right)} + t\right)\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

      9. lift-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(U* + \left(0 - U\right)\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om} + \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)}\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U} \]

   8. Applied egg-rr 47.6%

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

   10. Applied egg-rr 98.4%

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

   if 1.9999999999999999e-94 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 98.1%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. lower-/.f64 N/A

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

   5. Simplified 91.9%

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

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 20.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied egg-rr 26.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 26.5

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

   7. Simplified 26.5%

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

   8. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. associate-/l* N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 25.6

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

   10. Simplified 25.6%

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)\right)}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift--.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. associate-*l* N/A

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

   12. Applied egg-rr 20.1%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\frac{\ell}{Om}, n \cdot \left(U* - U\right), \ell \cdot -2\right), t\right)}\\ \mathbf{elif}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \sqrt{n \cdot \left(\mathsf{fma}\left(l\_m, -2 \cdot \frac{l\_m}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \frac{\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* 2.0 (* n U))
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
        (t_2 (sqrt (* n (* (fma l_m (* -2.0 (/ l_m Om)) t) (* 2.0 U))))))
   (if (<= t_1 0.0)
     t_2
     (if (<= t_1 INFINITY)
       t_2
       (sqrt
        (*
         -2.0
         (* (* n U) (/ (* (* l_m l_m) (fma n (/ (- U U*) Om) 2.0)) Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * (n * U)) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double t_2 = sqrt((n * (fma(l_m, (-2.0 * (l_m / Om)), t) * (2.0 * U))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = sqrt((-2.0 * ((n * U) * (((l_m * l_m) * fma(n, ((U - U_42_) / Om), 2.0)) / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	t_2 = sqrt(Float64(n * Float64(fma(l_m, Float64(-2.0 * Float64(l_m / Om)), t) * Float64(2.0 * U))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(n * U) * Float64(Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Om), 2.0)) / Om))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(n * N[(N[(l$95$m * N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, Infinity], t$95$2, N[Sqrt[N[(-2.0 * N[(N[(n * U), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 14.3%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 14.3

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

   5. Simplified 14.3%

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

   7. Applied egg-rr 50.1%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. sqrt-unprod N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   9. Applied egg-rr 50.2%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 66.3%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 57.0

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

   5. Simplified 57.0%

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

   7. Applied egg-rr 29.4%

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

   9. Applied egg-rr 60.5%

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. associate-*r/ N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

   5. Simplified 28.4%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_3 := n \cdot \left(U* - U\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(l\_m, -2 \cdot \frac{l\_m}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\frac{l\_m}{Om}, t\_3, l\_m \cdot -2\right), t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{t\_3}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U)))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        (t_3 (* n (- U* U))))
   (if (<= t_2 0.0)
     (sqrt (* n (* (fma l_m (* -2.0 (/ l_m Om)) t) (* 2.0 U))))
     (if (<= t_2 1e+307)
       (sqrt (* t_1 (fma (/ l_m Om) (fma (/ l_m Om) t_3 (* l_m -2.0)) t)))
       (*
        (sqrt (* (* n U) (- (/ t_3 (* Om Om)) (/ 2.0 Om))))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double t_3 = n * (U_42_ - U);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n * (fma(l_m, (-2.0 * (l_m / Om)), t) * (2.0 * U))));
	} else if (t_2 <= 1e+307) {
		tmp = sqrt((t_1 * fma((l_m / Om), fma((l_m / Om), t_3, (l_m * -2.0)), t)));
	} else {
		tmp = sqrt(((n * U) * ((t_3 / (Om * Om)) - (2.0 / Om)))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	t_3 = Float64(n * Float64(U_42_ - U))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(n * Float64(fma(l_m, Float64(-2.0 * Float64(l_m / Om)), t) * Float64(2.0 * U))));
	elseif (t_2 <= 1e+307)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m / Om), fma(Float64(l_m / Om), t_3, Float64(l_m * -2.0)), t)));
	else
		tmp = Float64(sqrt(Float64(Float64(n * U) * Float64(Float64(t_3 / Float64(Om * Om)) - Float64(2.0 / Om)))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(n * N[(N[(l$95$m * N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * t$95$3 + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(t$95$3 / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 12.2%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 18.1

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

   5. Simplified 18.1%

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

   7. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. sqrt-unprod N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   9. Applied egg-rr 48.8%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 98.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied egg-rr 99.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 99.2

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

   7. Simplified 99.2%

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

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 20.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied egg-rr 26.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   7. Simplified 21.0%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\frac{\ell}{Om}, n \cdot \left(U* - U\right), \ell \cdot -2\right), t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(l\_m, -2 \cdot \frac{l\_m}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{\left(n \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U)))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (sqrt (* n (* (fma l_m (* -2.0 (/ l_m Om)) t) (* 2.0 U))))
     (if (<= t_2 1e+307)
       (sqrt (* t_1 (- t (/ (* (* l_m l_m) (fma n (/ (- U U*) Om) 2.0)) Om))))
       (*
        l_m
        (sqrt
         (* (* n (fma n (/ (- U* U) (* Om Om)) (/ -2.0 Om))) (* 2.0 U))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n * (fma(l_m, (-2.0 * (l_m / Om)), t) * (2.0 * U))));
	} else if (t_2 <= 1e+307) {
		tmp = sqrt((t_1 * (t - (((l_m * l_m) * fma(n, ((U - U_42_) / Om), 2.0)) / Om))));
	} else {
		tmp = l_m * sqrt(((n * fma(n, ((U_42_ - U) / (Om * Om)), (-2.0 / Om))) * (2.0 * U)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(n * Float64(fma(l_m, Float64(-2.0 * Float64(l_m / Om)), t) * Float64(2.0 * U))));
	elseif (t_2 <= 1e+307)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Om), 2.0)) / Om))));
	else
		tmp = Float64(l_m * sqrt(Float64(Float64(n * fma(n, Float64(Float64(U_42_ - U) / Float64(Om * Om)), Float64(-2.0 / Om))) * Float64(2.0 * U))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(n * N[(N[(l$95$m * N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(N[(n * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 12.2%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 18.1

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

   5. Simplified 18.1%

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

   7. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. sqrt-unprod N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   9. Applied egg-rr 48.8%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 98.2%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. lower-/.f64 N/A

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

   5. Simplified 88.7%

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

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 20.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied egg-rr 26.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 26.5

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

   7. Simplified 26.5%

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

   8. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. associate-/l* N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 25.6

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

   10. Simplified 25.6%

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)\right)}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift--.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. associate-*l* N/A

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

   12. Applied egg-rr 20.1%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{elif}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0 \lor t\_1 \leq 10^{+307}:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(l\_m, -2 \cdot \frac{l\_m}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{\left(n \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* 2.0 (* n U))
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (or (<= t_1 0.0) (<= t_1 1e+307))
     (sqrt (* n (* (fma l_m (* -2.0 (/ l_m Om)) t) (* 2.0 U))))
     (*
      l_m
      (sqrt (* (* n (fma n (/ (- U* U) (* Om Om)) (/ -2.0 Om))) (* 2.0 U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * (n * U)) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if ((t_1 <= 0.0) || (t_1 <= 1e+307)) {
		tmp = sqrt((n * (fma(l_m, (-2.0 * (l_m / Om)), t) * (2.0 * U))));
	} else {
		tmp = l_m * sqrt(((n * fma(n, ((U_42_ - U) / (Om * Om)), (-2.0 / Om))) * (2.0 * U)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if ((t_1 <= 0.0) || (t_1 <= 1e+307))
		tmp = sqrt(Float64(n * Float64(fma(l_m, Float64(-2.0 * Float64(l_m / Om)), t) * Float64(2.0 * U))));
	else
		tmp = Float64(l_m * sqrt(Float64(Float64(n * fma(n, Float64(Float64(U_42_ - U) / Float64(Om * Om)), Float64(-2.0 / Om))) * Float64(2.0 * U))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.0], LessEqual[t$95$1, 1e+307]], N[Sqrt[N[(n * N[(N[(l$95$m * N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(N[(n * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 12.2%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 18.1

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

   5. Simplified 18.1%

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

   7. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. sqrt-unprod N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   9. Applied egg-rr 48.8%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 98.2%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 87.9

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

   5. Simplified 87.9%

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

   if 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 20.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied egg-rr 26.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(0 - \left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 26.5

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

   7. Simplified 26.5%

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

   8. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. associate-/l* N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 25.6

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

   10. Simplified 25.6%

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)\right)}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift--.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. associate-*l* N/A

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

   12. Applied egg-rr 20.1%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0 \lor \left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{+307}:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\left(n \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{-4 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* 2.0 (* n U))
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
        (t_2 (sqrt (* (* (* 2.0 U) t) n))))
   (if (<= t_1 0.0)
     t_2
     (if (<= t_1 5e+153) t_2 (sqrt (* n (/ (* -4.0 (* U (* l_m l_m))) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * (n * U)) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double t_2 = sqrt((((2.0 * U) * t) * n));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+153) {
		tmp = t_2;
	} else {
		tmp = sqrt((n * ((-4.0 * (U * (l_m * l_m))) / Om)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * (n * u)) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))))
    t_2 = sqrt((((2.0d0 * u) * t) * n))
    if (t_1 <= 0.0d0) then
        tmp = t_2
    else if (t_1 <= 5d+153) then
        tmp = t_2
    else
        tmp = sqrt((n * (((-4.0d0) * (u * (l_m * l_m))) / om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * (n * U)) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double t_2 = Math.sqrt((((2.0 * U) * t) * n));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+153) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt((n * ((-4.0 * (U * (l_m * l_m))) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt(((2.0 * (n * U)) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_)))))
	t_2 = math.sqrt((((2.0 * U) * t) * n))
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= 5e+153:
		tmp = t_2
	else:
		tmp = math.sqrt((n * ((-4.0 * (U * (l_m * l_m))) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * U) * t) * n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 5e+153)
		tmp = t_2;
	else
		tmp = sqrt(Float64(n * Float64(Float64(-4.0 * Float64(U * Float64(l_m * l_m))) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(((2.0 * (n * U)) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)))));
	t_2 = sqrt((((2.0 * U) * t) * n));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 5e+153)
		tmp = t_2;
	else
		tmp = sqrt((n * ((-4.0 * (U * (l_m * l_m))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 5e+153], t$95$2, N[Sqrt[N[(n * N[(N[(-4.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 14.3%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 14.3

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

   5. Simplified 14.3%

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

   7. Applied egg-rr 50.1%

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

   8. Taylor expanded in l around 0

      \[\leadsto \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]

      2. lower-*.f64 46.9

         \[\leadsto \left(\sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]

   10. Simplified 46.9%

       \[\leadsto \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000018e153

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 61.9

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 61.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot U\right)} \cdot n\right) \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(U \cdot 2\right)} \cdot n\right) \cdot t} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(2 \cdot n\right)}\right) \cdot t} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      9. lower-*.f64 73.2

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right) \cdot t} \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

      14. lower-*.f64 73.2

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

      17. lower-*.f64 73.2

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

   7. Applied egg-rr 73.2%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}} \]

   if 5.00000000000000018e153 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 19.1%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 21.9%

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

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{n \cdot \left(-4 \cdot \frac{U \cdot {\ell}^{2}}{Om} + 2 \cdot \left(U \cdot t\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 17.5

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

   8. Simplified 17.5%

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

   9. Taylor expanded in t around 0

      \[\leadsto \sqrt{n \cdot \color{blue}{\left(-4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

          \[\leadsto \sqrt{n \cdot \frac{-4 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{Om}} \]

       6. lower-*.f64 16.6

          \[\leadsto \sqrt{n \cdot \frac{-4 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{Om}} \]

   11. Simplified 16.6%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \mathbf{elif}\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(U \cdot \frac{\left(l\_m \cdot l\_m\right) \cdot n}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* 2.0 (* n U))
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
        (t_2 (sqrt (* (* (* 2.0 U) t) n))))
   (if (<= t_1 0.0)
     t_2
     (if (<= t_1 5e+153) t_2 (sqrt (* -4.0 (* U (/ (* (* l_m l_m) n) Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt(((2.0 * (n * U)) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double t_2 = sqrt((((2.0 * U) * t) * n));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+153) {
		tmp = t_2;
	} else {
		tmp = sqrt((-4.0 * (U * (((l_m * l_m) * n) / Om))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt(((2.0d0 * (n * u)) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))))
    t_2 = sqrt((((2.0d0 * u) * t) * n))
    if (t_1 <= 0.0d0) then
        tmp = t_2
    else if (t_1 <= 5d+153) then
        tmp = t_2
    else
        tmp = sqrt(((-4.0d0) * (u * (((l_m * l_m) * n) / om))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt(((2.0 * (n * U)) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double t_2 = Math.sqrt((((2.0 * U) * t) * n));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+153) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt((-4.0 * (U * (((l_m * l_m) * n) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt(((2.0 * (n * U)) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_)))))
	t_2 = math.sqrt((((2.0 * U) * t) * n))
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= 5e+153:
		tmp = t_2
	else:
		tmp = math.sqrt((-4.0 * (U * (((l_m * l_m) * n) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * U) * t) * n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 5e+153)
		tmp = t_2;
	else
		tmp = sqrt(Float64(-4.0 * Float64(U * Float64(Float64(Float64(l_m * l_m) * n) / Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(((2.0 * (n * U)) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)))));
	t_2 = sqrt((((2.0 * U) * t) * n));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 5e+153)
		tmp = t_2;
	else
		tmp = sqrt((-4.0 * (U * (((l_m * l_m) * n) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 5e+153], t$95$2, N[Sqrt[N[(-4.0 * N[(U * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 14.3%

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

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 34.0

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 34.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}} \]

      6. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot U\right) \cdot t} \cdot \sqrt{n}} \]

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow1/2 N/A

          \[\leadsto \sqrt{\left(U \cdot 2\right) \cdot t} \cdot \color{blue}{\sqrt{n}} \]

      15. lower-sqrt.f64 46.9

          \[\leadsto \sqrt{\left(U \cdot 2\right) \cdot t} \cdot \color{blue}{\sqrt{n}} \]

   7. Applied egg-rr 46.9%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot 2\right) \cdot t} \cdot \sqrt{n}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000018e153

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 61.9

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 61.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot U\right)} \cdot n\right) \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(U \cdot 2\right)} \cdot n\right) \cdot t} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(2 \cdot n\right)}\right) \cdot t} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      9. lower-*.f64 73.2

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right) \cdot t} \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

      14. lower-*.f64 73.2

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

      17. lower-*.f64 73.2

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

   7. Applied egg-rr 73.2%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}} \]

   if 5.00000000000000018e153 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 19.1%

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

   3. Taylor expanded in l around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 21.9%

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

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{n \cdot \left(-4 \cdot \frac{U \cdot {\ell}^{2}}{Om} + 2 \cdot \left(U \cdot t\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 17.5

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

   8. Simplified 17.5%

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

   9. Taylor expanded in t around 0

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

          \[\leadsto \sqrt{-4 \cdot \left(U \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot n}{Om}\right)} \]

       7. lower-*.f64 16.4

          \[\leadsto \sqrt{-4 \cdot \left(U \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot n}{Om}\right)} \]

   11. Simplified 16.4%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \mathbf{elif}\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \left(U \cdot \frac{\left(\ell \cdot \ell\right) \cdot n}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(l\_m, -2 \cdot \frac{l\_m}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \frac{\left(U* \cdot \left(l\_m \cdot l\_m\right)\right) \cdot \left(n \cdot n\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* 2.0 (* n U))
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (sqrt (* (* (* 2.0 U) t) n))
     (if (<= t_1 INFINITY)
       (sqrt (* n (* (fma l_m (* -2.0 (/ l_m Om)) t) (* 2.0 U))))
       (sqrt (* (* 2.0 U) (/ (* (* U* (* l_m l_m)) (* n n)) (* Om Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * (n * U)) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((((2.0 * U) * t) * n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((n * (fma(l_m, (-2.0 * (l_m / Om)), t) * (2.0 * U))));
	} else {
		tmp = sqrt(((2.0 * U) * (((U_42_ * (l_m * l_m)) * (n * n)) / (Om * Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(2.0 * U) * t) * n));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(n * Float64(fma(l_m, Float64(-2.0 * Float64(l_m / Om)), t) * Float64(2.0 * U))));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(Float64(Float64(U_42_ * Float64(l_m * l_m)) * Float64(n * n)) / Float64(Om * Om))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(N[(2.0 * U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(n * N[(N[(l$95$m * N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(N[(N[(U$42$ * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 12.2%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sqrt.f64 18.1

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

   5. Simplified 18.1%

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

   7. Applied egg-rr 48.7%

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

   8. Taylor expanded in l around 0

      \[\leadsto \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]

      2. lower-*.f64 42.7

         \[\leadsto \left(\sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]

   10. Simplified 42.7%

       \[\leadsto \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 66.3%

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

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \cdot \sqrt{2} \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \cdot \sqrt{2} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \cdot \sqrt{2} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \cdot \sqrt{2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \cdot \sqrt{2} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \cdot \sqrt{2} \]

      17. lower-sqrt.f64 57.0

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \color{blue}{\sqrt{2}} \]

   5. Simplified 57.0%

      \[\leadsto \color{blue}{\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \sqrt{2}} \]

   7. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]

   9. Applied egg-rr 60.5%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right)}} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 0.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \frac{\color{blue}{U* \cdot \left({\ell}^{2} \cdot n\right)}}{{Om}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \frac{U* \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}}{{Om}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \frac{U* \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)}{{Om}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \frac{U* \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)}{{Om}^{2}}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{\color{blue}{Om \cdot Om}}} \]

      7. lower-*.f64 21.7

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{\color{blue}{Om \cdot Om}}} \]

   5. Simplified 21.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om \cdot Om}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \mathbf{elif}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \frac{\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(l\_m, -2 \cdot \frac{l\_m}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\left(n \cdot \frac{\sqrt{2}}{Om}\right) \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* 2.0 (* n U))
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (sqrt (* (* (* 2.0 U) t) n))
     (if (<= t_1 INFINITY)
       (sqrt (* n (* (fma l_m (* -2.0 (/ l_m Om)) t) (* 2.0 U))))
       (* (sqrt (* U U*)) (* (* n (/ (sqrt 2.0) Om)) l_m))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * (n * U)) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((((2.0 * U) * t) * n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((n * (fma(l_m, (-2.0 * (l_m / Om)), t) * (2.0 * U))));
	} else {
		tmp = sqrt((U * U_42_)) * ((n * (sqrt(2.0) / Om)) * l_m);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(2.0 * U) * t) * n));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(n * Float64(fma(l_m, Float64(-2.0 * Float64(l_m / Om)), t) * Float64(2.0 * U))));
	else
		tmp = Float64(sqrt(Float64(U * U_42_)) * Float64(Float64(n * Float64(sqrt(2.0) / Om)) * l_m));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(N[(2.0 * U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(n * N[(N[(l$95$m * N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 12.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \cdot \sqrt{2} \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \cdot \sqrt{2} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \cdot \sqrt{2} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \cdot \sqrt{2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \cdot \sqrt{2} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \cdot \sqrt{2} \]

      17. lower-sqrt.f64 18.1

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \color{blue}{\sqrt{2}} \]

   5. Simplified 18.1%

      \[\leadsto \color{blue}{\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \sqrt{2}} \]

   7. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]

   8. Taylor expanded in l around 0

      \[\leadsto \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]

      2. lower-*.f64 42.7

         \[\leadsto \left(\sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]

   10. Simplified 42.7%

       \[\leadsto \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 66.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \cdot \sqrt{2} \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \cdot \sqrt{2} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \cdot \sqrt{2} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \cdot \sqrt{2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \cdot \sqrt{2} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \cdot \sqrt{2} \]

      17. lower-sqrt.f64 57.0

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \color{blue}{\sqrt{2}} \]

   5. Simplified 57.0%

      \[\leadsto \color{blue}{\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \sqrt{2}} \]

   7. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]

   9. Applied egg-rr 60.5%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right)}} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 0.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in U* around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\ell \cdot \left(n \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}{Om} \cdot \sqrt{U \cdot U*}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\ell \cdot \left(n \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}{Om} \cdot \sqrt{U \cdot U*}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{U \cdot U*} \cdot \frac{\ell \cdot \left(n \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}{Om}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot U*} \cdot \left(\mathsf{neg}\left(\frac{\ell \cdot \left(n \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}{Om}\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \color{blue}{\left(-1 \cdot \frac{\ell \cdot \left(n \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}{Om}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot U*} \cdot \left(-1 \cdot \frac{\ell \cdot \left(n \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}{Om}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot U*}} \cdot \left(-1 \cdot \frac{\ell \cdot \left(n \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}{Om}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{U \cdot U*}} \cdot \left(-1 \cdot \frac{\ell \cdot \left(n \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}{Om}\right) \]

      8. mul-1-neg N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\ell \cdot \left(n \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}{Om}\right)\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\ell \cdot \left(n \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}{Om}\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \sqrt{U \cdot U*} \cdot \left(\mathsf{neg}\left(\color{blue}{\ell \cdot \frac{n \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)}{Om}}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{U \cdot U*} \cdot \left(\mathsf{neg}\left(\color{blue}{\ell \cdot \frac{n \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)}{Om}}\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{U \cdot U*} \cdot \left(\mathsf{neg}\left(\ell \cdot \frac{\color{blue}{\left(n \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{2}}}{Om}\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \sqrt{U \cdot U*} \cdot \left(\mathsf{neg}\left(\ell \cdot \color{blue}{\left(\left(n \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \frac{\sqrt{2}}{Om}\right)}\right)\right) \]

   5. Simplified 31.3%

      \[\leadsto \color{blue}{\sqrt{U \cdot U*} \cdot \left(-\ell \cdot \left(\left(-n\right) \cdot \frac{\sqrt{2}}{Om}\right)\right)} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \left(\mathsf{neg}\left(\ell \cdot \left(\color{blue}{\left(\mathsf{neg}\left(n\right)\right)} \cdot \frac{\sqrt{2}}{Om}\right)\right)\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \left(\mathsf{neg}\left(\ell \cdot \left(\left(\mathsf{neg}\left(n\right)\right) \cdot \frac{\color{blue}{\sqrt{2}}}{Om}\right)\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \left(\mathsf{neg}\left(\ell \cdot \left(\left(\mathsf{neg}\left(n\right)\right) \cdot \color{blue}{\frac{\sqrt{2}}{Om}}\right)\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \left(\mathsf{neg}\left(\ell \cdot \color{blue}{\left(\left(\mathsf{neg}\left(n\right)\right) \cdot \frac{\sqrt{2}}{Om}\right)}\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \color{blue}{\left(\ell \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right) \cdot \frac{\sqrt{2}}{Om}\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(n\right)\right) \cdot \frac{\sqrt{2}}{Om}}\right)\right)\right) \]

      7. lift-neg.f64 N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(n\right)\right)} \cdot \frac{\sqrt{2}}{Om}\right)\right)\right) \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)}\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \left(\ell \cdot \color{blue}{\left(n \cdot \frac{\sqrt{2}}{Om}\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{U \cdot U*} \cdot \color{blue}{\left(\left(n \cdot \frac{\sqrt{2}}{Om}\right) \cdot \ell\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{U \cdot U*} \cdot \color{blue}{\left(\left(n \cdot \frac{\sqrt{2}}{Om}\right) \cdot \ell\right)} \]

      12. lower-*.f64 31.3

          \[\leadsto \sqrt{U \cdot U*} \cdot \left(\color{blue}{\left(n \cdot \frac{\sqrt{2}}{Om}\right)} \cdot \ell\right) \]

   7. Applied egg-rr 31.3%

      \[\leadsto \sqrt{U \cdot U*} \cdot \color{blue}{\left(\left(n \cdot \frac{\sqrt{2}}{Om}\right) \cdot \ell\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \mathbf{elif}\;\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\left(n \cdot \frac{\sqrt{2}}{Om}\right) \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(l\_m, -2 \cdot \frac{l\_m}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* 2.0 (* n U))
        (-
         (- t (* 2.0 (/ (* l_m l_m) Om)))
         (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
      0.0)
   (sqrt (* (* (* 2.0 U) t) n))
   (sqrt (* n (* (fma l_m (* -2.0 (/ l_m Om)) t) (* 2.0 U))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (sqrt(((2.0 * (n * U)) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = sqrt((((2.0 * U) * t) * n));
	} else {
		tmp = sqrt((n * (fma(l_m, (-2.0 * (l_m / Om)), t) * (2.0 * U))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))) <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(2.0 * U) * t) * n));
	else
		tmp = sqrt(Float64(n * Float64(fma(l_m, Float64(-2.0 * Float64(l_m / Om)), t) * Float64(2.0 * U))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[Sqrt[N[(N[(N[(2.0 * U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(l$95$m * N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 14.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \cdot \sqrt{2} \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \cdot \sqrt{2} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \cdot \sqrt{2} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \cdot \sqrt{2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \cdot \sqrt{2} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \cdot \sqrt{2} \]

      17. lower-sqrt.f64 14.3

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \color{blue}{\sqrt{2}} \]

   5. Simplified 14.3%

      \[\leadsto \color{blue}{\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \sqrt{2}} \]

   7. Applied egg-rr 50.1%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]

   8. Taylor expanded in l around 0

      \[\leadsto \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]

      2. lower-*.f64 46.9

         \[\leadsto \left(\sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]

   10. Simplified 46.9%

       \[\leadsto \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{n}\right) \cdot \sqrt{2} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 52.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \cdot \sqrt{2} \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \cdot \sqrt{2} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \cdot \sqrt{2} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \cdot \sqrt{2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \cdot \sqrt{2} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \cdot \sqrt{2} \]

      17. lower-sqrt.f64 46.6

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \color{blue}{\sqrt{2}} \]

   5. Simplified 46.6%

      \[\leadsto \color{blue}{\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \sqrt{2}} \]

   7. Applied egg-rr 24.8%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]

   9. Applied egg-rr 49.4%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(l\_m, -2 \cdot \frac{l\_m}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 1.15e+130)
   (sqrt (* n (* (fma l_m (* -2.0 (/ l_m Om)) t) (* 2.0 U))))
   (sqrt (* (* (* 2.0 U) t) n))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= 1.15e+130) {
		tmp = sqrt((n * (fma(l_m, (-2.0 * (l_m / Om)), t) * (2.0 * U))));
	} else {
		tmp = sqrt((((2.0 * U) * t) * n));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 1.15e+130)
		tmp = sqrt(Float64(n * Float64(fma(l_m, Float64(-2.0 * Float64(l_m / Om)), t) * Float64(2.0 * U))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * U) * t) * n));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, 1.15e+130], N[Sqrt[N[(n * N[(N[(l$95$m * N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.15000000000000011e130

   1. Initial program 45.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \cdot \sqrt{2} \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \cdot \sqrt{2} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \cdot \sqrt{2} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \cdot \sqrt{2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \cdot \sqrt{2} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \cdot \sqrt{2} \]

      17. lower-sqrt.f64 40.5

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \color{blue}{\sqrt{2}} \]

   5. Simplified 40.5%

      \[\leadsto \color{blue}{\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \sqrt{2}} \]

   7. Applied egg-rr 44.4%

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(\left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right) \cdot 2\right)}} \]

   if 1.15000000000000011e130 < t

   1. Initial program 57.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 43.3

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 43.3%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot U\right)} \cdot n\right) \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(U \cdot 2\right)} \cdot n\right) \cdot t} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(2 \cdot n\right)}\right) \cdot t} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      9. lower-*.f64 50.0

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right) \cdot t} \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

      14. lower-*.f64 50.0

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

      17. lower-*.f64 50.0

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

   7. Applied egg-rr 50.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot n\right)\right)} \cdot t} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{t \cdot \left(2 \cdot \left(U \cdot n\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \color{blue}{\left(2 \cdot \left(U \cdot n\right)\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(t \cdot 2\right) \cdot \left(U \cdot n\right)}} \]

      6. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{t \cdot 2} \cdot \sqrt{U \cdot n}} \]

      7. pow1/2 N/A

         \[\leadsto \sqrt{t \cdot 2} \cdot \color{blue}{{\left(U \cdot n\right)}^{\frac{1}{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{t \cdot 2} \cdot {\left(U \cdot n\right)}^{\frac{1}{2}}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{t \cdot 2}} \cdot {\left(U \cdot n\right)}^{\frac{1}{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{t \cdot 2}} \cdot {\left(U \cdot n\right)}^{\frac{1}{2}} \]

      11. pow1/2 N/A

          \[\leadsto \sqrt{t \cdot 2} \cdot \color{blue}{\sqrt{U \cdot n}} \]

      12. lower-sqrt.f64 66.1

          \[\leadsto \sqrt{t \cdot 2} \cdot \color{blue}{\sqrt{U \cdot n}} \]

   9. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{\sqrt{t \cdot 2} \cdot \sqrt{U \cdot n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 9.8 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(2, t, -4 \cdot \frac{l\_m \cdot l\_m}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 9.8e+129)
   (sqrt (* n (* U (fma 2.0 t (* -4.0 (/ (* l_m l_m) Om))))))
   (sqrt (* (* (* 2.0 U) t) n))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= 9.8e+129) {
		tmp = sqrt((n * (U * fma(2.0, t, (-4.0 * ((l_m * l_m) / Om))))));
	} else {
		tmp = sqrt((((2.0 * U) * t) * n));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 9.8e+129)
		tmp = sqrt(Float64(n * Float64(U * fma(2.0, t, Float64(-4.0 * Float64(Float64(l_m * l_m) / Om))))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * U) * t) * n));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, 9.8e+129], N[Sqrt[N[(n * N[(U * N[(2.0 * t + N[(-4.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.8e129

   1. Initial program 45.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-2 \cdot U, {\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right), 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}} \]

   5. Simplified 42.4%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-2 \cdot U, \left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right), \left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{n \cdot \left(-4 \cdot \frac{U \cdot {\ell}^{2}}{Om} + 2 \cdot \left(U \cdot t\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{n \cdot \left(-4 \cdot \frac{U \cdot {\ell}^{2}}{Om} + 2 \cdot \left(U \cdot t\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{n \cdot \color{blue}{\left(2 \cdot \left(U \cdot t\right) + -4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \sqrt{n \cdot \color{blue}{\mathsf{fma}\left(2, U \cdot t, -4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, \color{blue}{U \cdot t}, -4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \color{blue}{-4 \cdot \frac{U \cdot {\ell}^{2}}{Om}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, -4 \cdot \color{blue}{\frac{U \cdot {\ell}^{2}}{Om}}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, -4 \cdot \frac{\color{blue}{U \cdot {\ell}^{2}}}{Om}\right)} \]

      8. unpow2 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, -4 \cdot \frac{U \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om}\right)} \]

      9. lower-*.f64 39.9

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, -4 \cdot \frac{U \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om}\right)} \]

   8. Simplified 39.9%

      \[\leadsto \sqrt{\color{blue}{n \cdot \mathsf{fma}\left(2, U \cdot t, -4 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om}\right)}} \]

   9. Taylor expanded in U around 0

      \[\leadsto \sqrt{n \cdot \color{blue}{\left(U \cdot \left(-4 \cdot \frac{{\ell}^{2}}{Om} + 2 \cdot t\right)\right)}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \sqrt{n \cdot \color{blue}{\left(U \cdot \left(-4 \cdot \frac{{\ell}^{2}}{Om} + 2 \cdot t\right)\right)}} \]

       2. +-commutative N/A

          \[\leadsto \sqrt{n \cdot \left(U \cdot \color{blue}{\left(2 \cdot t + -4 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \sqrt{n \cdot \left(U \cdot \color{blue}{\mathsf{fma}\left(2, t, -4 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(2, t, \color{blue}{-4 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]

       5. lower-/.f64 N/A

          \[\leadsto \sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(2, t, -4 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)} \]

       6. unpow2 N/A

          \[\leadsto \sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(2, t, -4 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

       7. lower-*.f64 41.7

          \[\leadsto \sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(2, t, -4 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

   11. Simplified 41.7%

       \[\leadsto \sqrt{n \cdot \color{blue}{\left(U \cdot \mathsf{fma}\left(2, t, -4 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

   if 9.8e129 < t

   1. Initial program 57.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 43.3

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 43.3%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot U\right)} \cdot n\right) \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(U \cdot 2\right)} \cdot n\right) \cdot t} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(2 \cdot n\right)}\right) \cdot t} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      9. lower-*.f64 50.0

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right) \cdot t} \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

      14. lower-*.f64 50.0

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

      17. lower-*.f64 50.0

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

   7. Applied egg-rr 50.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot n\right)\right)} \cdot t} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{t \cdot \left(2 \cdot \left(U \cdot n\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \color{blue}{\left(2 \cdot \left(U \cdot n\right)\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(t \cdot 2\right) \cdot \left(U \cdot n\right)}} \]

      6. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{t \cdot 2} \cdot \sqrt{U \cdot n}} \]

      7. pow1/2 N/A

         \[\leadsto \sqrt{t \cdot 2} \cdot \color{blue}{{\left(U \cdot n\right)}^{\frac{1}{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{t \cdot 2} \cdot {\left(U \cdot n\right)}^{\frac{1}{2}}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{t \cdot 2}} \cdot {\left(U \cdot n\right)}^{\frac{1}{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{t \cdot 2}} \cdot {\left(U \cdot n\right)}^{\frac{1}{2}} \]

      11. pow1/2 N/A

          \[\leadsto \sqrt{t \cdot 2} \cdot \color{blue}{\sqrt{U \cdot n}} \]

      12. lower-sqrt.f64 66.1

          \[\leadsto \sqrt{t \cdot 2} \cdot \color{blue}{\sqrt{U \cdot n}} \]

   9. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{\sqrt{t \cdot 2} \cdot \sqrt{U \cdot n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.8 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \mathsf{fma}\left(2, t, -4 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{n \cdot \left(\mathsf{fma}\left(l\_m, -2 \cdot \frac{l\_m}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (sqrt (* n (* (fma l_m (* -2.0 (/ l_m Om)) t) (* 2.0 U)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt((n * (fma(l_m, (-2.0 * (l_m / Om)), t) * (2.0 * U))));
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(n * Float64(fma(l_m, Float64(-2.0 * Float64(l_m / Om)), t) * Float64(2.0 * U))))
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(n * N[(N[(l$95$m * N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if n < -5.00000000000023e-311

   1. Initial program 49.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \cdot \sqrt{2} \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \cdot \sqrt{2} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \cdot \sqrt{2} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \cdot \sqrt{2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \cdot \sqrt{2} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \cdot \sqrt{2} \]

      17. lower-sqrt.f64 43.7

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \color{blue}{\sqrt{2}} \]

   5. Simplified 43.7%

      \[\leadsto \color{blue}{\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \sqrt{2}} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\left(\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)} \cdot \sqrt{n}\right)} \cdot \sqrt{2} \]

   9. Applied egg-rr 47.6%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right)}} \]

   if -5.00000000000023e-311 < n

   1. Initial program 45.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \cdot \sqrt{2} \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \cdot \sqrt{2} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \cdot \sqrt{2} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \cdot \sqrt{2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \cdot \sqrt{2} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \cdot \sqrt{2} \]

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \cdot \sqrt{2} \]

      17. lower-sqrt.f64 40.9

          \[\leadsto \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \color{blue}{\sqrt{2}} \]

   5. Simplified 40.9%

      \[\leadsto \color{blue}{\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)} \cdot \sqrt{2}} \]

   7. Applied egg-rr 55.1%

      \[\leadsto \color{blue}{\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.3%

   \[\leadsto \sqrt{n \cdot \left(\mathsf{fma}\left(\ell, -2 \cdot \frac{\ell}{Om}, t\right) \cdot \left(2 \cdot U\right)\right)} \]
5. Add Preprocessing

Alternative 17: 34.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* (* 2.0 U) t) n)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt((((2.0 * U) * t) * n));
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * u) * t) * n))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt((((2.0 * U) * t) * n));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((((2.0 * U) * t) * n))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * U) * t) * n))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((((2.0 * U) * t) * n));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if n < 1.99999999999999982e-307

   1. Initial program 49.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 27.9

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 27.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot U\right)} \cdot n\right) \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(U \cdot 2\right)} \cdot n\right) \cdot t} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(2 \cdot n\right)}\right) \cdot t} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      9. lower-*.f64 30.2

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right) \cdot t} \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

      14. lower-*.f64 30.2

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

      17. lower-*.f64 30.2

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

   7. Applied egg-rr 30.2%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}} \]

   if 1.99999999999999982e-307 < n

   1. Initial program 45.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 33.0

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 33.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot U\right)} \cdot n\right) \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(U \cdot 2\right)} \cdot n\right) \cdot t} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(2 \cdot n\right)}\right) \cdot t} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      9. lower-*.f64 33.0

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right) \cdot t} \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

      14. lower-*.f64 33.0

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right)} \cdot t} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

      17. lower-*.f64 33.0

          \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

   7. Applied egg-rr 33.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot t} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(U \cdot n\right) \cdot 2\right)} \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(U \cdot n\right)} \cdot 2\right) \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot 2\right)\right)} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(n \cdot 2\right)}\right) \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot 2\right) \cdot U\right)} \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right) \cdot \left(U \cdot t\right)}} \]

      8. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}} \]

      9. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(n \cdot 2\right)}^{\frac{1}{2}}} \cdot \sqrt{U \cdot t} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{{\left(n \cdot 2\right)}^{\frac{1}{2}} \cdot \sqrt{U \cdot t}} \]

      11. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{n \cdot 2}} \cdot \sqrt{U \cdot t} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{n \cdot 2}} \cdot \sqrt{U \cdot t} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{n \cdot 2} \cdot \color{blue}{\sqrt{U \cdot t}} \]

      14. lower-*.f64 43.7

          \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{\color{blue}{U \cdot t}} \]

   9. Applied egg-rr 43.7%

      \[\leadsto \color{blue}{\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.4%

   \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot t\right) \cdot n} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))