Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 67.2%

Time: 19.6s

Alternatives: 25

Speedup: 0.7×

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: 50.2% 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: 67.2% accurate, 0.4× speedup

Math

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

FPCore

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

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 / Om), ((U_42_ - U) * ((l_m / Om) * n)), fma(((l_m / Om) * l_m), -2.0, t));
	double t_2 = U * (n * 2.0);
	double t_3 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_2;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((t_1 * U) * (n * 2.0)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * t_2));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt((((((U_42_ - U) / Om) * (n / Om)) - (2.0 / Om)) * (U * n)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(l_m / Om), Float64(Float64(U_42_ - U) * Float64(Float64(l_m / Om) * n)), fma(Float64(Float64(l_m / Om) * l_m), -2.0, t))
	t_2 = Float64(U * Float64(n * 2.0))
	t_3 = Float64(Float64(Float64(t - Float64(Float64(Float64(l_m * l_m) / Om) * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_2)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(Float64(t_1 * U) * Float64(n * 2.0)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_1 * t_2));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / Om)) - Float64(2.0 / Om)) * Float64(U * n))));
	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[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(t$95$1 * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$1 * t$95$2), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $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 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 17.1

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 48.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-+r+ N/A

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

   8. Applied rewrites 54.5%

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

   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 66.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 70.6

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 64.3%

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

   6. Applied rewrites 71.5%

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

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 0.2

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 0.2%

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

   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

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

   7. Applied rewrites 24.7%

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

4. Final simplification 62.2%

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

Alternative 2: 49.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \mathsf{fma}\left(-2, t\_2, t\right)\\ t_4 := \sqrt{\left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t\_3 \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_4 \leq 10^{+153}:\\ \;\;\;\;\sqrt{t\_3 \cdot t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\frac{l\_m}{Om} \cdot l\_m\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3 (fma -2.0 t_2 t))
        (t_4
         (sqrt
          (*
           (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
           t_1))))
   (if (<= t_4 0.0)
     (sqrt (* (* (* t_3 n) U) 2.0))
     (if (<= t_4 1e+153)
       (sqrt (* t_3 t_1))
       (if (<= t_4 INFINITY)
         (sqrt (* (* -2.0 (* (/ l_m Om) l_m)) (* (* U 2.0) n)))
         (sqrt (* (* (* t n) U) 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 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = fma(-2.0, t_2, t);
	double t_4 = sqrt((((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((((t_3 * n) * U) * 2.0));
	} else if (t_4 <= 1e+153) {
		tmp = sqrt((t_3 * t_1));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((-2.0 * ((l_m / Om) * l_m)) * ((U * 2.0) * n)));
	} else {
		tmp = sqrt((((t * n) * U) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(l_m * l_m) / Om)
	t_3 = fma(-2.0, t_2, t)
	t_4 = sqrt(Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t_3 * n) * U) * 2.0));
	elseif (t_4 <= 1e+153)
		tmp = sqrt(Float64(t_3 * t_1));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(l_m / Om) * l_m)) * Float64(Float64(U * 2.0) * n)));
	else
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * t$95$2 + t), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(N[(t$95$3 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 1e+153], N[Sqrt[N[(t$95$3 * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(N[(-2.0 * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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 15.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.3

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

   5. Applied rewrites 50.3%

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

   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*))))) < 1e153

   1. Initial program 97.6%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if 1e153 < (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 25.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

   4. Applied rewrites 15.2%

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

   5. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 12.2

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

   7. Applied rewrites 12.2%

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

   8. Taylor expanded in Om around inf

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

   10. Applied rewrites 9.0%

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

   12. Applied rewrites 24.0%

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

   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 inf

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 11.8

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 11.8%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ t_4 := \sqrt{\mathsf{fma}\left(\left(\frac{l\_m}{Om} \cdot U\right) \cdot \left(l\_m \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 10^{+306}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{U* \cdot \left(l\_m \cdot l\_m\right)}{Om} \cdot \frac{n}{Om}\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3
         (* (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n))) t_1))
        (t_4
         (sqrt
          (fma (* (* (/ l_m Om) U) (* l_m n)) -4.0 (* (* (* t n) U) 2.0)))))
   (if (<= t_3 0.0)
     t_4
     (if (<= t_3 1e+306)
       (sqrt (* (fma -2.0 t_2 t) t_1))
       (if (<= t_3 INFINITY)
         t_4
         (sqrt (* (* (* (/ (* U* (* l_m l_m)) Om) (/ n Om)) U) (* n 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 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double t_4 = sqrt(fma((((l_m / Om) * U) * (l_m * n)), -4.0, (((t * n) * U) * 2.0)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = t_4;
	} else if (t_3 <= 1e+306) {
		tmp = sqrt((fma(-2.0, t_2, t) * t_1));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = sqrt((((((U_42_ * (l_m * l_m)) / Om) * (n / Om)) * U) * (n * 2.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(l_m * l_m) / Om)
	t_3 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	t_4 = sqrt(fma(Float64(Float64(Float64(l_m / Om) * U) * Float64(l_m * n)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = t_4;
	elseif (t_3 <= 1e+306)
		tmp = sqrt(Float64(fma(-2.0, t_2, t) * t_1));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ * Float64(l_m * l_m)) / Om) * Float64(n / Om)) * U) * Float64(n * 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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * U), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], t$95$4, If[LessEqual[t$95$3, 1e+306], N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[Sqrt[N[(N[(N[(N[(N[(U$42$ * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * N[(n * 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 or 1.00000000000000002e306 < (*.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 22.1%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 25.8

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

   5. Applied rewrites 25.8%

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

   7. Applied rewrites 33.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 37.0%

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

   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*)))) < 1.00000000000000002e306

   1. Initial program 97.6%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 19.8

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 8.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-+r+ N/A

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

   8. Applied rewrites 8.6%

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

   9. Taylor expanded in U* around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. times-frac N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 36.8

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

   11. Applied rewrites 36.8%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot U\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 10^{+306}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot U\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n}{Om}\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ t_4 := \sqrt{\mathsf{fma}\left(\left(\frac{l\_m}{Om} \cdot U\right) \cdot \left(l\_m \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 10^{+306}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(l\_m \cdot l\_m\right) \cdot n\right) \cdot U*\right) \cdot U}{Om \cdot Om} \cdot \left(n \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3
         (* (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n))) t_1))
        (t_4
         (sqrt
          (fma (* (* (/ l_m Om) U) (* l_m n)) -4.0 (* (* (* t n) U) 2.0)))))
   (if (<= t_3 0.0)
     t_4
     (if (<= t_3 1e+306)
       (sqrt (* (fma -2.0 t_2 t) t_1))
       (if (<= t_3 INFINITY)
         t_4
         (sqrt (* (/ (* (* (* (* l_m l_m) n) U*) U) (* Om Om)) (* n 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 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double t_4 = sqrt(fma((((l_m / Om) * U) * (l_m * n)), -4.0, (((t * n) * U) * 2.0)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = t_4;
	} else if (t_3 <= 1e+306) {
		tmp = sqrt((fma(-2.0, t_2, t) * t_1));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = sqrt(((((((l_m * l_m) * n) * U_42_) * U) / (Om * Om)) * (n * 2.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(l_m * l_m) / Om)
	t_3 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	t_4 = sqrt(fma(Float64(Float64(Float64(l_m / Om) * U) * Float64(l_m * n)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = t_4;
	elseif (t_3 <= 1e+306)
		tmp = sqrt(Float64(fma(-2.0, t_2, t) * t_1));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(l_m * l_m) * n) * U_42_) * U) / Float64(Om * Om)) * Float64(n * 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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * U), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], t$95$4, If[LessEqual[t$95$3, 1e+306], N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[Sqrt[N[(N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] * U$42$), $MachinePrecision] * U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(n * 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 or 1.00000000000000002e306 < (*.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 22.1%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 25.8

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

   5. Applied rewrites 25.8%

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

   7. Applied rewrites 33.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 37.0%

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

   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*)))) < 1.00000000000000002e306

   1. Initial program 97.6%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 19.8

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 8.3%

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

   7. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 36.5

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

   9. Applied rewrites 36.5%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot U\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 10^{+306}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot U\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U*\right) \cdot U}{Om \cdot Om} \cdot \left(n \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m}{Om} \cdot n\\ t_2 := U \cdot \left(n \cdot 2\right)\\ t_3 := \frac{l\_m \cdot l\_m}{Om}\\ t_4 := \left(\left(t - t\_3 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_2\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, \left(U* - U\right) \cdot t\_1, \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot \frac{l\_m}{Om}, t\_1, \mathsf{fma}\left(-2, t\_3, t\right)\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (/ l_m Om) n))
        (t_2 (* U (* n 2.0)))
        (t_3 (/ (* l_m l_m) Om))
        (t_4
         (*
          (- (- t (* t_3 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_2)))
   (if (<= t_4 0.0)
     (sqrt
      (*
       (* (fma (/ l_m Om) (* (- U* U) t_1) (fma (* (/ l_m Om) l_m) -2.0 t)) U)
       (* n 2.0)))
     (if (<= t_4 5e+306)
       (sqrt (* (fma (* (- U* U) (/ l_m Om)) t_1 (fma -2.0 t_3 t)) t_2))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (- (* (/ (- U* U) Om) (/ n Om)) (/ 2.0 Om)) (* U n))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m / Om) * n;
	double t_2 = U * (n * 2.0);
	double t_3 = (l_m * l_m) / Om;
	double t_4 = ((t - (t_3 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_2;
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((fma((l_m / Om), ((U_42_ - U) * t_1), fma(((l_m / Om) * l_m), -2.0, t)) * U) * (n * 2.0)));
	} else if (t_4 <= 5e+306) {
		tmp = sqrt((fma(((U_42_ - U) * (l_m / Om)), t_1, fma(-2.0, t_3, t)) * t_2));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt((((((U_42_ - U) / Om) * (n / Om)) - (2.0 / Om)) * (U * n)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m / Om) * n)
	t_2 = Float64(U * Float64(n * 2.0))
	t_3 = Float64(Float64(l_m * l_m) / Om)
	t_4 = Float64(Float64(Float64(t - Float64(t_3 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_2)
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(fma(Float64(l_m / Om), Float64(Float64(U_42_ - U) * t_1), fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)) * U) * Float64(n * 2.0)));
	elseif (t_4 <= 5e+306)
		tmp = sqrt(Float64(fma(Float64(Float64(U_42_ - U) * Float64(l_m / Om)), t_1, fma(-2.0, t_3, t)) * t_2));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / Om)) - Float64(2.0 / Om)) * Float64(U * n))));
	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[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t - N[(t$95$3 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 5e+306], N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(-2.0 * t$95$3 + t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $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 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 17.1

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 48.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-+r+ N/A

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

   8. Applied rewrites 54.5%

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

   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*)))) < 4.99999999999999993e306

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 98.3

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in U* around 0

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

      1. lower--.f64 98.3

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

   7. Applied rewrites 98.3%

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

   if 4.99999999999999993e306 < (*.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 16.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 18.2

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 18.2%

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

   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

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

   7. Applied rewrites 20.0%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{l\_m}{Om} \cdot U\right) \cdot \left(l\_m \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot \frac{l\_m}{Om}, \frac{l\_m}{Om} \cdot n, \mathsf{fma}\left(-2, t\_2, t\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3
         (*
          (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_3 0.0)
     (sqrt (fma (* (* (/ l_m Om) U) (* l_m n)) -4.0 (* (* (* t n) U) 2.0)))
     (if (<= t_3 5e+306)
       (sqrt
        (*
         (fma (* (- U* U) (/ l_m Om)) (* (/ l_m Om) n) (fma -2.0 t_2 t))
         t_1))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (- (* (/ (- U* U) Om) (/ n Om)) (/ 2.0 Om)) (* U n))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(fma((((l_m / Om) * U) * (l_m * n)), -4.0, (((t * n) * U) * 2.0)));
	} else if (t_3 <= 5e+306) {
		tmp = sqrt((fma(((U_42_ - U) * (l_m / Om)), ((l_m / Om) * n), fma(-2.0, t_2, t)) * t_1));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt((((((U_42_ - U) / Om) * (n / Om)) - (2.0 / Om)) * (U * n)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(l_m * l_m) / Om)
	t_3 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(fma(Float64(Float64(Float64(l_m / Om) * U) * Float64(l_m * n)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	elseif (t_3 <= 5e+306)
		tmp = sqrt(Float64(fma(Float64(Float64(U_42_ - U) * Float64(l_m / Om)), Float64(Float64(l_m / Om) * n), fma(-2.0, t_2, t)) * t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / Om)) - Float64(2.0 / Om)) * Float64(U * n))));
	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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * U), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 5e+306], N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * t$95$2 + t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $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 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 Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 49.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.9%

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

   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*)))) < 4.99999999999999993e306

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 98.3

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in U* around 0

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

      1. lower--.f64 98.3

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

   7. Applied rewrites 98.3%

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

   if 4.99999999999999993e306 < (*.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 16.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 18.2

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 18.2%

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

   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

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

   7. Applied rewrites 20.0%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot U\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{l\_m}{Om} \cdot U\right) \cdot \left(l\_m \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U* \cdot l\_m}{Om}, \frac{l\_m}{Om} \cdot n, \mathsf{fma}\left(-2, t\_2, t\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3
         (*
          (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_3 0.0)
     (sqrt (fma (* (* (/ l_m Om) U) (* l_m n)) -4.0 (* (* (* t n) U) 2.0)))
     (if (<= t_3 5e+306)
       (sqrt (* (fma (/ (* U* l_m) Om) (* (/ l_m Om) n) (fma -2.0 t_2 t)) t_1))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (- (* (/ (- U* U) Om) (/ n Om)) (/ 2.0 Om)) (* U n))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(fma((((l_m / Om) * U) * (l_m * n)), -4.0, (((t * n) * U) * 2.0)));
	} else if (t_3 <= 5e+306) {
		tmp = sqrt((fma(((U_42_ * l_m) / Om), ((l_m / Om) * n), fma(-2.0, t_2, t)) * t_1));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt((((((U_42_ - U) / Om) * (n / Om)) - (2.0 / Om)) * (U * n)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(l_m * l_m) / Om)
	t_3 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(fma(Float64(Float64(Float64(l_m / Om) * U) * Float64(l_m * n)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	elseif (t_3 <= 5e+306)
		tmp = sqrt(Float64(fma(Float64(Float64(U_42_ * l_m) / Om), Float64(Float64(l_m / Om) * n), fma(-2.0, t_2, t)) * t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / Om)) - Float64(2.0 / Om)) * Float64(U * n))));
	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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * U), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 5e+306], N[Sqrt[N[(N[(N[(N[(U$42$ * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * t$95$2 + t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $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 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 Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 49.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.9%

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

   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*)))) < 4.99999999999999993e306

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 98.3

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 97.3

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

   7. Applied rewrites 97.3%

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

   if 4.99999999999999993e306 < (*.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 16.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 18.2

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 18.2%

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

   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

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

   7. Applied rewrites 20.0%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot U\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U* \cdot \ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{l\_m}{Om} \cdot U\right) \cdot \left(l\_m \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot 2 - \left(\left(U* - U\right) \cdot \frac{l\_m}{Om}\right) \cdot \left(l\_m \cdot n\right)}{Om}\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2
         (*
          (-
           (- t (* (/ (* l_m l_m) Om) 2.0))
           (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_2 0.0)
     (sqrt (fma (* (* (/ l_m Om) U) (* l_m n)) -4.0 (* (* (* t n) U) 2.0)))
     (if (<= t_2 5e+306)
       (sqrt
        (*
         (-
          t
          (/ (- (* (* l_m l_m) 2.0) (* (* (- U* U) (/ l_m Om)) (* l_m n))) Om))
         t_1))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (- (* (/ (- U* U) Om) (/ n Om)) (/ 2.0 Om)) (* U n))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(fma((((l_m / Om) * U) * (l_m * n)), -4.0, (((t * n) * U) * 2.0)));
	} else if (t_2 <= 5e+306) {
		tmp = sqrt(((t - ((((l_m * l_m) * 2.0) - (((U_42_ - U) * (l_m / Om)) * (l_m * n))) / Om)) * t_1));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt((((((U_42_ - U) / Om) * (n / Om)) - (2.0 / Om)) * (U * n)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(Float64(t - Float64(Float64(Float64(l_m * l_m) / Om) * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(fma(Float64(Float64(Float64(l_m / Om) * U) * Float64(l_m * n)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	elseif (t_2 <= 5e+306)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(Float64(Float64(l_m * l_m) * 2.0) - Float64(Float64(Float64(U_42_ - U) * Float64(l_m / Om)) * Float64(l_m * n))) / Om)) * t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / Om)) - Float64(2.0 / Om)) * Float64(U * n))));
	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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * U), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+306], N[Sqrt[N[(N[(t - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $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 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 Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 49.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.9%

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

   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*)))) < 4.99999999999999993e306

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 97.6

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 87.0%

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

   6. Applied rewrites 96.4%

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

   if 4.99999999999999993e306 < (*.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 16.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 18.2

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 18.2%

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

   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

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

   7. Applied rewrites 20.0%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot U\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot 2 - \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\ell \cdot n\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{l\_m}{Om} \cdot U\right) \cdot \left(l\_m \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot 2 - \left(\left(U* - U\right) \cdot \frac{l\_m}{Om}\right) \cdot \left(l\_m \cdot n\right)}{Om}\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left(U* - U\right) \cdot n}{Om \cdot Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)} \cdot \left(\sqrt{2} \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2
         (*
          (-
           (- t (* (/ (* l_m l_m) Om) 2.0))
           (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_2 0.0)
     (sqrt (fma (* (* (/ l_m Om) U) (* l_m n)) -4.0 (* (* (* t n) U) 2.0)))
     (if (<= t_2 5e+306)
       (sqrt
        (*
         (-
          t
          (/ (- (* (* l_m l_m) 2.0) (* (* (- U* U) (/ l_m Om)) (* l_m n))) Om))
         t_1))
       (*
        (sqrt (* (- (/ (* (- U* U) n) (* Om Om)) (/ 2.0 Om)) (* U n)))
        (* (sqrt 2.0) 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 = U * (n * 2.0);
	double t_2 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(fma((((l_m / Om) * U) * (l_m * n)), -4.0, (((t * n) * U) * 2.0)));
	} else if (t_2 <= 5e+306) {
		tmp = sqrt(((t - ((((l_m * l_m) * 2.0) - (((U_42_ - U) * (l_m / Om)) * (l_m * n))) / Om)) * t_1));
	} else {
		tmp = sqrt((((((U_42_ - U) * n) / (Om * Om)) - (2.0 / Om)) * (U * n))) * (sqrt(2.0) * l_m);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(Float64(t - Float64(Float64(Float64(l_m * l_m) / Om) * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(fma(Float64(Float64(Float64(l_m / Om) * U) * Float64(l_m * n)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	elseif (t_2 <= 5e+306)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(Float64(Float64(l_m * l_m) * 2.0) - Float64(Float64(Float64(U_42_ - U) * Float64(l_m / Om)) * Float64(l_m * n))) / Om)) * t_1));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) * n) / Float64(Om * Om)) - Float64(2.0 / Om)) * Float64(U * n))) * Float64(sqrt(2.0) * 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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * U), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+306], N[Sqrt[N[(N[(t - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * n), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $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 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 Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 49.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.9%

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

   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*)))) < 4.99999999999999993e306

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 97.6

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 87.0%

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

   6. Applied rewrites 96.4%

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

   if 4.99999999999999993e306 < (*.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 16.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 29.8

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 29.4%

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

   6. Applied rewrites 26.8%

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

   7. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   8. 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

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

   9. Applied rewrites 18.5%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot U\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot 2 - \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\ell \cdot n\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left(U* - U\right) \cdot n}{Om \cdot Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)} \cdot \left(\sqrt{2} \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{l\_m}{Om} \cdot U\right) \cdot \left(l\_m \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\frac{U* - U}{Om} \cdot l\_m, n, -2 \cdot l\_m\right), t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left(U* - U\right) \cdot n}{Om \cdot Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)} \cdot \left(\sqrt{2} \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (-
           (- t (* (/ (* l_m l_m) Om) 2.0))
           (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          (* U (* n 2.0)))))
   (if (<= t_1 0.0)
     (sqrt (fma (* (* (/ l_m Om) U) (* l_m n)) -4.0 (* (* (* t n) U) 2.0)))
     (if (<= t_1 INFINITY)
       (sqrt
        (*
         (* (* U 2.0) n)
         (fma (/ l_m Om) (fma (* (/ (- U* U) Om) l_m) n (* -2.0 l_m)) t)))
       (*
        (sqrt (* (- (/ (* (- U* U) n) (* Om Om)) (/ 2.0 Om)) (* U n)))
        (* (sqrt 2.0) 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 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * (U * (n * 2.0));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(fma((((l_m / Om) * U) * (l_m * n)), -4.0, (((t * n) * U) * 2.0)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((((U * 2.0) * n) * fma((l_m / Om), fma((((U_42_ - U) / Om) * l_m), n, (-2.0 * l_m)), t)));
	} else {
		tmp = sqrt((((((U_42_ - U) * n) / (Om * Om)) - (2.0 / Om)) * (U * n))) * (sqrt(2.0) * l_m);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(Float64(t - Float64(Float64(Float64(l_m * l_m) / Om) * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * Float64(U * Float64(n * 2.0)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(fma(Float64(Float64(Float64(l_m / Om) * U) * Float64(l_m * n)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(U * 2.0) * n) * fma(Float64(l_m / Om), fma(Float64(Float64(Float64(U_42_ - U) / Om) * l_m), n, Float64(-2.0 * l_m)), t)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) * n) / Float64(Om * Om)) - Float64(2.0 / Om)) * Float64(U * n))) * Float64(sqrt(2.0) * 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[(N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * U), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * n + N[(-2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * n), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $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 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 Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 49.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.9%

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

   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 66.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 70.6

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 64.3%

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

   6. Applied rewrites 66.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-+r+ N/A

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

   8. Applied rewrites 67.2%

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

   10. Applied rewrites 68.7%

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

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 19.8

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 8.3%

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

   7. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   8. 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

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

   9. Applied rewrites 24.3%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot U\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\frac{U* - U}{Om} \cdot \ell, n, -2 \cdot \ell\right), t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left(U* - U\right) \cdot n}{Om \cdot Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)} \cdot \left(\sqrt{2} \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \mathsf{fma}\left(-2, t\_2, t\right)\\ t_4 := \sqrt{\left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t\_3 \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{t\_3 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{U* \cdot U} \cdot \frac{\sqrt{2} \cdot l\_m}{Om}\right) \cdot n\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3 (fma -2.0 t_2 t))
        (t_4
         (sqrt
          (*
           (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
           t_1))))
   (if (<= t_4 0.0)
     (sqrt (* (* (* t_3 n) U) 2.0))
     (if (<= t_4 5e+152)
       (sqrt (* t_3 t_1))
       (* (* (sqrt (* U* U)) (/ (* (sqrt 2.0) l_m) Om)) n)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = fma(-2.0, t_2, t);
	double t_4 = sqrt((((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((((t_3 * n) * U) * 2.0));
	} else if (t_4 <= 5e+152) {
		tmp = sqrt((t_3 * t_1));
	} else {
		tmp = (sqrt((U_42_ * U)) * ((sqrt(2.0) * l_m) / Om)) * n;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(l_m * l_m) / Om)
	t_3 = fma(-2.0, t_2, t)
	t_4 = sqrt(Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t_3 * n) * U) * 2.0));
	elseif (t_4 <= 5e+152)
		tmp = sqrt(Float64(t_3 * t_1));
	else
		tmp = Float64(Float64(sqrt(Float64(U_42_ * U)) * Float64(Float64(sqrt(2.0) * l_m) / Om)) * n);
	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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * t$95$2 + t), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(N[(t$95$3 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 5e+152], N[Sqrt[N[(t$95$3 * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * n), $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 15.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.3

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

   5. Applied rewrites 50.3%

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

   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*))))) < 5e152

   1. Initial program 97.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if 5e152 < (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 17.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 20.8

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 20.8%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 9.6%

      \[\leadsto \color{blue}{n \cdot \mathsf{fma}\left(0.5, \left(\frac{Om \cdot \sqrt{2}}{\ell} \cdot \frac{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}{n}\right) \cdot \sqrt{\frac{U}{U* - U}}, \frac{\ell \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot \left(U* - U\right)}\right)} \]

   8. Taylor expanded in U* around inf

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

   10. Applied rewrites 24.1%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{U* \cdot U} \cdot \frac{\sqrt{2} \cdot \ell}{Om}\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{l\_m}{Om} \cdot U\right) \cdot \left(l\_m \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(l\_m \cdot l\_m, \frac{U*}{Om} \cdot \frac{n}{Om}, t\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3
         (*
          (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_3 0.0)
     (sqrt (fma (* (* (/ l_m Om) U) (* l_m n)) -4.0 (* (* (* t n) U) 2.0)))
     (if (<= t_3 5e+14)
       (sqrt (* (fma -2.0 t_2 t) t_1))
       (sqrt (* (fma (* l_m l_m) (* (/ U* Om) (/ n Om)) t) t_1))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(fma((((l_m / Om) * U) * (l_m * n)), -4.0, (((t * n) * U) * 2.0)));
	} else if (t_3 <= 5e+14) {
		tmp = sqrt((fma(-2.0, t_2, t) * t_1));
	} else {
		tmp = sqrt((fma((l_m * l_m), ((U_42_ / Om) * (n / Om)), t) * t_1));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(l_m * l_m) / Om)
	t_3 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(fma(Float64(Float64(Float64(l_m / Om) * U) * Float64(l_m * n)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	elseif (t_3 <= 5e+14)
		tmp = sqrt(Float64(fma(-2.0, t_2, t) * t_1));
	else
		tmp = sqrt(Float64(fma(Float64(l_m * l_m), Float64(Float64(U_42_ / Om) * Float64(n / Om)), t) * t_1));
	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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * U), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 5e+14], N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(U$42$ / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * t$95$1), $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 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 Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 49.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.9%

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

   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*)))) < 5e14

   1. Initial program 98.9%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if 5e14 < (*.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 40.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-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. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 42.4

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 42.4%

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

   5. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 46.5

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

   7. Applied rewrites 46.5%

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

   8. Taylor expanded in U* around inf

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

   10. Applied rewrites 45.6%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot U\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \ell, \frac{U*}{Om} \cdot \frac{n}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \mathsf{fma}\left(-2, t\_2, t\right)\\ t_4 := \left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(t\_3 \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{t\_3 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(l\_m \cdot l\_m\right) \cdot n\right) \cdot U*\right) \cdot U}{Om \cdot Om} \cdot \left(n \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3 (fma -2.0 t_2 t))
        (t_4
         (*
          (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_4 0.0)
     (sqrt (* (* t_3 U) (* n 2.0)))
     (if (<= t_4 4e+305)
       (sqrt (* t_3 t_1))
       (sqrt (* (/ (* (* (* (* l_m l_m) n) U*) U) (* Om Om)) (* n 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 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = fma(-2.0, t_2, t);
	double t_4 = ((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((t_3 * U) * (n * 2.0)));
	} else if (t_4 <= 4e+305) {
		tmp = sqrt((t_3 * t_1));
	} else {
		tmp = sqrt(((((((l_m * l_m) * n) * U_42_) * U) / (Om * Om)) * (n * 2.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(l_m * l_m) / Om)
	t_3 = fma(-2.0, t_2, t)
	t_4 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(t_3 * U) * Float64(n * 2.0)));
	elseif (t_4 <= 4e+305)
		tmp = sqrt(Float64(t_3 * t_1));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(l_m * l_m) * n) * U_42_) * U) / Float64(Om * Om)) * Float64(n * 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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * t$95$2 + t), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(t$95$3 * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 4e+305], N[Sqrt[N[(t$95$3 * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] * U$42$), $MachinePrecision] * U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(n * 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 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 17.1

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 48.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+r+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-+r+ N/A

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

   8. Applied rewrites 54.5%

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

   9. Taylor expanded in Om around inf

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 47.2

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

   11. Applied rewrites 47.2%

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

   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*)))) < 3.9999999999999998e305

   1. Initial program 97.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if 3.9999999999999998e305 < (*.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 18.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 31.0

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 29.8%

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

   6. Applied rewrites 28.1%

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

   7. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)}{{Om}^{2}}} \cdot \left(2 \cdot n\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)}}{{Om}^{2}} \cdot \left(2 \cdot n\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{U \cdot \color{blue}{\left(U* \cdot \left({\ell}^{2} \cdot n\right)\right)}}{{Om}^{2}} \cdot \left(2 \cdot n\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{U \cdot \left(U* \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right)}{{Om}^{2}} \cdot \left(2 \cdot n\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{U \cdot \left(U* \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right)}{{Om}^{2}} \cdot \left(2 \cdot n\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{U \cdot \left(U* \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right)}{{Om}^{2}} \cdot \left(2 \cdot n\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{U \cdot \left(U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{\color{blue}{Om \cdot Om}} \cdot \left(2 \cdot n\right)} \]

      8. lower-*.f64 29.1

         \[\leadsto \sqrt{\frac{U \cdot \left(U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{\color{blue}{Om \cdot Om}} \cdot \left(2 \cdot n\right)} \]

   9. Applied rewrites 29.1%

      \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left(U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om \cdot Om}} \cdot \left(2 \cdot n\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U*\right) \cdot U}{Om \cdot Om} \cdot \left(n \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \mathsf{fma}\left(-2, t\_2, t\right)\\ t_4 := \left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(t\_3 \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;t\_4 \leq 10^{+306}:\\ \;\;\;\;\sqrt{t\_3 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U}{Om \cdot Om} \cdot \left(\left(n \cdot n\right) \cdot \left(U* \cdot \left(l\_m \cdot l\_m\right)\right)\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3 (fma -2.0 t_2 t))
        (t_4
         (*
          (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_4 0.0)
     (sqrt (* (* t_3 U) (* n 2.0)))
     (if (<= t_4 1e+306)
       (sqrt (* t_3 t_1))
       (sqrt (* (* (/ U (* Om Om)) (* (* n n) (* U* (* l_m l_m)))) 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 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = fma(-2.0, t_2, t);
	double t_4 = ((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((t_3 * U) * (n * 2.0)));
	} else if (t_4 <= 1e+306) {
		tmp = sqrt((t_3 * t_1));
	} else {
		tmp = sqrt((((U / (Om * Om)) * ((n * n) * (U_42_ * (l_m * l_m)))) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(l_m * l_m) / Om)
	t_3 = fma(-2.0, t_2, t)
	t_4 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(t_3 * U) * Float64(n * 2.0)));
	elseif (t_4 <= 1e+306)
		tmp = sqrt(Float64(t_3 * t_1));
	else
		tmp = sqrt(Float64(Float64(Float64(U / Float64(Om * Om)) * Float64(Float64(n * n) * Float64(U_42_ * Float64(l_m * l_m)))) * 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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * t$95$2 + t), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(t$95$3 * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 1e+306], N[Sqrt[N[(t$95$3 * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(U / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(N[(n * n), $MachinePrecision] * N[(U$42$ * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $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 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. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. associate--l+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{Om}}\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \ell} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{-2} \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower--.f64 17.1

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, \color{blue}{t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

   4. Applied rewrites 19.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}} \]

   6. Applied rewrites 48.9%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-\left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{2} \cdot n, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot -2 + t\right)}\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \left(\frac{\ell}{Om} \cdot \ell\right) \cdot -2\right) + t\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \left(\color{blue}{\frac{\ell}{Om}} \cdot \ell\right) \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{\frac{\ell \cdot \ell}{Om}} \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{\frac{\ell \cdot \ell}{Om}} \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{-2 \cdot \frac{\ell \cdot \ell}{Om}}\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      10. associate-+r+ N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \left(-2 \cdot \frac{\ell \cdot \ell}{Om} + t\right)\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

   8. Applied rewrites 54.5%

      \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

   9. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

       3. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

       4. unpow2 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

       5. lower-*.f64 47.2

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

   11. Applied rewrites 47.2%

       \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

   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*)))) < 1.00000000000000002e306

   1. Initial program 97.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right)} \]

      4. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)} \]

      5. lower-*.f64 87.8

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)} \]

   5. Applied rewrites 87.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}} \]

   if 1.00000000000000002e306 < (*.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 17.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}} \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}} \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right) \cdot U}}{{Om}^{2}} \cdot 2} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right) \cdot \frac{U}{{Om}^{2}}\right)} \cdot 2} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right) \cdot \frac{U}{{Om}^{2}}\right)} \cdot 2} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(U* \cdot {\ell}^{2}\right) \cdot {n}^{2}\right)} \cdot \frac{U}{{Om}^{2}}\right) \cdot 2} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(U* \cdot {\ell}^{2}\right) \cdot {n}^{2}\right)} \cdot \frac{U}{{Om}^{2}}\right) \cdot 2} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\left(U* \cdot {\ell}^{2}\right)} \cdot {n}^{2}\right) \cdot \frac{U}{{Om}^{2}}\right) \cdot 2} \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(\left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot {n}^{2}\right) \cdot \frac{U}{{Om}^{2}}\right) \cdot 2} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot {n}^{2}\right) \cdot \frac{U}{{Om}^{2}}\right) \cdot 2} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\left(n \cdot n\right)}\right) \cdot \frac{U}{{Om}^{2}}\right) \cdot 2} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\left(n \cdot n\right)}\right) \cdot \frac{U}{{Om}^{2}}\right) \cdot 2} \]

      13. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)\right) \cdot \color{blue}{\frac{U}{{Om}^{2}}}\right) \cdot 2} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)\right) \cdot \frac{U}{\color{blue}{Om \cdot Om}}\right) \cdot 2} \]

      15. lower-*.f64 24.6

          \[\leadsto \sqrt{\left(\left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)\right) \cdot \frac{U}{\color{blue}{Om \cdot Om}}\right) \cdot 2} \]

   5. Applied rewrites 24.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)\right) \cdot \frac{U}{Om \cdot Om}\right) \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 10^{+306}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U}{Om \cdot Om} \cdot \left(\left(n \cdot n\right) \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_2 \leq 10^{+306}:\\ \;\;\;\;\sqrt{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left(l\_m \cdot l\_m\right) \cdot n}{Om} \cdot U\right) \cdot -4}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2
         (*
          (-
           (- t (* (/ (* l_m l_m) Om) 2.0))
           (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_2 0.0)
     (sqrt (* (* (* t n) U) 2.0))
     (if (<= t_2 1e+306)
       (sqrt (* t t_1))
       (sqrt (* (* (/ (* (* l_m l_m) n) Om) U) -4.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else if (t_2 <= 1e+306) {
		tmp = sqrt((t * t_1));
	} else {
		tmp = sqrt((((((l_m * l_m) * n) / Om) * U) * -4.0));
	}
	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 = u * (n * 2.0d0)
    t_2 = ((t - (((l_m * l_m) / om) * 2.0d0)) - ((u - u_42) * (((l_m / om) ** 2.0d0) * n))) * t_1
    if (t_2 <= 0.0d0) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else if (t_2 <= 1d+306) then
        tmp = sqrt((t * t_1))
    else
        tmp = sqrt((((((l_m * l_m) * n) / om) * u) * (-4.0d0)))
    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 = U * (n * 2.0);
	double t_2 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (Math.pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else if (t_2 <= 1e+306) {
		tmp = Math.sqrt((t * t_1));
	} else {
		tmp = Math.sqrt((((((l_m * l_m) * n) / Om) * U) * -4.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (n * 2.0)
	t_2 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (math.pow((l_m / Om), 2.0) * n))) * t_1
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	elif t_2 <= 1e+306:
		tmp = math.sqrt((t * t_1))
	else:
		tmp = math.sqrt((((((l_m * l_m) * n) / Om) * U) * -4.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(Float64(t - Float64(Float64(Float64(l_m * l_m) / Om) * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	elseif (t_2 <= 1e+306)
		tmp = sqrt(Float64(t * t_1));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l_m * l_m) * n) / Om) * U) * -4.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U * (n * 2.0);
	t_2 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (((l_m / Om) ^ 2.0) * n))) * t_1;
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt((((t * n) * U) * 2.0));
	elseif (t_2 <= 1e+306)
		tmp = sqrt((t * t_1));
	else
		tmp = sqrt((((((l_m * l_m) * n) / Om) * U) * -4.0));
	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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+306], N[Sqrt[N[(t * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * U), $MachinePrecision] * -4.0), $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 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. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 46.5

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 46.5%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]

   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*)))) < 1.00000000000000002e306

   1. Initial program 97.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 62.6

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 62.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.2%

      \[\leadsto \sqrt{t \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)}} \]

   if 1.00000000000000002e306 < (*.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 17.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} \cdot -4} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left(n \cdot {\ell}^{2}\right)} \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left(n \cdot {\ell}^{2}\right)} \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      14. lower-*.f64 14.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}, -4, \left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2\right)} \]

   5. Applied rewrites 14.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}, -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \sqrt{-4 \cdot \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.3%

      \[\leadsto \sqrt{-4 \cdot \color{blue}{\left(U \cdot \frac{\left(\ell \cdot \ell\right) \cdot n}{Om}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 10^{+306}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om} \cdot U\right) \cdot -4}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 41.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;t\_2 \leq 10^{+306}:\\ \;\;\;\;\sqrt{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(l\_m \cdot l\_m\right) \cdot n\right) \cdot U}{Om} \cdot -4}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2
         (*
          (-
           (- t (* (/ (* l_m l_m) Om) 2.0))
           (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_2 0.0)
     (sqrt (* (* (* t n) U) 2.0))
     (if (<= t_2 1e+306)
       (sqrt (* t t_1))
       (sqrt (* (/ (* (* (* l_m l_m) n) U) Om) -4.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else if (t_2 <= 1e+306) {
		tmp = sqrt((t * t_1));
	} else {
		tmp = sqrt((((((l_m * l_m) * n) * U) / Om) * -4.0));
	}
	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 = u * (n * 2.0d0)
    t_2 = ((t - (((l_m * l_m) / om) * 2.0d0)) - ((u - u_42) * (((l_m / om) ** 2.0d0) * n))) * t_1
    if (t_2 <= 0.0d0) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else if (t_2 <= 1d+306) then
        tmp = sqrt((t * t_1))
    else
        tmp = sqrt((((((l_m * l_m) * n) * u) / om) * (-4.0d0)))
    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 = U * (n * 2.0);
	double t_2 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (Math.pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else if (t_2 <= 1e+306) {
		tmp = Math.sqrt((t * t_1));
	} else {
		tmp = Math.sqrt((((((l_m * l_m) * n) * U) / Om) * -4.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (n * 2.0)
	t_2 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (math.pow((l_m / Om), 2.0) * n))) * t_1
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	elif t_2 <= 1e+306:
		tmp = math.sqrt((t * t_1))
	else:
		tmp = math.sqrt((((((l_m * l_m) * n) * U) / Om) * -4.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(Float64(t - Float64(Float64(Float64(l_m * l_m) / Om) * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	elseif (t_2 <= 1e+306)
		tmp = sqrt(Float64(t * t_1));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l_m * l_m) * n) * U) / Om) * -4.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U * (n * 2.0);
	t_2 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (((l_m / Om) ^ 2.0) * n))) * t_1;
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt((((t * n) * U) * 2.0));
	elseif (t_2 <= 1e+306)
		tmp = sqrt((t * t_1));
	else
		tmp = sqrt((((((l_m * l_m) * n) * U) / Om) * -4.0));
	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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+306], N[Sqrt[N[(t * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $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 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. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 46.5

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 46.5%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]

   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*)))) < 1.00000000000000002e306

   1. Initial program 97.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 62.6

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 62.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.2%

      \[\leadsto \sqrt{t \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)}} \]

   if 1.00000000000000002e306 < (*.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 17.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} \cdot -4} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left(n \cdot {\ell}^{2}\right)} \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left(n \cdot {\ell}^{2}\right)} \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      14. lower-*.f64 14.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}, -4, \left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2\right)} \]

   5. Applied rewrites 14.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om}, -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.0%

      \[\leadsto \sqrt{\mathsf{fma}\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto \sqrt{-4 \cdot \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 13.5%

       \[\leadsto \sqrt{-4 \cdot \color{blue}{\frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 10^{+306}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om} \cdot -4}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ \mathbf{if}\;\sqrt{\left(\left(t - t\_1 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, t\_1, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\frac{U* - U}{Om} \cdot l\_m, n, -2 \cdot l\_m\right), t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om)))
   (if (<=
        (sqrt
         (*
          (- (- t (* t_1 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          (* U (* n 2.0))))
        0.0)
     (sqrt (* (* (* (fma -2.0 t_1 t) n) U) 2.0))
     (sqrt
      (*
       (* (* U 2.0) n)
       (fma (/ l_m Om) (fma (* (/ (- U* U) Om) l_m) n (* -2.0 l_m)) t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double tmp;
	if (sqrt((((t - (t_1 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * (U * (n * 2.0)))) <= 0.0) {
		tmp = sqrt((((fma(-2.0, t_1, t) * n) * U) * 2.0));
	} else {
		tmp = sqrt((((U * 2.0) * n) * fma((l_m / Om), fma((((U_42_ - U) / Om) * l_m), n, (-2.0 * l_m)), t)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(t - Float64(t_1 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * Float64(U * Float64(n * 2.0)))) <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, t_1, t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(U * 2.0) * n) * fma(Float64(l_m / Om), fma(Float64(Float64(Float64(U_42_ - U) / Om) * l_m), n, Float64(-2.0 * l_m)), t)));
	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[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[(N[(t - N[(t$95$1 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[Sqrt[N[(N[(N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * n + N[(-2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] + t), $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 15.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U\right) \cdot 2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U\right) \cdot 2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      13. lower-*.f64 50.3

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

   5. Applied rewrites 50.3%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}} \]

   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 54.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. associate--l+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{Om}}\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \ell} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{-2} \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower--.f64 61.4

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, \color{blue}{t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

   4. Applied rewrites 56.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}} \]

   6. Applied rewrites 55.5%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-\left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{2} \cdot n, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot -2 + t\right)}\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \left(\frac{\ell}{Om} \cdot \ell\right) \cdot -2\right) + t\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \left(\color{blue}{\frac{\ell}{Om}} \cdot \ell\right) \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{\frac{\ell \cdot \ell}{Om}} \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{\frac{\ell \cdot \ell}{Om}} \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{-2 \cdot \frac{\ell \cdot \ell}{Om}}\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      10. associate-+r+ N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \left(-2 \cdot \frac{\ell \cdot \ell}{Om} + t\right)\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

   8. Applied rewrites 57.2%

      \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

   10. Applied rewrites 63.6%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\ell \cdot \frac{U* - U}{Om}, n, -2 \cdot \ell\right), t\right) \cdot \left(\left(2 \cdot U\right) \cdot n\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\frac{U* - U}{Om} \cdot \ell, n, -2 \cdot \ell\right), t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 38.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;\sqrt{\left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1} \leq 0:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0))))
   (if (<=
        (sqrt
         (*
          (-
           (- t (* (/ (* l_m l_m) Om) 2.0))
           (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1))
        0.0)
     (sqrt (* (* t U) (* n 2.0)))
     (sqrt (* t t_1)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double tmp;
	if (sqrt((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1)) <= 0.0) {
		tmp = sqrt(((t * U) * (n * 2.0)));
	} else {
		tmp = sqrt((t * t_1));
	}
	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) :: tmp
    t_1 = u * (n * 2.0d0)
    if (sqrt((((t - (((l_m * l_m) / om) * 2.0d0)) - ((u - u_42) * (((l_m / om) ** 2.0d0) * n))) * t_1)) <= 0.0d0) then
        tmp = sqrt(((t * u) * (n * 2.0d0)))
    else
        tmp = sqrt((t * t_1))
    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 = U * (n * 2.0);
	double tmp;
	if (Math.sqrt((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (Math.pow((l_m / Om), 2.0) * n))) * t_1)) <= 0.0) {
		tmp = Math.sqrt(((t * U) * (n * 2.0)));
	} else {
		tmp = Math.sqrt((t * t_1));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (n * 2.0)
	tmp = 0
	if math.sqrt((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (math.pow((l_m / Om), 2.0) * n))) * t_1)) <= 0.0:
		tmp = math.sqrt(((t * U) * (n * 2.0)))
	else:
		tmp = math.sqrt((t * t_1))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(t - Float64(Float64(Float64(l_m * l_m) / Om) * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)) <= 0.0)
		tmp = sqrt(Float64(Float64(t * U) * Float64(n * 2.0)));
	else
		tmp = sqrt(Float64(t * t_1));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U * (n * 2.0);
	tmp = 0.0;
	if (sqrt((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (((l_m / Om) ^ 2.0) * n))) * t_1)) <= 0.0)
		tmp = sqrt(((t * U) * (n * 2.0)));
	else
		tmp = sqrt((t * t_1));
	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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[(N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], 0.0], N[Sqrt[N[(N[(t * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t * t$95$1), $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 15.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 47.5

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 47.5%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]

   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 54.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 34.4

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 34.4%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 38.9%

      \[\leadsto \sqrt{t \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 38.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;\left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0))))
   (if (<=
        (*
         (-
          (- t (* (/ (* l_m l_m) Om) 2.0))
          (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
         t_1)
        0.0)
     (sqrt (* (* (* t n) U) 2.0))
     (sqrt (* t t_1)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double tmp;
	if ((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1) <= 0.0) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((t * t_1));
	}
	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) :: tmp
    t_1 = u * (n * 2.0d0)
    if ((((t - (((l_m * l_m) / om) * 2.0d0)) - ((u - u_42) * (((l_m / om) ** 2.0d0) * n))) * t_1) <= 0.0d0) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((t * t_1))
    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 = U * (n * 2.0);
	double tmp;
	if ((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (Math.pow((l_m / Om), 2.0) * n))) * t_1) <= 0.0) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((t * t_1));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (n * 2.0)
	tmp = 0
	if (((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (math.pow((l_m / Om), 2.0) * n))) * t_1) <= 0.0:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((t * t_1))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(t - Float64(Float64(Float64(l_m * l_m) / Om) * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1) <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(t * t_1));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U * (n * 2.0);
	tmp = 0.0;
	if ((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (((l_m / Om) ^ 2.0) * n))) * t_1) <= 0.0)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((t * t_1));
	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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 0.0], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t * t$95$1), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 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 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. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 46.5

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 46.5%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]

   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*))))

   1. Initial program 54.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 34.4

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 34.4%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.0%

      \[\leadsto \sqrt{t \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 47.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;l\_m \leq 2.9 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\frac{l\_m}{Om} \cdot l\_m\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5e-181)
   (sqrt (* (* t U) (* n 2.0)))
   (if (<= l_m 2.9e+140)
     (sqrt (* (* (* (fma -2.0 (/ (* l_m l_m) Om) t) n) U) 2.0))
     (sqrt (* (* -2.0 (* (/ l_m Om) l_m)) (* (* U 2.0) 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 (l_m <= 5e-181) {
		tmp = sqrt(((t * U) * (n * 2.0)));
	} else if (l_m <= 2.9e+140) {
		tmp = sqrt((((fma(-2.0, ((l_m * l_m) / Om), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(((-2.0 * ((l_m / Om) * l_m)) * ((U * 2.0) * n)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 5e-181)
		tmp = sqrt(Float64(Float64(t * U) * Float64(n * 2.0)));
	elseif (l_m <= 2.9e+140)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(Float64(l_m * l_m) / Om), t) * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(l_m / Om) * l_m)) * Float64(Float64(U * 2.0) * n)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 5e-181], N[Sqrt[N[(N[(t * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 2.9e+140], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-2.0 * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.0000000000000001e-181

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 38.9

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 38.9%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.5%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]

   if 5.0000000000000001e-181 < l < 2.8999999999999999e140

   1. Initial program 58.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U\right) \cdot 2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U\right) \cdot 2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      13. lower-*.f64 51.6

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

   5. Applied rewrites 51.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}} \]

   if 2.8999999999999999e140 < l

   1. Initial program 19.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      8. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{n}} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   4. Applied rewrites 8.2%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}} \]

   5. Taylor expanded in Om around 0

      \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\frac{-2 \cdot \left(Om \cdot {\ell}^{2}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\frac{-2 \cdot \left(Om \cdot {\ell}^{2}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}}} \]

      2. lower--.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{\color{blue}{-2 \cdot \left(Om \cdot {\ell}^{2}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}}{{Om}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{\color{blue}{-2 \cdot \left(Om \cdot {\ell}^{2}\right)} - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \color{blue}{\left(Om \cdot {\ell}^{2}\right)} - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}} \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \color{blue}{\left({\ell}^{2} \cdot n\right) \cdot \left(U - U*\right)}}{{Om}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \color{blue}{\left({\ell}^{2} \cdot n\right) \cdot \left(U - U*\right)}}{{Om}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot \left(U - U*\right)}{{Om}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(U - U*\right)}{{Om}^{2}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(U - U*\right)}{{Om}^{2}}} \]

      12. lower--.f64 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}} \]

      14. lower-*.f64 7.3

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}} \]

   7. Applied rewrites 7.3%

      \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U - U*\right)}{Om \cdot Om}}} \]

   8. Taylor expanded in Om around inf

      \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \left(-2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 8.5%

       \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \left(-2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)} \]

   12. Applied rewrites 30.9%

       \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot -2\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 47.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.25 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\frac{l\_m}{Om} \cdot l\_m\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.25e+153)
   (sqrt (* (* (fma -2.0 (/ (* l_m l_m) Om) t) U) (* n 2.0)))
   (sqrt (* (* -2.0 (* (/ l_m Om) l_m)) (* (* U 2.0) 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 (l_m <= 1.25e+153) {
		tmp = sqrt(((fma(-2.0, ((l_m * l_m) / Om), t) * U) * (n * 2.0)));
	} else {
		tmp = sqrt(((-2.0 * ((l_m / Om) * l_m)) * ((U * 2.0) * n)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.25e+153)
		tmp = sqrt(Float64(Float64(fma(-2.0, Float64(Float64(l_m * l_m) / Om), t) * U) * Float64(n * 2.0)));
	else
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(l_m / Om) * l_m)) * Float64(Float64(U * 2.0) * n)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.25e+153], N[Sqrt[N[(N[(N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-2.0 * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.25000000000000005e153

   1. Initial program 52.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. associate--l+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{Om}}\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \ell} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{-2} \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower--.f64 57.2

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, \color{blue}{t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

   4. Applied rewrites 52.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}} \]

   6. Applied rewrites 57.0%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-\left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{2} \cdot n, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot -2 + t\right)}\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \left(\frac{\ell}{Om} \cdot \ell\right) \cdot -2\right) + t\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \left(\color{blue}{\frac{\ell}{Om}} \cdot \ell\right) \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{\frac{\ell \cdot \ell}{Om}} \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \frac{\color{blue}{\ell \cdot \ell}}{Om} \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{\frac{\ell \cdot \ell}{Om}} \cdot -2\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \color{blue}{-2 \cdot \frac{\ell \cdot \ell}{Om}}\right) + t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

      10. associate-+r+ N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(\left(-\left(U - U*\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) + \left(-2 \cdot \frac{\ell \cdot \ell}{Om} + t\right)\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

   8. Applied rewrites 58.6%

      \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

   9. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

       3. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

       4. unpow2 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

       5. lower-*.f64 47.6

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot U\right) \cdot \left(2 \cdot n\right)} \]

   11. Applied rewrites 47.6%

       \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)} \cdot U\right) \cdot \left(2 \cdot n\right)} \]

   if 1.25000000000000005e153 < l

   1. Initial program 17.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      8. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{n}} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   4. Applied rewrites 8.9%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}} \]

   5. Taylor expanded in Om around 0

      \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\frac{-2 \cdot \left(Om \cdot {\ell}^{2}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\frac{-2 \cdot \left(Om \cdot {\ell}^{2}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}}} \]

      2. lower--.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{\color{blue}{-2 \cdot \left(Om \cdot {\ell}^{2}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}}{{Om}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{\color{blue}{-2 \cdot \left(Om \cdot {\ell}^{2}\right)} - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \color{blue}{\left(Om \cdot {\ell}^{2}\right)} - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}} \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \color{blue}{\left({\ell}^{2} \cdot n\right) \cdot \left(U - U*\right)}}{{Om}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \color{blue}{\left({\ell}^{2} \cdot n\right) \cdot \left(U - U*\right)}}{{Om}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot \left(U - U*\right)}{{Om}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(U - U*\right)}{{Om}^{2}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(U - U*\right)}{{Om}^{2}}} \]

      12. lower--.f64 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}} \]

      14. lower-*.f64 7.9

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}} \]

   7. Applied rewrites 7.9%

      \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U - U*\right)}{Om \cdot Om}}} \]

   8. Taylor expanded in Om around inf

      \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \left(-2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 9.1%

       \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \left(-2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)} \]

   12. Applied rewrites 29.5%

       \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot -2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.95 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\frac{l\_m}{Om} \cdot l\_m\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.95e+26)
   (sqrt (* (* t U) (* n 2.0)))
   (sqrt (* (* -2.0 (* (/ l_m Om) l_m)) (* (* U 2.0) 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 (l_m <= 2.95e+26) {
		tmp = sqrt(((t * U) * (n * 2.0)));
	} else {
		tmp = sqrt(((-2.0 * ((l_m / Om) * l_m)) * ((U * 2.0) * n)));
	}
	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) :: tmp
    if (l_m <= 2.95d+26) then
        tmp = sqrt(((t * u) * (n * 2.0d0)))
    else
        tmp = sqrt((((-2.0d0) * ((l_m / om) * l_m)) * ((u * 2.0d0) * n)))
    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 tmp;
	if (l_m <= 2.95e+26) {
		tmp = Math.sqrt(((t * U) * (n * 2.0)));
	} else {
		tmp = Math.sqrt(((-2.0 * ((l_m / Om) * l_m)) * ((U * 2.0) * n)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.95e+26:
		tmp = math.sqrt(((t * U) * (n * 2.0)))
	else:
		tmp = math.sqrt(((-2.0 * ((l_m / Om) * l_m)) * ((U * 2.0) * n)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.95e+26)
		tmp = sqrt(Float64(Float64(t * U) * Float64(n * 2.0)));
	else
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(l_m / Om) * l_m)) * Float64(Float64(U * 2.0) * n)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 2.95e+26)
		tmp = sqrt(((t * U) * (n * 2.0)));
	else
		tmp = sqrt(((-2.0 * ((l_m / Om) * l_m)) * ((U * 2.0) * n)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.95e+26], N[Sqrt[N[(N[(t * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-2.0 * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(U * 2.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.95000000000000015e26

   1. Initial program 53.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 42.1

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 42.1%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]

   if 2.95000000000000015e26 < l

   1. Initial program 33.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      8. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{n}} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}} \]

   5. Taylor expanded in Om around 0

      \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\frac{-2 \cdot \left(Om \cdot {\ell}^{2}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\frac{-2 \cdot \left(Om \cdot {\ell}^{2}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}}} \]

      2. lower--.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{\color{blue}{-2 \cdot \left(Om \cdot {\ell}^{2}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}}{{Om}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{\color{blue}{-2 \cdot \left(Om \cdot {\ell}^{2}\right)} - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \color{blue}{\left(Om \cdot {\ell}^{2}\right)} - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) - {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}} \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \color{blue}{\left({\ell}^{2} \cdot n\right) \cdot \left(U - U*\right)}}{{Om}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \color{blue}{\left({\ell}^{2} \cdot n\right) \cdot \left(U - U*\right)}}{{Om}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot \left(U - U*\right)}{{Om}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(U - U*\right)}{{Om}^{2}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(U - U*\right)}{{Om}^{2}}} \]

      12. lower--.f64 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}} \]

      14. lower-*.f64 10.1

          \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}} \]

   7. Applied rewrites 10.1%

      \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\frac{-2 \cdot \left(Om \cdot \left(\ell \cdot \ell\right)\right) - \left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U - U*\right)}{Om \cdot Om}}} \]

   8. Taylor expanded in Om around inf

      \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \left(-2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 16.0%

       \[\leadsto \sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \left(-2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right)} \]

   12. Applied rewrites 32.8%

       \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot -2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.95 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 38.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 4.5 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(t \cdot U\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 4.5e-282)
   (sqrt (* (* t U) (* n 2.0)))
   (* (sqrt n) (sqrt (* (* t U) 2.0)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 4.5e-282) {
		tmp = sqrt(((t * U) * (n * 2.0)));
	} else {
		tmp = sqrt(n) * sqrt(((t * U) * 2.0));
	}
	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) :: tmp
    if (n <= 4.5d-282) then
        tmp = sqrt(((t * u) * (n * 2.0d0)))
    else
        tmp = sqrt(n) * sqrt(((t * u) * 2.0d0))
    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 tmp;
	if (n <= 4.5e-282) {
		tmp = Math.sqrt(((t * U) * (n * 2.0)));
	} else {
		tmp = Math.sqrt(n) * Math.sqrt(((t * U) * 2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 4.5e-282:
		tmp = math.sqrt(((t * U) * (n * 2.0)))
	else:
		tmp = math.sqrt(n) * math.sqrt(((t * U) * 2.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 4.5e-282)
		tmp = sqrt(Float64(Float64(t * U) * Float64(n * 2.0)));
	else
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(t * U) * 2.0)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= 4.5e-282)
		tmp = sqrt(((t * U) * (n * 2.0)));
	else
		tmp = sqrt(n) * sqrt(((t * U) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 4.5e-282], N[Sqrt[N[(N[(t * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(t * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 4.50000000000000008e-282

   1. Initial program 43.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 36.3

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 36.3%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 36.9%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]

   if 4.50000000000000008e-282 < n

   1. Initial program 55.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. associate--l+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{Om}}\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \ell} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{-2} \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower--.f64 60.2

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, \color{blue}{t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

   4. Applied rewrites 53.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}} \]

   6. Applied rewrites 58.9%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-\left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{2} \cdot n, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}} \]

   7. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot t\right)} \cdot \left(2 \cdot n\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 37.7

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot t\right)} \cdot \left(2 \cdot n\right)} \]

   9. Applied rewrites 37.7%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot t\right)} \cdot \left(2 \cdot n\right)} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right)}} \]

       2. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot t\right) \cdot \left(2 \cdot n\right)}} \]

       3. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot t\right) \cdot \color{blue}{\left(2 \cdot n\right)}} \]

       4. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(U \cdot t\right) \cdot 2\right) \cdot n}} \]

       5. sqrt-prod N/A

          \[\leadsto \color{blue}{\sqrt{\left(U \cdot t\right) \cdot 2} \cdot \sqrt{n}} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot t\right) \cdot 2} \cdot \color{blue}{\sqrt{n}} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\left(U \cdot t\right) \cdot 2} \cdot \sqrt{n}} \]

   11. Applied rewrites 43.6%

       \[\leadsto \color{blue}{\sqrt{\left(t \cdot U\right) \cdot 2} \cdot \sqrt{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.5 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(t \cdot U\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 38.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 2.8 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot U} \cdot \sqrt{n \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 2.8e-278)
   (sqrt (* (* t U) (* n 2.0)))
   (* (sqrt (* t U)) (sqrt (* n 2.0)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 2.8e-278) {
		tmp = sqrt(((t * U) * (n * 2.0)));
	} else {
		tmp = sqrt((t * U)) * sqrt((n * 2.0));
	}
	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) :: tmp
    if (n <= 2.8d-278) then
        tmp = sqrt(((t * u) * (n * 2.0d0)))
    else
        tmp = sqrt((t * u)) * sqrt((n * 2.0d0))
    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 tmp;
	if (n <= 2.8e-278) {
		tmp = Math.sqrt(((t * U) * (n * 2.0)));
	} else {
		tmp = Math.sqrt((t * U)) * Math.sqrt((n * 2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 2.8e-278:
		tmp = math.sqrt(((t * U) * (n * 2.0)))
	else:
		tmp = math.sqrt((t * U)) * math.sqrt((n * 2.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 2.8e-278)
		tmp = sqrt(Float64(Float64(t * U) * Float64(n * 2.0)));
	else
		tmp = Float64(sqrt(Float64(t * U)) * sqrt(Float64(n * 2.0)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= 2.8e-278)
		tmp = sqrt(((t * U) * (n * 2.0)));
	else
		tmp = sqrt((t * U)) * sqrt((n * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 2.8e-278], N[Sqrt[N[(N[(t * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(t * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.80000000000000008e-278

   1. Initial program 43.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 36.5

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 36.5%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]

   if 2.80000000000000008e-278 < n

   1. Initial program 55.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. associate--l+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{Om}}\right) + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \ell} + \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell}{Om}}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{-2} \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      17. lower--.f64 60.3

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, \color{blue}{t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

   4. Applied rewrites 53.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(-2 \cdot \frac{\ell}{Om}, \ell, t - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}} \]

   6. Applied rewrites 59.1%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-\left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{2} \cdot n, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}} \]

   7. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot t\right)} \cdot \left(2 \cdot n\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 37.5

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot t\right)} \cdot \left(2 \cdot n\right)} \]

   9. Applied rewrites 37.5%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot t\right)} \cdot \left(2 \cdot n\right)} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right)}} \]

       2. pow1/2 N/A

          \[\leadsto \color{blue}{{\left(\left(U \cdot t\right) \cdot \left(2 \cdot n\right)\right)}^{\frac{1}{2}}} \]

       3. lift-*.f64 N/A

          \[\leadsto {\color{blue}{\left(\left(U \cdot t\right) \cdot \left(2 \cdot n\right)\right)}}^{\frac{1}{2}} \]

       4. *-commutative N/A

          \[\leadsto {\color{blue}{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}}^{\frac{1}{2}} \]

       5. unpow-prod-down N/A

          \[\leadsto \color{blue}{{\left(2 \cdot n\right)}^{\frac{1}{2}} \cdot {\left(U \cdot t\right)}^{\frac{1}{2}}} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{{\left(2 \cdot n\right)}^{\frac{1}{2}} \cdot {\left(U \cdot t\right)}^{\frac{1}{2}}} \]

       7. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{2 \cdot n}} \cdot {\left(U \cdot t\right)}^{\frac{1}{2}} \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{2 \cdot n}} \cdot {\left(U \cdot t\right)}^{\frac{1}{2}} \]

       9. pow1/2 N/A

          \[\leadsto \sqrt{2 \cdot n} \cdot \color{blue}{\sqrt{U \cdot t}} \]

   11. Applied rewrites 43.5%

       \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{t \cdot U}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.8 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot U} \cdot \sqrt{n \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 36.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* t (* U (* n 2.0)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt((t * (U * (n * 2.0))));
}

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((t * (u * (n * 2.0d0))))
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((t * (U * (n * 2.0))));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((t * (U * (n * 2.0))))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(t * Float64(U * Float64(n * 2.0))))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((t * (U * (n * 2.0))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(t * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.2%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
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. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

   3. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

   5. lower-*.f64 36.1

      \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

5. Applied rewrites 36.1%

   \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
6. Step-by-step derivation

7. Applied rewrites 35.9%

   \[\leadsto \sqrt{t \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)}} \]

8. Final simplification 35.9%

   \[\leadsto \sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(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*))))))