Toniolo and Linder, Equation (13)

Percentage Accurate: 50.4% → 66.9%

Time: 15.6s

Alternatives: 19

Speedup: 2.2×

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: 50.4% 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: 66.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := -\left(U - U*\right)\\ t_2 := \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_5 := t\_3 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot t\_4\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_5 \leq 10^{-311}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(t\_4 \cdot n, t\_1, t\_2\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_3 \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, t\_1 \cdot \left(n \cdot \frac{l\_m}{Om}\right), t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (- (- U U*)))
        (t_2 (fma (* (/ l_m Om) l_m) -2.0 t))
        (t_3 (* (* 2.0 n) U))
        (t_4 (pow (/ l_m Om) 2.0))
        (t_5
         (* t_3 (- (- t (* 2.0 (/ (* l_m l_m) Om))) (* (* n t_4) (- U U*))))))
   (if (<= t_5 1e-311)
     (sqrt (* (* (fma (* t_4 n) t_1 t_2) (* 2.0 n)) U))
     (if (<= t_5 5e+306)
       (sqrt (* t_3 (fma (/ l_m Om) (* t_1 (* n (/ l_m Om))) t_2)))
       (*
        (sqrt (* (* U n) (fma (/ n Om) (/ (- U* U) Om) (/ -2.0 Om))))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = -(U - U_42_);
	double t_2 = fma(((l_m / Om) * l_m), -2.0, t);
	double t_3 = (2.0 * n) * U;
	double t_4 = pow((l_m / Om), 2.0);
	double t_5 = t_3 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * t_4) * (U - U_42_)));
	double tmp;
	if (t_5 <= 1e-311) {
		tmp = sqrt(((fma((t_4 * n), t_1, t_2) * (2.0 * n)) * U));
	} else if (t_5 <= 5e+306) {
		tmp = sqrt((t_3 * fma((l_m / Om), (t_1 * (n * (l_m / Om))), t_2)));
	} else {
		tmp = sqrt(((U * n) * fma((n / Om), ((U_42_ - U) / Om), (-2.0 / Om)))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(-Float64(U - U_42_))
	t_2 = fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = Float64(l_m / Om) ^ 2.0
	t_5 = Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * t_4) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_5 <= 1e-311)
		tmp = sqrt(Float64(Float64(fma(Float64(t_4 * n), t_1, t_2) * Float64(2.0 * n)) * U));
	elseif (t_5 <= 5e+306)
		tmp = sqrt(Float64(t_3 * fma(Float64(l_m / Om), Float64(t_1 * Float64(n * Float64(l_m / Om))), t_2)));
	else
		tmp = Float64(sqrt(Float64(Float64(U * n) * fma(Float64(n / Om), Float64(Float64(U_42_ - U) / Om), Float64(-2.0 / Om)))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = (-N[(U - U$42$), $MachinePrecision])}, Block[{t$95$2 = N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * t$95$4), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 1e-311], N[Sqrt[N[(N[(N[(N[(t$95$4 * n), $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, 5e+306], N[Sqrt[N[(t$95$3 * N[(N[(l$95$m / Om), $MachinePrecision] * N[(t$95$1 * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(U * n), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999948e-312

   1. Initial program 20.7%

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

      1. lift--.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 20.7

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

      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 20.7%

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

   6. Applied rewrites 49.7%

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

   if 9.99999999999948e-312 < (*.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 98.7%

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

      1. lift--.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 98.2

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

      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 98.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 99.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.1

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 99.1

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if 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 22.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 23.7

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

      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 23.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 23.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 23.9

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 23.9

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 29.2

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

   6. Applied rewrites 29.2%

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

   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

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

   9. Applied rewrites 25.1%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{-311}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n, -\left(U - U*\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_4 := \sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, -2 \cdot l\_m, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{if}\;t\_3 \leq 10^{-285}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(\left(\left(2 \cdot U\right) \cdot U*\right) \cdot \left(l\_m \cdot n\right)\right) \cdot \left(l\_m \cdot n\right)}{Om}}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        (t_4 (sqrt (* (* (fma (/ l_m Om) (* -2.0 l_m) t) (* 2.0 n)) U))))
   (if (<= t_3 1e-285)
     t_4
     (if (<= t_3 5e+306)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (if (<= t_3 INFINITY)
         t_4
         (sqrt (/ (/ (* (* (* (* 2.0 U) U*) (* l_m n)) (* l_m n)) Om) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double t_4 = sqrt(((fma((l_m / Om), (-2.0 * l_m), t) * (2.0 * n)) * U));
	double tmp;
	if (t_3 <= 1e-285) {
		tmp = t_4;
	} else if (t_3 <= 5e+306) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = sqrt(((((((2.0 * U) * U_42_) * (l_m * n)) * (l_m * n)) / Om) / Om));
	}
	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)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	t_4 = sqrt(Float64(Float64(fma(Float64(l_m / Om), Float64(-2.0 * l_m), t) * Float64(2.0 * n)) * U))
	tmp = 0.0
	if (t_3 <= 1e-285)
		tmp = t_4;
	elseif (t_3 <= 5e+306)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * U) * U_42_) * Float64(l_m * n)) * Float64(l_m * n)) / Om) / Om));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(-2.0 * l$95$m), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1e-285], t$95$4, If[LessEqual[t$95$3, 5e+306], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[Sqrt[N[(N[(N[(N[(N[(N[(2.0 * U), $MachinePrecision] * U$42$), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] / Om), $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*)))) < 1.00000000000000007e-285 or 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*)))) < +inf.0

   1. Initial program 29.7%

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

      1. lift--.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 31.2

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

      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 31.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 31.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 31.3

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 31.3

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 37.0

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

   6. Applied rewrites 37.0%

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

   8. Applied rewrites 40.1%

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

   9. Taylor expanded in n around 0

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

       1. lower-*.f64 38.0

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

   11. Applied rewrites 38.0%

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

   if 1.00000000000000007e-285 < (*.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 98.7%

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

   3. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   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. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 6.0%

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

   6. Taylor expanded in U* around inf

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

   8. Applied rewrites 37.0%

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

   10. Applied rewrites 46.8%

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

   12. Applied rewrites 46.8%

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

Alternative 3: 55.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_4 := \sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, -2 \cdot l\_m, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{if}\;t\_3 \leq 10^{-285}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(\left(\left(U* \cdot U\right) \cdot n\right) \cdot l\_m\right) \cdot \left(l\_m \cdot n\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        (t_4 (sqrt (* (* (fma (/ l_m Om) (* -2.0 l_m) t) (* 2.0 n)) U))))
   (if (<= t_3 1e-285)
     t_4
     (if (<= t_3 5e+306)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (if (<= t_3 INFINITY)
         t_4
         (sqrt (* 2.0 (/ (* (* (* (* U* U) n) l_m) (* l_m n)) (* Om Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double t_4 = sqrt(((fma((l_m / Om), (-2.0 * l_m), t) * (2.0 * n)) * U));
	double tmp;
	if (t_3 <= 1e-285) {
		tmp = t_4;
	} else if (t_3 <= 5e+306) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = sqrt((2.0 * (((((U_42_ * U) * n) * l_m) * (l_m * n)) / (Om * Om))));
	}
	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)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	t_4 = sqrt(Float64(Float64(fma(Float64(l_m / Om), Float64(-2.0 * l_m), t) * Float64(2.0 * n)) * U))
	tmp = 0.0
	if (t_3 <= 1e-285)
		tmp = t_4;
	elseif (t_3 <= 5e+306)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(Float64(Float64(U_42_ * U) * n) * l_m) * Float64(l_m * n)) / Float64(Om * Om))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(-2.0 * l$95$m), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1e-285], t$95$4, If[LessEqual[t$95$3, 5e+306], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[Sqrt[N[(2.0 * N[(N[(N[(N[(N[(U$42$ * U), $MachinePrecision] * n), $MachinePrecision] * l$95$m), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.00000000000000007e-285 or 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*)))) < +inf.0

   1. Initial program 29.7%

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

      1. lift--.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 31.2

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

      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 31.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 31.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 31.3

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 31.3

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 37.0

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

   6. Applied rewrites 37.0%

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

   8. Applied rewrites 40.1%

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

   9. Taylor expanded in n around 0

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

       1. lower-*.f64 38.0

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

   11. Applied rewrites 38.0%

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

   if 1.00000000000000007e-285 < (*.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 98.7%

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

   3. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   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. 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 0.3

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

      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 0.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 0.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 0.7

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 0.7

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 1.1

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

   6. Applied rewrites 1.1%

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

   7. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.6

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

   9. Applied rewrites 40.6%

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

   11. Applied rewrites 40.7%

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

Alternative 4: 55.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_4 := \sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, -2 \cdot l\_m, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{if}\;t\_3 \leq 10^{-285}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(U \cdot U*\right) \cdot \left(\left(l\_m \cdot n\right) \cdot \left(l\_m \cdot n\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        (t_4 (sqrt (* (* (fma (/ l_m Om) (* -2.0 l_m) t) (* 2.0 n)) U))))
   (if (<= t_3 1e-285)
     t_4
     (if (<= t_3 5e+306)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (if (<= t_3 INFINITY)
         t_4
         (sqrt (* 2.0 (/ (* (* U U*) (* (* l_m n) (* l_m n))) (* Om Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double t_4 = sqrt(((fma((l_m / Om), (-2.0 * l_m), t) * (2.0 * n)) * U));
	double tmp;
	if (t_3 <= 1e-285) {
		tmp = t_4;
	} else if (t_3 <= 5e+306) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = sqrt((2.0 * (((U * U_42_) * ((l_m * n) * (l_m * n))) / (Om * Om))));
	}
	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)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	t_4 = sqrt(Float64(Float64(fma(Float64(l_m / Om), Float64(-2.0 * l_m), t) * Float64(2.0 * n)) * U))
	tmp = 0.0
	if (t_3 <= 1e-285)
		tmp = t_4;
	elseif (t_3 <= 5e+306)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(U * U_42_) * Float64(Float64(l_m * n) * Float64(l_m * n))) / Float64(Om * Om))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(-2.0 * l$95$m), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1e-285], t$95$4, If[LessEqual[t$95$3, 5e+306], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[Sqrt[N[(2.0 * N[(N[(N[(U * U$42$), $MachinePrecision] * N[(N[(l$95$m * n), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.00000000000000007e-285 or 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*)))) < +inf.0

   1. Initial program 29.7%

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

      1. lift--.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 31.2

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

      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 31.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 31.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 31.3

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 31.3

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 37.0

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

   6. Applied rewrites 37.0%

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

   8. Applied rewrites 40.1%

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

   9. Taylor expanded in n around 0

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

       1. lower-*.f64 38.0

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

   11. Applied rewrites 38.0%

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

   if 1.00000000000000007e-285 < (*.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 98.7%

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

   3. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   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. 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 0.3

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

      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 0.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 0.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 0.7

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 0.7

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 1.1

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

   6. Applied rewrites 1.1%

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

   7. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.6

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

   9. Applied rewrites 40.6%

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

Alternative 5: 54.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_4 := \sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, -2 \cdot l\_m, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{if}\;t\_3 \leq 10^{-285}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(U \cdot U*\right) \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \left(n \cdot n\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        (t_4 (sqrt (* (* (fma (/ l_m Om) (* -2.0 l_m) t) (* 2.0 n)) U))))
   (if (<= t_3 1e-285)
     t_4
     (if (<= t_3 5e+306)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (if (<= t_3 INFINITY)
         t_4
         (sqrt (* 2.0 (/ (* (* U U*) (* (* l_m l_m) (* n n))) (* Om Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double t_4 = sqrt(((fma((l_m / Om), (-2.0 * l_m), t) * (2.0 * n)) * U));
	double tmp;
	if (t_3 <= 1e-285) {
		tmp = t_4;
	} else if (t_3 <= 5e+306) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = sqrt((2.0 * (((U * U_42_) * ((l_m * l_m) * (n * n))) / (Om * Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	t_4 = sqrt(Float64(Float64(fma(Float64(l_m / Om), Float64(-2.0 * l_m), t) * Float64(2.0 * n)) * U))
	tmp = 0.0
	if (t_3 <= 1e-285)
		tmp = t_4;
	elseif (t_3 <= 5e+306)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(U * U_42_) * Float64(Float64(l_m * l_m) * Float64(n * n))) / Float64(Om * Om))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(-2.0 * l$95$m), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1e-285], t$95$4, If[LessEqual[t$95$3, 5e+306], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[Sqrt[N[(2.0 * N[(N[(N[(U * U$42$), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.00000000000000007e-285 or 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*)))) < +inf.0

   1. Initial program 29.7%

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

      1. lift--.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 31.2

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

      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 31.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 31.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 31.3

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 31.3

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 37.0

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

   6. Applied rewrites 37.0%

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

   8. Applied rewrites 40.1%

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

   9. Taylor expanded in n around 0

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

       1. lower-*.f64 38.0

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

   11. Applied rewrites 38.0%

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

   if 1.00000000000000007e-285 < (*.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 98.7%

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

   3. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   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. 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 0.3

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

      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 0.3%

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

   5. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.2

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

   7. Applied rewrites 33.2%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{-285}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, -2 \cdot \ell, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, -2 \cdot \ell, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(U \cdot U*\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_4 := \sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, -2 \cdot l\_m, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{if}\;t\_3 \leq 10^{-285}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(n \cdot \sqrt{2}\right) \cdot \frac{l\_m}{Om}\right) \cdot \sqrt{U* \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        (t_4 (sqrt (* (* (fma (/ l_m Om) (* -2.0 l_m) t) (* 2.0 n)) U))))
   (if (<= t_3 1e-285)
     t_4
     (if (<= t_3 5e+306)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (if (<= t_3 INFINITY)
         t_4
         (* (* (* n (sqrt 2.0)) (/ l_m Om)) (sqrt (* U* U))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double t_4 = sqrt(((fma((l_m / Om), (-2.0 * l_m), t) * (2.0 * n)) * U));
	double tmp;
	if (t_3 <= 1e-285) {
		tmp = t_4;
	} else if (t_3 <= 5e+306) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = ((n * sqrt(2.0)) * (l_m / Om)) * sqrt((U_42_ * U));
	}
	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)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	t_4 = sqrt(Float64(Float64(fma(Float64(l_m / Om), Float64(-2.0 * l_m), t) * Float64(2.0 * n)) * U))
	tmp = 0.0
	if (t_3 <= 1e-285)
		tmp = t_4;
	elseif (t_3 <= 5e+306)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(Float64(Float64(n * sqrt(2.0)) * Float64(l_m / Om)) * sqrt(Float64(U_42_ * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(-2.0 * l$95$m), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1e-285], t$95$4, If[LessEqual[t$95$3, 5e+306], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[(N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U$42$ * U), $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*)))) < 1.00000000000000007e-285 or 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*)))) < +inf.0

   1. Initial program 29.7%

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

      1. lift--.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 31.2

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

      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 31.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 31.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 31.3

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 31.3

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 37.0

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

   6. Applied rewrites 37.0%

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

   8. Applied rewrites 40.1%

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

   9. Taylor expanded in n around 0

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

       1. lower-*.f64 38.0

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

   11. Applied rewrites 38.0%

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

   if 1.00000000000000007e-285 < (*.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 98.7%

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

   3. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   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. Taylor expanded in U* around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 27.5%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{-285}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, -2 \cdot \ell, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, -2 \cdot \ell, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(n \cdot \sqrt{2}\right) \cdot \frac{\ell}{Om}\right) \cdot \sqrt{U* \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_4 := \sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, -2 \cdot l\_m, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{if}\;t\_3 \leq 10^{-285}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U* \cdot U} \cdot \frac{\left(\sqrt{2} \cdot n\right) \cdot l\_m}{Om}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        (t_4 (sqrt (* (* (fma (/ l_m Om) (* -2.0 l_m) t) (* 2.0 n)) U))))
   (if (<= t_3 1e-285)
     t_4
     (if (<= t_3 5e+306)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (if (<= t_3 INFINITY)
         t_4
         (* (sqrt (* U* U)) (/ (* (* (sqrt 2.0) n) l_m) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double t_4 = sqrt(((fma((l_m / Om), (-2.0 * l_m), t) * (2.0 * n)) * U));
	double tmp;
	if (t_3 <= 1e-285) {
		tmp = t_4;
	} else if (t_3 <= 5e+306) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l_m) / Om);
	}
	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)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	t_4 = sqrt(Float64(Float64(fma(Float64(l_m / Om), Float64(-2.0 * l_m), t) * Float64(2.0 * n)) * U))
	tmp = 0.0
	if (t_3 <= 1e-285)
		tmp = t_4;
	elseif (t_3 <= 5e+306)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(sqrt(Float64(U_42_ * U)) * Float64(Float64(Float64(sqrt(2.0) * n) * l_m) / Om));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(-2.0 * l$95$m), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1e-285], t$95$4, If[LessEqual[t$95$3, 5e+306], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * l$95$m), $MachinePrecision] / Om), $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*)))) < 1.00000000000000007e-285 or 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*)))) < +inf.0

   1. Initial program 29.7%

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

      1. lift--.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 31.2

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

      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 31.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 31.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 31.3

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 31.3

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 37.0

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

   6. Applied rewrites 37.0%

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

   8. Applied rewrites 40.1%

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

   9. Taylor expanded in n around 0

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

       1. lower-*.f64 38.0

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

   11. Applied rewrites 38.0%

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

   if 1.00000000000000007e-285 < (*.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 98.7%

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

   3. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   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. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 24.6

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

   5. Applied rewrites 24.6%

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

Alternative 8: 67.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 10^{-311}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\left(-n\right) \cdot \left(U - U*\right), \frac{l\_m}{Om}, -2 \cdot l\_m\right), t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{l\_m}{Om}\right), \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 1e-311)
     (sqrt
      (*
       (*
        (fma (/ l_m Om) (fma (* (- n) (- U U*)) (/ l_m Om) (* -2.0 l_m)) t)
        (* 2.0 n))
       U))
     (if (<= t_2 5e+306)
       (sqrt
        (*
         t_1
         (fma
          (/ l_m Om)
          (* (- (- U U*)) (* n (/ l_m Om)))
          (fma (* (/ l_m Om) l_m) -2.0 t))))
       (*
        (sqrt (* (* U n) (fma (/ n Om) (/ (- U* U) Om) (/ -2.0 Om))))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 1e-311) {
		tmp = sqrt(((fma((l_m / Om), fma((-n * (U - U_42_)), (l_m / Om), (-2.0 * l_m)), t) * (2.0 * n)) * U));
	} else if (t_2 <= 5e+306) {
		tmp = sqrt((t_1 * fma((l_m / Om), (-(U - U_42_) * (n * (l_m / Om))), fma(((l_m / Om) * l_m), -2.0, t))));
	} else {
		tmp = sqrt(((U * n) * fma((n / Om), ((U_42_ - U) / Om), (-2.0 / Om)))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 1e-311)
		tmp = sqrt(Float64(Float64(fma(Float64(l_m / Om), fma(Float64(Float64(-n) * Float64(U - U_42_)), Float64(l_m / Om), Float64(-2.0 * l_m)), t) * Float64(2.0 * n)) * U));
	elseif (t_2 <= 5e+306)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m / Om), Float64(Float64(-Float64(U - U_42_)) * Float64(n * Float64(l_m / Om))), fma(Float64(Float64(l_m / Om) * l_m), -2.0, t))));
	else
		tmp = Float64(sqrt(Float64(Float64(U * n) * fma(Float64(n / Om), Float64(Float64(U_42_ - U) / Om), Float64(-2.0 / Om)))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-311], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[((-n) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + N[(-2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+306], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m / Om), $MachinePrecision] * N[((-N[(U - U$42$), $MachinePrecision]) * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(U * n), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999948e-312

   1. Initial program 20.7%

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

      1. lift--.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 20.7

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

      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 20.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 20.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 20.7

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 20.7

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 23.4

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

   6. Applied rewrites 23.4%

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

   8. Applied rewrites 49.5%

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

   if 9.99999999999948e-312 < (*.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 98.7%

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

      1. lift--.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 98.2

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

      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 98.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 99.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.1

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 99.1

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if 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 22.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 23.7

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

      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 23.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 23.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 23.9

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 23.9

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 29.2

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

   6. Applied rewrites 29.2%

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

   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

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

   9. Applied rewrites 25.1%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{-311}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\left(-n\right) \cdot \left(U - U*\right), \frac{\ell}{Om}, -2 \cdot \ell\right), t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(Float64(fma(Float64(l_m / Om), fma(Float64(Float64(-n) * Float64(U - U_42_)), Float64(l_m / Om), Float64(-2.0 * l_m)), t) * Float64(2.0 * n)) * U));
	elseif (t_3 <= 5e+306)
		tmp = sqrt(Float64(t_2 * fma(Float64(Float64(-Float64(U - U_42_)) * Float64(l_m / Om)), Float64(Float64(l_m / Om) * n), fma(-2.0, t_1, t))));
	else
		tmp = Float64(sqrt(Float64(Float64(U * n) * fma(Float64(n / Om), Float64(Float64(U_42_ - U) / Om), Float64(-2.0 / Om)))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[((-n) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + N[(-2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 5e+306], N[Sqrt[N[(t$95$2 * N[(N[((-N[(U - U$42$), $MachinePrecision]) * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(U * n), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 17.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 17.2

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

      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 17.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 17.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 17.2

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 17.2

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 20.1

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

   6. Applied rewrites 20.1%

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

   8. Applied rewrites 47.6%

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

   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 98.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. 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 98.0

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

      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 98.0%

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

   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 22.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 23.7

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

      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 23.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 23.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 23.9

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 23.9

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 29.2

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

   6. Applied rewrites 29.2%

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

   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

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

   9. Applied rewrites 25.1%

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

4. Final simplification 58.9%

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

Alternative 10: 62.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\left(-n\right) \cdot \left(U - U*\right), \frac{l\_m}{Om}, -2 \cdot l\_m\right), t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_3 5e-11)
     (sqrt
      (*
       (*
        (fma (/ l_m Om) (fma (* (- n) (- U U*)) (/ l_m Om) (* -2.0 l_m)) t)
        (* 2.0 n))
       U))
     (if (<= t_3 5e+306)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (*
        (sqrt (* (* U n) (fma (/ n Om) (/ (- U* U) Om) (/ -2.0 Om))))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 5e-11) {
		tmp = sqrt(((fma((l_m / Om), fma((-n * (U - U_42_)), (l_m / Om), (-2.0 * l_m)), t) * (2.0 * n)) * U));
	} else if (t_3 <= 5e+306) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = sqrt(((U * n) * fma((n / Om), ((U_42_ - U) / Om), (-2.0 / Om)))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 5e-11)
		tmp = sqrt(Float64(Float64(fma(Float64(l_m / Om), fma(Float64(Float64(-n) * Float64(U - U_42_)), Float64(l_m / Om), Float64(-2.0 * l_m)), t) * Float64(2.0 * n)) * U));
	elseif (t_3 <= 5e+306)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = Float64(sqrt(Float64(Float64(U * n) * fma(Float64(n / Om), Float64(Float64(U_42_ - U) / Om), Float64(-2.0 / Om)))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-11], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[((-n) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + N[(-2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 5e+306], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(U * n), $MachinePrecision] * N[(N[(n / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.00000000000000018e-11

   1. Initial program 64.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 64.7

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

      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 64.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 64.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 64.7

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 64.7

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 65.9

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

   6. Applied rewrites 65.9%

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

   8. Applied rewrites 71.9%

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

   if 5.00000000000000018e-11 < (*.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 99.7%

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

   3. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if 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 22.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 23.7

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

      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 23.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 23.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 23.9

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 23.9

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 29.2

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

   6. Applied rewrites 29.2%

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

   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

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

   9. Applied rewrites 25.1%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\left(-n\right) \cdot \left(U - U*\right), \frac{\ell}{Om}, -2 \cdot \ell\right), t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U* - U}{Om}, \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\left(-n\right) \cdot \left(U - U*\right), \frac{l\_m}{Om}, -2 \cdot l\_m\right), t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_3 5e-11)
     (sqrt
      (*
       (*
        (fma (/ l_m Om) (fma (* (- n) (- U U*)) (/ l_m Om) (* -2.0 l_m)) t)
        (* 2.0 n))
       U))
     (if (<= t_3 5e+306)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (*
        (sqrt (* (* U n) (- (/ (* n (- U* U)) (* Om Om)) (/ 2.0 Om))))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 5e-11) {
		tmp = sqrt(((fma((l_m / Om), fma((-n * (U - U_42_)), (l_m / Om), (-2.0 * l_m)), t) * (2.0 * n)) * U));
	} else if (t_3 <= 5e+306) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = sqrt(((U * n) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om)))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 5e-11)
		tmp = sqrt(Float64(Float64(fma(Float64(l_m / Om), fma(Float64(Float64(-n) * Float64(U - U_42_)), Float64(l_m / Om), Float64(-2.0 * l_m)), t) * Float64(2.0 * n)) * U));
	elseif (t_3 <= 5e+306)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = Float64(sqrt(Float64(Float64(U * n) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) - Float64(2.0 / Om)))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-11], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[((-n) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + N[(-2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 5e+306], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(U * n), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.00000000000000018e-11

   1. Initial program 64.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 64.7

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

      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 64.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 64.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 64.7

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 64.7

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 65.9

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

   6. Applied rewrites 65.9%

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

   8. Applied rewrites 71.9%

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

   if 5.00000000000000018e-11 < (*.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 99.7%

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

   3. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if 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 22.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 23.7

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

      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 23.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 23.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 23.9

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 23.9

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 29.2

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

   6. Applied rewrites 29.2%

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

   8. Applied rewrites 38.1%

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

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   10. 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

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

   11. Applied rewrites 25.9%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\left(-n\right) \cdot \left(U - U*\right), \frac{\ell}{Om}, -2 \cdot \ell\right), t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 10^{-285}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, -2 \cdot l\_m, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, \frac{U* \cdot \left(l\_m \cdot n\right)}{Om}, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_3 1e-285)
     (sqrt (* (* (fma (/ l_m Om) (* -2.0 l_m) t) (* 2.0 n)) U))
     (if (<= t_3 5e+306)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (sqrt
        (* (* (fma (/ l_m Om) (/ (* U* (* l_m n)) Om) t) (* 2.0 n)) U))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 1e-285) {
		tmp = sqrt(((fma((l_m / Om), (-2.0 * l_m), t) * (2.0 * n)) * U));
	} else if (t_3 <= 5e+306) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = sqrt(((fma((l_m / Om), ((U_42_ * (l_m * n)) / Om), t) * (2.0 * n)) * U));
	}
	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)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 1e-285)
		tmp = sqrt(Float64(Float64(fma(Float64(l_m / Om), Float64(-2.0 * l_m), t) * Float64(2.0 * n)) * U));
	elseif (t_3 <= 5e+306)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = sqrt(Float64(Float64(fma(Float64(l_m / Om), Float64(Float64(U_42_ * Float64(l_m * n)) / Om), t) * Float64(2.0 * n)) * U));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 1e-285], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(-2.0 * l$95$m), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 5e+306], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $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*)))) < 1.00000000000000007e-285

   1. Initial program 26.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 26.2

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

      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 26.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 26.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 26.2

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 26.2

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 28.7

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

   6. Applied rewrites 28.7%

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

   8. Applied rewrites 50.8%

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

   9. Taylor expanded in n around 0

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

       1. lower-*.f64 46.9

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

   11. Applied rewrites 46.9%

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

   if 1.00000000000000007e-285 < (*.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 98.7%

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

   3. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if 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 22.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 23.7

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

      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 23.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 23.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 23.9

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 23.9

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 29.2

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

   6. Applied rewrites 29.2%

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

   8. Applied rewrites 38.1%

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

   9. Taylor expanded in U* around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 37.9

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

   11. Applied rewrites 37.9%

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

Alternative 13: 51.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 10^{-285} \lor \neg \left(t\_3 \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, -2 \cdot l\_m, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (or (<= t_3 1e-285) (not (<= t_3 5e+306)))
     (sqrt (* (* (fma (/ l_m Om) (* -2.0 l_m) t) (* 2.0 n)) U))
     (sqrt (* t_2 (fma -2.0 t_1 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 t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if ((t_3 <= 1e-285) || !(t_3 <= 5e+306)) {
		tmp = sqrt(((fma((l_m / Om), (-2.0 * l_m), t) * (2.0 * n)) * U));
	} else {
		tmp = sqrt((t_2 * fma(-2.0, t_1, 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)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if ((t_3 <= 1e-285) || !(t_3 <= 5e+306))
		tmp = sqrt(Float64(Float64(fma(Float64(l_m / Om), Float64(-2.0 * l_m), t) * Float64(2.0 * n)) * U));
	else
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, 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]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$3, 1e-285], N[Not[LessEqual[t$95$3, 5e+306]], $MachinePrecision]], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(-2.0 * l$95$m), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $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*)))) < 1.00000000000000007e-285 or 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 23.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. 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 24.4

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

      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 24.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 24.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 24.5

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 24.5

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 29.0

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

   6. Applied rewrites 29.0%

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

   8. Applied rewrites 41.7%

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

   9. Taylor expanded in n around 0

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

       1. lower-*.f64 30.5

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

   11. Applied rewrites 30.5%

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

   if 1.00000000000000007e-285 < (*.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 98.7%

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

   3. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{-285} \lor \neg \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, -2 \cdot \ell, t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.2% accurate, 0.8× speedup

Math

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

FPCore

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

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 t_2 = fma(-2.0, t_1, t);
	double t_3 = (2.0 * n) * U;
	double tmp;
	if ((t_3 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 0.0) {
		tmp = sqrt((((t_2 * n) * U) * 2.0));
	} else {
		tmp = sqrt((t_3 * t_2));
	}
	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)
	t_2 = fma(-2.0, t_1, t)
	t_3 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))) <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t_2 * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(t_3 * t_2));
	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]}, Block[{t$95$2 = N[(-2.0 * t$95$1 + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[Sqrt[N[(N[(N[(t$95$2 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$3 * t$95$2), $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 17.2%

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

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   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 45.0

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

   5. Applied rewrites 45.0%

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

   if 0.0 < (*.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 59.5%

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

   3. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 50.9

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

   5. Applied rewrites 50.9%

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

Alternative 15: 38.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{-311}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (*
       (* (* 2.0 n) U)
       (-
        (- t (* 2.0 (/ (* l_m l_m) Om)))
        (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))
      1e-311)
   (sqrt (* (* 2.0 n) (* t U)))
   (sqrt (* (* (* n U) t) 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 ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 1e-311) {
		tmp = sqrt(((2.0 * n) * (t * U)));
	} else {
		tmp = sqrt((((n * U) * t) * 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 ((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))) <= 1d-311) then
        tmp = sqrt(((2.0d0 * n) * (t * u)))
    else
        tmp = sqrt((((n * u) * t) * 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 ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 1e-311) {
		tmp = Math.sqrt(((2.0 * n) * (t * U)));
	} else {
		tmp = Math.sqrt((((n * U) * t) * 2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 1e-311:
		tmp = math.sqrt(((2.0 * n) * (t * U)))
	else:
		tmp = math.sqrt((((n * U) * t) * 2.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))) <= 1e-311)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(t * U)));
	else
		tmp = sqrt(Float64(Float64(Float64(n * U) * t) * 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 ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)))) <= 1e-311)
		tmp = sqrt(((2.0 * n) * (t * U)));
	else
		tmp = sqrt((((n * U) * t) * 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[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-311], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(t * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(n * U), $MachinePrecision] * t), $MachinePrecision] * 2.0), $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*)))) < 9.99999999999948e-312

   1. Initial program 20.7%

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

   3. Taylor expanded in 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 40.5

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

   5. Applied rewrites 40.5%

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

   7. Applied rewrites 42.9%

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

   if 9.99999999999948e-312 < (*.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 59.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 40.5

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

   5. Applied rewrites 40.5%

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

   7. Applied rewrites 46.5%

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

Alternative 16: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 5.8 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\left(-n\right) \cdot \left(U - U*\right), \frac{l\_m}{Om}, -2 \cdot l\_m\right), t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\frac{l\_m}{Om} \cdot n, \mathsf{fma}\left(-1, U, U*\right), -2 \cdot l\_m\right), t\right)\right)} \cdot \sqrt{n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 5.8e-307)
   (sqrt
    (*
     (*
      (fma (/ l_m Om) (fma (* (- n) (- U U*)) (/ l_m Om) (* -2.0 l_m)) t)
      (* 2.0 n))
     U))
   (*
    (sqrt
     (*
      U
      (*
       2.0
       (fma
        (/ l_m Om)
        (fma (* (/ l_m Om) n) (fma -1.0 U U*) (* -2.0 l_m))
        t))))
    (sqrt 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 (n <= 5.8e-307) {
		tmp = sqrt(((fma((l_m / Om), fma((-n * (U - U_42_)), (l_m / Om), (-2.0 * l_m)), t) * (2.0 * n)) * U));
	} else {
		tmp = sqrt((U * (2.0 * fma((l_m / Om), fma(((l_m / Om) * n), fma(-1.0, U, U_42_), (-2.0 * l_m)), t)))) * sqrt(n);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 5.8e-307)
		tmp = sqrt(Float64(Float64(fma(Float64(l_m / Om), fma(Float64(Float64(-n) * Float64(U - U_42_)), Float64(l_m / Om), Float64(-2.0 * l_m)), t) * Float64(2.0 * n)) * U));
	else
		tmp = Float64(sqrt(Float64(U * Float64(2.0 * fma(Float64(l_m / Om), fma(Float64(Float64(l_m / Om) * n), fma(-1.0, U, U_42_), Float64(-2.0 * l_m)), t)))) * sqrt(n));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 5.8e-307], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[((-n) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + N[(-2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(2.0 * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] * N[(-1.0 * U + U$42$), $MachinePrecision] + N[(-2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 5.8000000000000001e-307

   1. Initial program 53.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 53.2

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

      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 53.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 53.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.3

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 53.3

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 54.7

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

   6. Applied rewrites 54.7%

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

   8. Applied rewrites 57.2%

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

   if 5.8000000000000001e-307 < n

   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. 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 54.5

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

      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 54.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 55.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.4

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 55.4

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 59.5

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

   6. Applied rewrites 59.5%

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

   8. Applied rewrites 57.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 73.9%

       \[\leadsto \color{blue}{\sqrt{U \cdot \left(2 \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-1, U, U*\right), -2 \cdot \ell\right), t\right)\right)} \cdot \sqrt{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5.8 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\left(-n\right) \cdot \left(U - U*\right), \frac{\ell}{Om}, -2 \cdot \ell\right), t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-1, U, U*\right), -2 \cdot \ell\right), t\right)\right)} \cdot \sqrt{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.26 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(\left(-n\right) \cdot \left(U - U*\right), \frac{l\_m}{Om}, -2 \cdot l\_m\right), t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.26e-166)
   (sqrt (* (* (* n U) t) 2.0))
   (sqrt
    (*
     (*
      (fma (/ l_m Om) (fma (* (- n) (- U U*)) (/ l_m Om) (* -2.0 l_m)) t)
      (* 2.0 n))
     U))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.26e-166) {
		tmp = sqrt((((n * U) * t) * 2.0));
	} else {
		tmp = sqrt(((fma((l_m / Om), fma((-n * (U - U_42_)), (l_m / Om), (-2.0 * l_m)), t) * (2.0 * n)) * U));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.26e-166)
		tmp = sqrt(Float64(Float64(Float64(n * U) * t) * 2.0));
	else
		tmp = sqrt(Float64(Float64(fma(Float64(l_m / Om), fma(Float64(Float64(-n) * Float64(U - U_42_)), Float64(l_m / Om), Float64(-2.0 * l_m)), t) * Float64(2.0 * n)) * U));
	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, 1.26e-166], N[Sqrt[N[(N[(N[(n * U), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[((-n) * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + N[(-2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.26e-166

   1. Initial program 53.8%

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

   3. 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 45.2

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

   5. Applied rewrites 45.2%

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

   7. Applied rewrites 47.1%

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

   if 1.26e-166 < l

   1. Initial program 52.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. 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 52.0

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

      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 52.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \left(\frac{\ell}{Om} \cdot n\right) + \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \left(-\left(U - U*\right)\right)\right)} \cdot \left(\frac{\ell}{Om} \cdot n\right) + \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\left(-\left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right), \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

      6. lower-*.f64 53.2

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\left(-\left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot n\right)}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

      9. lower-*.f64 53.2

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

      10. lift-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{-2 \cdot \frac{\ell \cdot \ell}{Om} + t}\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\frac{\ell \cdot \ell}{Om} \cdot -2} + t\right)} \]

      12. lower-fma.f64 53.2

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \color{blue}{\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)}\right)} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(\color{blue}{\frac{\ell \cdot \ell}{Om}}, -2, t\right)\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)\right)} \]

      15. associate-/l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(\color{blue}{\ell \cdot \frac{\ell}{Om}}, -2, t\right)\right)} \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(\ell \cdot \color{blue}{\frac{\ell}{Om}}, -2, t\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(\color{blue}{\frac{\ell}{Om} \cdot \ell}, -2, t\right)\right)} \]

      18. lower-*.f64 55.6

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(\color{blue}{\frac{\ell}{Om} \cdot \ell}, -2, t\right)\right)} \]

   6. Applied rewrites 55.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}} \]

   8. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(n \cdot \left(-\left(U - U*\right)\right), \frac{\ell}{Om}, -2 \cdot \ell\right), t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.26 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\left(-n\right) \cdot \left(U - U*\right), \frac{\ell}{Om}, -2 \cdot \ell\right), t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 41.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq 3.5 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om 3.5e-159)
   (sqrt (* (* (* (fma -2.0 (/ (* l_m l_m) Om) t) n) U) 2.0))
   (sqrt (* (* (* n U) t) 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 (Om <= 3.5e-159) {
		tmp = sqrt((((fma(-2.0, ((l_m * l_m) / Om), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt((((n * U) * t) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Om <= 3.5e-159)
		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(Float64(n * U) * t) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[Om, 3.5e-159], 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[(N[(n * U), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 3.50000000000000002e-159

   1. Initial program 45.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 42.7

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

   5. Applied rewrites 42.7%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}} \]

   if 3.50000000000000002e-159 < Om

   1. Initial program 63.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 49.8

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 49.8%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 57.8%

      \[\leadsto \sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 35.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(2 \cdot n\right) \cdot \left(t \cdot U\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* 2.0 n) (* t U))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt(((2.0 * n) * (t * U)));
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt(((2.0d0 * n) * (t * u)))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt(((2.0 * n) * (t * U)));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt(((2.0 * n) * (t * U)))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(2.0 * n) * Float64(t * U)))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt(((2.0 * n) * (t * U)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(t * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.2%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
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 40.5

      \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

5. Applied rewrites 40.5%

   \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
6. Step-by-step derivation

7. Applied rewrites 40.0%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024323 
(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*))))))