Toniolo and Linder, Equation (13)

Percentage Accurate: 49.9% → 68.4%

Time: 20.8s

Alternatives: 19

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.9% 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: 68.4% accurate, 0.4× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (/ l_m Om) (fma (- U* U) (* n (/ l_m Om)) (* l_m -2.0)) t))
        (t_2 (* U (* 2.0 n)))
        (t_3
         (*
          t_2
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_3 0.0)
     (sqrt (* U (* t_1 (* 2.0 n))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 t_1))
       (*
        (sqrt (* n (* U (/ (- (* n (/ (- 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 = fma((l_m / Om), fma((U_42_ - U), (n * (l_m / Om)), (l_m * -2.0)), t);
	double t_2 = U * (2.0 * n);
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((U * (t_1 * (2.0 * n))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * t_1));
	} else {
		tmp = sqrt((n * (U * (((n * ((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 = fma(Float64(l_m / Om), fma(Float64(U_42_ - U), Float64(n * Float64(l_m / Om)), Float64(l_m * -2.0)), t)
	t_2 = Float64(U * Float64(2.0 * n))
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(U * Float64(t_1 * Float64(2.0 * n))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = Float64(sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n * Float64(Float64(U_42_ - U) / Om)) - 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 / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(U * N[(t$95$1 * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * N[(U * N[(N[(N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / 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 7.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. 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. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 20.0

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 23.1%

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

   6. Applied rewrites 51.8%

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

   7. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 51.8

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

   9. Applied rewrites 51.8%

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

   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*)))) < +inf.0

   1. Initial program 68.0%

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

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 68.9

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 74.5%

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

   6. Applied rewrites 74.5%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 0.9

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 4.1%

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

   6. Applied rewrites 18.5%

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

   7. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 32.9%

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

   11. Applied rewrites 47.6%

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

4. Final simplification 68.0%

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

Alternative 2: 55.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ t_3 := \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(l\_m \cdot -2\right) \cdot \left(U \cdot \frac{l\_m}{Om}\right), 2 \cdot n, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot t\_3}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(l\_m \cdot \left(l\_m \cdot \frac{t\_3}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{U \cdot \left(\left(n \cdot \left(l\_m \cdot l\_m\right)\right) \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))
        (t_3 (fma n (/ (- U U*) Om) 2.0)))
   (if (<= t_2 0.0)
     (sqrt
      (fma (* (* l_m -2.0) (* U (/ l_m Om))) (* 2.0 n) (* U (* t (* 2.0 n)))))
     (if (<= t_2 2e+261)
       (sqrt (* t_1 (- t (/ (* (* l_m l_m) t_3) Om))))
       (if (<= t_2 INFINITY)
         (sqrt (* -2.0 (* (* n U) (* l_m (* l_m (/ t_3 Om))))))
         (sqrt
          (*
           -2.0
           (/ (* U (* (* n (* l_m l_m)) (- 2.0 (/ (* n U*) 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 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double t_3 = fma(n, ((U - U_42_) / Om), 2.0);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(fma(((l_m * -2.0) * (U * (l_m / Om))), (2.0 * n), (U * (t * (2.0 * n)))));
	} else if (t_2 <= 2e+261) {
		tmp = sqrt((t_1 * (t - (((l_m * l_m) * t_3) / Om))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((-2.0 * ((n * U) * (l_m * (l_m * (t_3 / Om))))));
	} else {
		tmp = sqrt((-2.0 * ((U * ((n * (l_m * l_m)) * (2.0 - ((n * U_42_) / Om)))) / Om)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	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_42_ - U))))
	t_3 = fma(n, Float64(Float64(U - U_42_) / Om), 2.0)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(fma(Float64(Float64(l_m * -2.0) * Float64(U * Float64(l_m / Om))), Float64(2.0 * n), Float64(U * Float64(t * Float64(2.0 * n)))));
	elseif (t_2 <= 2e+261)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64(Float64(l_m * l_m) * t_3) / Om))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(-2.0 * Float64(Float64(n * U) * Float64(l_m * Float64(l_m * Float64(t_3 / Om))))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(U * Float64(Float64(n * Float64(l_m * l_m)) * Float64(2.0 - Float64(Float64(n * U_42_) / 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[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(N[(l$95$m * -2.0), $MachinePrecision] * N[(U * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * n), $MachinePrecision] + N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+261], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * t$95$3), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(-2.0 * N[(N[(n * U), $MachinePrecision] * N[(l$95$m * N[(l$95$m * N[(t$95$3 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[(N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 - N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 7.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. 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. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 20.0

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 23.1%

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

   6. Applied rewrites 51.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

   8. Applied rewrites 48.8%

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

   9. Taylor expanded in Om around inf

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

       1. lower-*.f64 38.6

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

   11. Applied rewrites 38.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.9999999999999999e261

   1. Initial program 96.4%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 93.9%

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

   if 1.9999999999999999e261 < (*.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 31.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 25.4%

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

   7. Applied rewrites 37.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 31.6%

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

   6. Taylor expanded in U around 0

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

   8. Applied rewrites 46.9%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\ell \cdot -2\right) \cdot \left(U \cdot \frac{\ell}{Om}\right), 2 \cdot n, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{U \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(l\_m \cdot -2\right) \cdot \left(U \cdot \frac{l\_m}{Om}\right), 2 \cdot n, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+259}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(l\_m \cdot \left(l\_m \cdot \frac{\mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{U \cdot \left(\left(n \cdot \left(l\_m \cdot l\_m\right)\right) \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* U (* 2.0 n)))
        (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 -2.0) (* U (/ l_m Om))) (* 2.0 n) (* U (* t (* 2.0 n)))))
     (if (<= t_3 1e+259)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (if (<= t_3 INFINITY)
         (sqrt
          (*
           -2.0
           (* (* n U) (* l_m (* l_m (/ (fma n (/ (- U U*) Om) 2.0) Om))))))
         (sqrt
          (*
           -2.0
           (/ (* U (* (* n (* l_m l_m)) (- 2.0 (/ (* n U*) 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 = U * (2.0 * n);
	double t_3 = t_2 * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(fma(((l_m * -2.0) * (U * (l_m / Om))), (2.0 * n), (U * (t * (2.0 * n)))));
	} else if (t_3 <= 1e+259) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((-2.0 * ((n * U) * (l_m * (l_m * (fma(n, ((U - U_42_) / Om), 2.0) / Om))))));
	} else {
		tmp = sqrt((-2.0 * ((U * ((n * (l_m * l_m)) * (2.0 - ((n * U_42_) / 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(U * Float64(2.0 * n))
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(fma(Float64(Float64(l_m * -2.0) * Float64(U * Float64(l_m / Om))), Float64(2.0 * n), Float64(U * Float64(t * Float64(2.0 * n)))));
	elseif (t_3 <= 1e+259)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(-2.0 * Float64(Float64(n * U) * Float64(l_m * Float64(l_m * Float64(fma(n, Float64(Float64(U - U_42_) / Om), 2.0) / Om))))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(U * Float64(Float64(n * Float64(l_m * l_m)) * Float64(2.0 - Float64(Float64(n * U_42_) / 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[(U * N[(2.0 * n), $MachinePrecision]), $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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(N[(l$95$m * -2.0), $MachinePrecision] * N[(U * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * n), $MachinePrecision] + N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+259], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(-2.0 * N[(N[(n * U), $MachinePrecision] * N[(l$95$m * N[(l$95$m * N[(N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[(N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 - N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 7.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. 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. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 20.0

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 23.1%

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

   6. Applied rewrites 51.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

   8. Applied rewrites 48.8%

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

   9. Taylor expanded in Om around inf

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

       1. lower-*.f64 38.6

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

   11. Applied rewrites 38.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.999999999999999e258

   1. Initial program 96.4%

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

   3. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 6.4

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 90.5

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

   8. Applied rewrites 90.5%

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

   if 9.999999999999999e258 < (*.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 32.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 25.9%

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

   7. Applied rewrites 38.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 31.6%

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

   6. Taylor expanded in U around 0

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

   8. Applied rewrites 46.9%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\ell \cdot -2\right) \cdot \left(U \cdot \frac{\ell}{Om}\right), 2 \cdot n, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+259}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{U \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \mathsf{fma}\left(-2, t\_1, t\right)\\ t_3 := U \cdot \left(2 \cdot n\right)\\ t_4 := \sqrt{t\_3 \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\_2\right)}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{t\_3 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(l\_m \cdot \left(l\_m \cdot \frac{\mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

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 (fma -2.0 t_1 t))
        (t_3 (* U (* 2.0 n)))
        (t_4
         (sqrt
          (*
           t_3
           (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_4 0.0)
     (sqrt (* (* 2.0 U) (* n t_2)))
     (if (<= t_4 5e+129)
       (sqrt (* t_3 t_2))
       (sqrt
        (*
         -2.0
         (* (* n U) (* l_m (* l_m (/ (fma n (/ (- U U*) Om) 2.0) Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = fma(-2.0, t_1, t);
	double t_3 = U * (2.0 * n);
	double t_4 = sqrt((t_3 * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((2.0 * U) * (n * t_2)));
	} else if (t_4 <= 5e+129) {
		tmp = sqrt((t_3 * t_2));
	} else {
		tmp = sqrt((-2.0 * ((n * U) * (l_m * (l_m * (fma(n, ((U - U_42_) / Om), 2.0) / Om))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = fma(-2.0, t_1, t)
	t_3 = Float64(U * Float64(2.0 * n))
	t_4 = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t_2)));
	elseif (t_4 <= 5e+129)
		tmp = sqrt(Float64(t_3 * t_2));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(n * U) * Float64(l_m * Float64(l_m * Float64(fma(n, Float64(Float64(U - U_42_) / Om), 2.0) / Om))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(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[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 5e+129], N[Sqrt[N[(t$95$3 * t$95$2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(n * U), $MachinePrecision] * N[(l$95$m * N[(l$95$m * N[(N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 9.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 27.6

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

   5. Applied rewrites 27.6%

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

   6. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

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

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 34.5

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

   8. Applied rewrites 34.5%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.0000000000000003e129

   1. Initial program 96.4%

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

   3. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 6.4

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 90.5

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

   8. Applied rewrites 90.5%

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

   if 5.0000000000000003e129 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 22.2%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 28.9%

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

   7. Applied rewrites 37.3%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(n \cdot U\right) \cdot \left(\ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(l\_m \cdot -2\right) \cdot \left(U \cdot \frac{l\_m}{Om}\right), 2 \cdot n, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \frac{\mathsf{fma}\left(-2, Om, n \cdot \left(U* - U\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 0.0)
     (sqrt
      (fma (* (* l_m -2.0) (* U (/ l_m Om))) (* 2.0 n) (* U (* t (* 2.0 n)))))
     (if (<= t_2 2e+261)
       (sqrt (* t_1 (- t (/ (* (* l_m l_m) (fma n (/ (- U U*) Om) 2.0)) Om))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* (* n U) (/ (fma -2.0 Om (* n (- U* U))) (* 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 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(fma(((l_m * -2.0) * (U * (l_m / Om))), (2.0 * n), (U * (t * (2.0 * n)))));
	} else if (t_2 <= 2e+261) {
		tmp = sqrt((t_1 * (t - (((l_m * l_m) * fma(n, ((U - U_42_) / Om), 2.0)) / Om))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * (fma(-2.0, Om, (n * (U_42_ - U))) / (Om * Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	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_42_ - U))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(fma(Float64(Float64(l_m * -2.0) * Float64(U * Float64(l_m / Om))), Float64(2.0 * n), Float64(U * Float64(t * Float64(2.0 * n)))));
	elseif (t_2 <= 2e+261)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Om), 2.0)) / Om))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * Float64(fma(-2.0, Om, Float64(n * Float64(U_42_ - U))) / 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[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(N[(l$95$m * -2.0), $MachinePrecision] * N[(U * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * n), $MachinePrecision] + N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+261], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(-2.0 * Om + N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. 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. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 20.0

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 23.1%

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

   6. Applied rewrites 51.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

   8. Applied rewrites 48.8%

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

   9. Taylor expanded in Om around inf

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

       1. lower-*.f64 38.6

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

   11. Applied rewrites 38.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.9999999999999999e261

   1. Initial program 96.4%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 93.9%

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

   if 1.9999999999999999e261 < (*.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 22.5%

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

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 22.7

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 32.7%

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

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 25.2%

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

   10. Taylor expanded in Om around 0

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

   12. Applied rewrites 24.3%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\ell \cdot -2\right) \cdot \left(U \cdot \frac{\ell}{Om}\right), 2 \cdot n, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \frac{\mathsf{fma}\left(-2, Om, n \cdot \left(U* - U\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(l\_m \cdot -2\right) \cdot \left(U \cdot \frac{l\_m}{Om}\right), 2 \cdot n, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+259}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \frac{n \cdot U*}{Om \cdot Om}}\\ \end{array} \end{array} \]

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 (* U (* 2.0 n)))
        (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 -2.0) (* U (/ l_m Om))) (* 2.0 n) (* U (* t (* 2.0 n)))))
     (if (<= t_3 1e+259)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (* (* l_m (sqrt 2.0)) (sqrt (* (* n U) (/ (* n U*) (* 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 = U * (2.0 * n);
	double t_3 = t_2 * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(fma(((l_m * -2.0) * (U * (l_m / Om))), (2.0 * n), (U * (t * (2.0 * n)))));
	} else if (t_3 <= 1e+259) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((n * U_42_) / (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(U * Float64(2.0 * n))
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(fma(Float64(Float64(l_m * -2.0) * Float64(U * Float64(l_m / Om))), Float64(2.0 * n), Float64(U * Float64(t * Float64(2.0 * n)))));
	elseif (t_3 <= 1e+259)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * Float64(Float64(n * U_42_) / 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[(U * N[(2.0 * n), $MachinePrecision]), $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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(N[(l$95$m * -2.0), $MachinePrecision] * N[(U * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * n), $MachinePrecision] + N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+259], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(n * U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. 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. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 20.0

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 23.1%

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

   6. Applied rewrites 51.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

   8. Applied rewrites 48.8%

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

   9. Taylor expanded in Om around inf

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

       1. lower-*.f64 38.6

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

   11. Applied rewrites 38.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.999999999999999e258

   1. Initial program 96.4%

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

   3. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 6.4

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 90.5

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

   8. Applied rewrites 90.5%

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

   if 9.999999999999999e258 < (*.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. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 23.4

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 33.3%

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

   6. Applied rewrites 23.2%

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

   7. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 25.5%

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

   10. Taylor expanded in U* around inf

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

   12. Applied rewrites 20.5%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\ell \cdot -2\right) \cdot \left(U \cdot \frac{\ell}{Om}\right), 2 \cdot n, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+259}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \frac{n \cdot U*}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(-4, \frac{U \cdot \left(l\_m \cdot l\_m\right)}{Om}, 2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+259}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \frac{n \cdot U*}{Om \cdot Om}}\\ \end{array} \end{array} \]

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 (* U (* 2.0 n)))
        (t_3
         (*
          t_2
          (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_3 0.0)
     (sqrt (* n (fma -4.0 (/ (* U (* l_m l_m)) Om) (* 2.0 (* U t)))))
     (if (<= t_3 1e+259)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (* (* l_m (sqrt 2.0)) (sqrt (* (* n U) (/ (* n U*) (* 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 = U * (2.0 * n);
	double t_3 = t_2 * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((n * fma(-4.0, ((U * (l_m * l_m)) / Om), (2.0 * (U * t)))));
	} else if (t_3 <= 1e+259) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * ((n * U_42_) / (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(U * Float64(2.0 * n))
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(n * fma(-4.0, Float64(Float64(U * Float64(l_m * l_m)) / Om), Float64(2.0 * Float64(U * t)))));
	elseif (t_3 <= 1e+259)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * Float64(Float64(n * U_42_) / 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[(U * N[(2.0 * n), $MachinePrecision]), $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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(n * N[(-4.0 * N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+259], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(n * U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. 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. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 20.0

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 23.1%

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

   6. Applied rewrites 51.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

   8. Applied rewrites 48.8%

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

   9. Taylor expanded in n around 0

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 38.4

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

   11. Applied rewrites 38.4%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.999999999999999e258

   1. Initial program 96.4%

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

   3. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 6.4

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 90.5

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

   8. Applied rewrites 90.5%

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

   if 9.999999999999999e258 < (*.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. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 23.4

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 33.3%

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

   6. Applied rewrites 23.2%

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

   7. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 25.5%

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

   10. Taylor expanded in U* around inf

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

   12. Applied rewrites 20.5%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(-4, \frac{U \cdot \left(\ell \cdot \ell\right)}{Om}, 2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+259}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \frac{n \cdot U*}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(-4, \frac{U \cdot \left(l\_m \cdot l\_m\right)}{Om}, 2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+259}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_2 \cdot \frac{U* \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

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 (* U (* 2.0 n)))
        (t_3
         (*
          t_2
          (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_3 0.0)
     (sqrt (* n (fma -4.0 (/ (* U (* l_m l_m)) Om) (* 2.0 (* U t)))))
     (if (<= t_3 1e+259)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (sqrt (* t_2 (/ (* U* (* n (* l_m l_m))) (* 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 = U * (2.0 * n);
	double t_3 = t_2 * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((n * fma(-4.0, ((U * (l_m * l_m)) / Om), (2.0 * (U * t)))));
	} else if (t_3 <= 1e+259) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = sqrt((t_2 * ((U_42_ * (n * (l_m * l_m))) / (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(U * Float64(2.0 * n))
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(n * fma(-4.0, Float64(Float64(U * Float64(l_m * l_m)) / Om), Float64(2.0 * Float64(U * t)))));
	elseif (t_3 <= 1e+259)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = sqrt(Float64(t_2 * Float64(Float64(U_42_ * Float64(n * Float64(l_m * l_m))) / 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[(U * N[(2.0 * n), $MachinePrecision]), $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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(n * N[(-4.0 * N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+259], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$2 * N[(N[(U$42$ * N[(n * N[(l$95$m * l$95$m), $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*)))) < 0.0

   1. Initial program 7.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. 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. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 20.0

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 23.1%

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

   6. Applied rewrites 51.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

   8. Applied rewrites 48.8%

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

   9. Taylor expanded in n around 0

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 38.4

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

   11. Applied rewrites 38.4%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.999999999999999e258

   1. Initial program 96.4%

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

   3. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 6.4

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 90.5

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

   8. Applied rewrites 90.5%

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

   if 9.999999999999999e258 < (*.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. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(-4, \frac{U \cdot \left(\ell \cdot \ell\right)}{Om}, 2 \cdot \left(U \cdot t\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+259}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.9% accurate, 0.7× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (*
       (* U (* 2.0 n))
       (+
        (- t (* 2.0 (/ (* l_m l_m) Om)))
        (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))
      INFINITY)
   (sqrt
    (*
     U
     (*
      (fma (/ l_m Om) (fma (- U* U) (* n (/ l_m Om)) (* l_m -2.0)) t)
      (* 2.0 n))))
   (*
    (sqrt (* n (* U (/ (- (* n (/ (- 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 tmp;
	if (((U * (2.0 * n)) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))) <= ((double) INFINITY)) {
		tmp = sqrt((U * (fma((l_m / Om), fma((U_42_ - U), (n * (l_m / Om)), (l_m * -2.0)), t) * (2.0 * n))));
	} else {
		tmp = sqrt((n * (U * (((n * ((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_)
	tmp = 0.0
	if (Float64(Float64(U * Float64(2.0 * n)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))) <= Inf)
		tmp = sqrt(Float64(U * Float64(fma(Float64(l_m / Om), fma(Float64(U_42_ - U), Float64(n * Float64(l_m / Om)), Float64(l_m * -2.0)), t) * Float64(2.0 * n))));
	else
		tmp = Float64(sqrt(Float64(n * Float64(U * Float64(Float64(Float64(n * Float64(Float64(U_42_ - U) / Om)) - 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$42$_] := If[LessEqual[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[Sqrt[N[(U * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * N[(U * N[(N[(N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $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*)))) < +inf.0

   1. Initial program 59.1%

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

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 61.6

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 66.9%

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

   6. Applied rewrites 68.3%

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

   7. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 68.3

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

   9. Applied rewrites 68.3%

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

   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. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 0.9

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 4.1%

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

   6. Applied rewrites 18.5%

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

   7. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 32.9%

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

   11. Applied rewrites 47.6%

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

4. Final simplification 65.5%

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

Alternative 10: 61.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[Sqrt[N[(U * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(-2.0 * Om + N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $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*)))) < +inf.0

   1. Initial program 59.1%

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

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 61.6

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 66.9%

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

   6. Applied rewrites 68.3%

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

   7. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 68.3

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

   9. Applied rewrites 68.3%

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

   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. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 0.9

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 4.1%

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

   6. Applied rewrites 18.5%

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

   7. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 32.9%

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

   10. Taylor expanded in Om around 0

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

   12. Applied rewrites 33.0%

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

4. Final simplification 63.6%

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

Alternative 11: 46.9% accurate, 0.8× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

   1. Initial program 59.1%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 43.7

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

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

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 55.5

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

   8. Applied rewrites 55.5%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.9%

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

   5. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 30.5

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

   7. Applied rewrites 30.5%

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

   9. Applied rewrites 27.9%

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

   11. Applied rewrites 27.8%

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

4. Final simplification 51.8%

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

Alternative 12: 46.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Wolfram

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

   1. Initial program 59.1%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 43.7

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

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

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 55.5

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

   8. Applied rewrites 55.5%

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

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

   1. Initial program 0.0%

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

   3. 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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 27.9

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

   5. Applied rewrites 27.9%

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

4. Final simplification 51.8%

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

Alternative 13: 48.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.12 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \frac{-2}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.12e+151)
   (sqrt (* (* 2.0 U) (* n (fma -2.0 (/ (* l_m l_m) Om) t))))
   (* (* l_m (sqrt 2.0)) (sqrt (* (* n U) (/ -2.0 Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.12e+151) {
		tmp = sqrt(((2.0 * U) * (n * fma(-2.0, ((l_m * l_m) / Om), t))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * (-2.0 / Om)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.12e+151)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * fma(-2.0, Float64(Float64(l_m * l_m) / Om), t))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * Float64(-2.0 / Om))));
	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, 1.12e+151], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.12000000000000004e151

   1. Initial program 56.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

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

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 53.8

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

   8. Applied rewrites 53.8%

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

   if 1.12000000000000004e151 < l

   1. Initial program 14.0%

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

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. lower-*.f64 14.3

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. lift--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \color{blue}{t - 2 \cdot \frac{\ell \cdot \ell}{Om}}\right)} \]

      17. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

   4. Applied rewrites 38.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   6. Applied rewrites 22.8%

      \[\leadsto \color{blue}{\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(-\left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right)}} \]

   7. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right)} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\color{blue}{\frac{n \cdot \left(U* - U\right)}{{Om}^{2}}} - 2 \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{\color{blue}{n \cdot \left(U* - U\right)}}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      9. lower--.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \color{blue}{\left(U* - U\right)}}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{\color{blue}{2}}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \color{blue}{\frac{2}{Om}}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \color{blue}{\left(\ell \cdot \sqrt{2}\right)} \]

   9. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]

   10. Taylor expanded in n around 0

       \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \frac{-2}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 23.8%

       \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \frac{-2}{Om}} \cdot \left(\ell \cdot \sqrt{2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.12 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \frac{-2}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \frac{2 \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5.5e+24)
   (sqrt (* (* 2.0 U) (* n t)))
   (sqrt (* (* U -2.0) (/ (* 2.0 (* n (* l_m l_m))) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 5.5e+24) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = sqrt(((U * -2.0) * ((2.0 * (n * (l_m * l_m))) / Om)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 5.5d+24) then
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    else
        tmp = sqrt(((u * (-2.0d0)) * ((2.0d0 * (n * (l_m * l_m))) / om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 5.5e+24) {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = Math.sqrt(((U * -2.0) * ((2.0 * (n * (l_m * l_m))) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 5.5e+24:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	else:
		tmp = math.sqrt(((U * -2.0) * ((2.0 * (n * (l_m * l_m))) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 5.5e+24)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	else
		tmp = sqrt(Float64(Float64(U * -2.0) * Float64(Float64(2.0 * Float64(n * Float64(l_m * l_m))) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 5.5e+24)
		tmp = sqrt(((2.0 * U) * (n * t)));
	else
		tmp = sqrt(((U * -2.0) * ((2.0 * (n * (l_m * l_m))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 5.5e+24], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * -2.0), $MachinePrecision] * N[(N[(2.0 * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.5000000000000002e24

   1. Initial program 56.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 47.2

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Applied rewrites 47.2%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 5.5000000000000002e24 < l

   1. Initial program 36.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right)} \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)}\right)} \]

      10. associate-/l* N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\color{blue}{n \cdot \frac{U - U*}{{Om}^{2}}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(n, \frac{U - U*}{{Om}^{2}}, 2 \cdot \frac{1}{Om}\right)}\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \color{blue}{\frac{U - U*}{{Om}^{2}}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{\color{blue}{U - U*}}{{Om}^{2}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      16. associate-*r/ N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{\color{blue}{2}}{Om}\right)\right)} \]

      18. lower-/.f64 35.3

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \color{blue}{\frac{2}{Om}}\right)\right)} \]

   5. Applied rewrites 35.3%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right)}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot n}{Om}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 26.3%

      \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{\color{blue}{Om}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \frac{2 \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(U \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5.5e+24)
   (sqrt (* (* 2.0 U) (* n t)))
   (sqrt (/ (* -4.0 (* U (* n (* l_m l_m)))) Om))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 5.5e+24) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = sqrt(((-4.0 * (U * (n * (l_m * l_m)))) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 5.5d+24) then
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    else
        tmp = sqrt((((-4.0d0) * (u * (n * (l_m * l_m)))) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 5.5e+24) {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = Math.sqrt(((-4.0 * (U * (n * (l_m * l_m)))) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 5.5e+24:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	else:
		tmp = math.sqrt(((-4.0 * (U * (n * (l_m * l_m)))) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 5.5e+24)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(U * Float64(n * Float64(l_m * l_m)))) / Om));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 5.5e+24)
		tmp = sqrt(((2.0 * U) * (n * t)));
	else
		tmp = sqrt(((-4.0 * (U * (n * (l_m * l_m)))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 5.5e+24], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-4.0 * N[(U * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.5000000000000002e24

   1. Initial program 56.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 47.2

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Applied rewrites 47.2%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 5.5000000000000002e24 < l

   1. Initial program 36.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right)} \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)}\right)} \]

      10. associate-/l* N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\color{blue}{n \cdot \frac{U - U*}{{Om}^{2}}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(n, \frac{U - U*}{{Om}^{2}}, 2 \cdot \frac{1}{Om}\right)}\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \color{blue}{\frac{U - U*}{{Om}^{2}}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{\color{blue}{U - U*}}{{Om}^{2}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      16. associate-*r/ N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{\color{blue}{2}}{Om}\right)\right)} \]

      18. lower-/.f64 35.3

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \color{blue}{\frac{2}{Om}}\right)\right)} \]

   5. Applied rewrites 35.3%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right)}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{-4 \cdot \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.6%

      \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{\color{blue}{Om}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (sqrt (* (* 2.0 U) (* n (fma -2.0 (/ (* l_m l_m) Om) t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt(((2.0 * U) * (n * fma(-2.0, ((l_m * l_m) / Om), t))));
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(2.0 * U) * Float64(n * fma(-2.0, Float64(Float64(l_m * l_m) / Om), t))))
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

   4. lower-*.f64 39.8

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

5. Applied rewrites 39.8%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

6. Taylor expanded in n around 0

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

   2. cancel-sign-sub-inv N/A

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

   7. +-commutative N/A

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)}\right)} \]

   9. lower-/.f64 N/A

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right)\right)} \]

   10. unpow2 N/A

       \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)} \]

   11. lower-*.f64 48.8

       \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)} \]

8. Applied rewrites 48.8%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]
9. Add Preprocessing

Alternative 17: 38.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 2.8 \cdot 10^{-262}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 2.8e-262)
   (sqrt (* (* 2.0 U) (* n t)))
   (* (sqrt (* 2.0 n)) (sqrt (* U t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 2.8e-262) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 2.8d-262) then
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    else
        tmp = sqrt((2.0d0 * n)) * sqrt((u * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 2.8e-262) {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 2.8e-262:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	else:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 2.8e-262)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	else
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= 2.8e-262)
		tmp = sqrt(((2.0 * U) * (n * t)));
	else
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 2.8e-262], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.79999999999999975e-262

   1. Initial program 51.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 44.1

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Applied rewrites 44.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 2.79999999999999975e-262 < n

   1. Initial program 51.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. lower-*.f64 52.5

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \color{blue}{\left(n \cdot \frac{\ell}{Om}\right)}, \frac{\ell}{Om}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. lift--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \color{blue}{t - 2 \cdot \frac{\ell \cdot \ell}{Om}}\right)} \]

      17. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

   4. Applied rewrites 57.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   6. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(-\left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right)}} \]

   7. Taylor expanded in l around 0

      \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{\color{blue}{U \cdot t}} \]
   8. Step-by-step derivation

      1. lower-*.f64 38.6

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{\color{blue}{U \cdot t}} \]

   9. Applied rewrites 38.6%

      \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{\color{blue}{U \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.8 \cdot 10^{-262}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -6 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -6e-288) (sqrt (* (* 2.0 U) (* n t))) (sqrt (* (* U (* 2.0 n)) t))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -6e-288) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = sqrt(((U * (2.0 * n)) * t));
	}
	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 (u <= (-6d-288)) then
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    else
        tmp = sqrt(((u * (2.0d0 * n)) * t))
    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 (U <= -6e-288) {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = Math.sqrt(((U * (2.0 * n)) * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= -6e-288:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	else:
		tmp = math.sqrt(((U * (2.0 * n)) * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= -6e-288)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	else
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * t));
	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 (U <= -6e-288)
		tmp = sqrt(((2.0 * U) * (n * t)));
	else
		tmp = sqrt(((U * (2.0 * n)) * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, -6e-288], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -5.99999999999999998e-288

   1. Initial program 44.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 40.7

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Applied rewrites 40.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if -5.99999999999999998e-288 < U

   1. Initial program 56.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 39.0

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Applied rewrites 39.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.6%

      \[\leadsto \sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -6 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* 2.0 U) (* n t))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt(((2.0 * U) * (n * t)));
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt(((2.0d0 * u) * (n * t)))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt(((2.0 * U) * (n * t)));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt(((2.0 * U) * (n * t)))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(2.0 * U) * Float64(n * t)))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt(((2.0 * U) * (n * t)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

   4. lower-*.f64 39.8

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

5. Applied rewrites 39.8%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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*))))))