Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 64.1%

Time: 18.6s

Alternatives: 15

Speedup: 2.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \left(\left(U* - U\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right) - \left(\frac{l\_m \cdot l\_m}{Om} \cdot 2 - t\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot \frac{l\_m}{Om}, \frac{l\_m}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{l\_m}{Om} \cdot l\_m, t\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\frac{\left(U* - U\right) \cdot n}{Om \cdot Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2
         (*
          (-
           (* (- U* U) (* (pow (/ l_m Om) 2.0) n))
           (- (* (/ (* l_m l_m) Om) 2.0) t))
          t_1)))
   (if (<= t_2 5e-319)
     (* (sqrt U) (sqrt (* (fma (* -2.0 l_m) (/ l_m Om) t) (* n 2.0))))
     (if (<= t_2 4e+288)
       (sqrt
        (*
         (fma
          (* (- U* U) (/ l_m Om))
          (* (/ l_m Om) n)
          (fma -2.0 (* (/ l_m Om) l_m) t))
         t_1))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (- (/ (* (- U* U) n) (* Om Om)) (/ 2.0 Om)) (* U n))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (((U_42_ - U) * (pow((l_m / Om), 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * t_1;
	double tmp;
	if (t_2 <= 5e-319) {
		tmp = sqrt(U) * sqrt((fma((-2.0 * l_m), (l_m / Om), t) * (n * 2.0)));
	} else if (t_2 <= 4e+288) {
		tmp = sqrt((fma(((U_42_ - U) * (l_m / Om)), ((l_m / Om) * n), fma(-2.0, ((l_m / Om) * l_m), t)) * t_1));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt((((((U_42_ - U) * n) / (Om * Om)) - (2.0 / Om)) * (U * n)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l_m / Om) ^ 2.0) * n)) - Float64(Float64(Float64(Float64(l_m * l_m) / Om) * 2.0) - t)) * t_1)
	tmp = 0.0
	if (t_2 <= 5e-319)
		tmp = Float64(sqrt(U) * sqrt(Float64(fma(Float64(-2.0 * l_m), Float64(l_m / Om), t) * Float64(n * 2.0))));
	elseif (t_2 <= 4e+288)
		tmp = sqrt(Float64(fma(Float64(Float64(U_42_ - U) * Float64(l_m / Om)), Float64(Float64(l_m / Om) * n), fma(-2.0, Float64(Float64(l_m / Om) * l_m), t)) * t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) * n) / Float64(Om * Om)) - Float64(2.0 / Om)) * Float64(U * n))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-319], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+288], N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * n), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999937e-319

   1. Initial program 13.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 18.8

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 24.1%

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

   6. Applied rewrites 24.2%

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

   7. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 24.2

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

   9. Applied rewrites 24.2%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. sqrt-prod N/A

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

       6. pow1/2 N/A

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

   11. Applied rewrites 44.2%

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

   if 4.9999937e-319 < (*.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*)))) < 4e288

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 97.6

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 97.7%

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

   if 4e288 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 19.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 21.0

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 31.7%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 23.0%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\frac{\left(U* - U\right) \cdot n}{Om \cdot Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \left(\left(U* - U\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right) - \left(\frac{l\_m \cdot l\_m}{Om} \cdot 2 - t\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U - U*, \left(\frac{l\_m}{Om \cdot Om} \cdot l\_m\right) \cdot \left(-n\right), \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\frac{\left(U* - U\right) \cdot n}{Om \cdot Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2
         (*
          (-
           (* (- U* U) (* (pow (/ l_m Om) 2.0) n))
           (- (* (/ (* l_m l_m) Om) 2.0) t))
          t_1)))
   (if (<= t_2 5e-319)
     (* (sqrt U) (sqrt (* (fma (* -2.0 l_m) (/ l_m Om) t) (* n 2.0))))
     (if (<= t_2 4e+288)
       (sqrt
        (*
         (fma
          (- U U*)
          (* (* (/ l_m (* Om Om)) l_m) (- n))
          (fma (* (/ l_m Om) l_m) -2.0 t))
         t_1))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (- (/ (* (- U* U) n) (* Om Om)) (/ 2.0 Om)) (* U n))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (((U_42_ - U) * (pow((l_m / Om), 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * t_1;
	double tmp;
	if (t_2 <= 5e-319) {
		tmp = sqrt(U) * sqrt((fma((-2.0 * l_m), (l_m / Om), t) * (n * 2.0)));
	} else if (t_2 <= 4e+288) {
		tmp = sqrt((fma((U - U_42_), (((l_m / (Om * Om)) * l_m) * -n), fma(((l_m / Om) * l_m), -2.0, t)) * t_1));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt((((((U_42_ - U) * n) / (Om * Om)) - (2.0 / Om)) * (U * n)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l_m / Om) ^ 2.0) * n)) - Float64(Float64(Float64(Float64(l_m * l_m) / Om) * 2.0) - t)) * t_1)
	tmp = 0.0
	if (t_2 <= 5e-319)
		tmp = Float64(sqrt(U) * sqrt(Float64(fma(Float64(-2.0 * l_m), Float64(l_m / Om), t) * Float64(n * 2.0))));
	elseif (t_2 <= 4e+288)
		tmp = sqrt(Float64(fma(Float64(U - U_42_), Float64(Float64(Float64(l_m / Float64(Om * Om)) * l_m) * Float64(-n)), fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)) * t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) * n) / Float64(Om * Om)) - Float64(2.0 / Om)) * Float64(U * n))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-319], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+288], N[Sqrt[N[(N[(N[(U - U$42$), $MachinePrecision] * N[(N[(N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision] * (-n)), $MachinePrecision] + N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * n), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999937e-319

   1. Initial program 13.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 18.8

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 24.1%

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

   6. Applied rewrites 24.2%

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

   7. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 24.2

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

   9. Applied rewrites 24.2%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. sqrt-prod N/A

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

       6. pow1/2 N/A

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

   11. Applied rewrites 44.2%

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

   if 4.9999937e-319 < (*.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*)))) < 4e288

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 97.6

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 97.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 92.7%

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

   if 4e288 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 19.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 21.0

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 31.7%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 23.0%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U - U*, \left(\frac{\ell}{Om \cdot Om} \cdot \ell\right) \cdot \left(-n\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\frac{\left(U* - U\right) \cdot n}{Om \cdot Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right)\\ t_2 := U \cdot \left(n \cdot 2\right)\\ t_3 := \left(\left(U* - U\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right) - \left(\frac{l\_m \cdot l\_m}{Om} \cdot 2 - t\right)\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{t\_1 \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{t\_1 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\frac{\left(U* - U\right) \cdot n}{Om \cdot Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* -2.0 l_m) (/ l_m Om) t))
        (t_2 (* U (* n 2.0)))
        (t_3
         (*
          (-
           (* (- U* U) (* (pow (/ l_m Om) 2.0) n))
           (- (* (/ (* l_m l_m) Om) 2.0) t))
          t_2)))
   (if (<= t_3 5e-319)
     (* (sqrt U) (sqrt (* t_1 (* n 2.0))))
     (if (<= t_3 5e+215)
       (sqrt (* t_1 t_2))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (- (/ (* (- U* U) n) (* Om Om)) (/ 2.0 Om)) (* U n))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma((-2.0 * l_m), (l_m / Om), t);
	double t_2 = U * (n * 2.0);
	double t_3 = (((U_42_ - U) * (pow((l_m / Om), 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * t_2;
	double tmp;
	if (t_3 <= 5e-319) {
		tmp = sqrt(U) * sqrt((t_1 * (n * 2.0)));
	} else if (t_3 <= 5e+215) {
		tmp = sqrt((t_1 * t_2));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt((((((U_42_ - U) * n) / (Om * Om)) - (2.0 / Om)) * (U * n)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(-2.0 * l_m), Float64(l_m / Om), t)
	t_2 = Float64(U * Float64(n * 2.0))
	t_3 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l_m / Om) ^ 2.0) * n)) - Float64(Float64(Float64(Float64(l_m * l_m) / Om) * 2.0) - t)) * t_2)
	tmp = 0.0
	if (t_3 <= 5e-319)
		tmp = Float64(sqrt(U) * sqrt(Float64(t_1 * Float64(n * 2.0))));
	elseif (t_3 <= 5e+215)
		tmp = sqrt(Float64(t_1 * t_2));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) * n) / Float64(Om * Om)) - Float64(2.0 / Om)) * Float64(U * n))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-319], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+215], N[Sqrt[N[(t$95$1 * t$95$2), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * n), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999937e-319

   1. Initial program 13.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 18.8

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 24.1%

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

   6. Applied rewrites 24.2%

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

   7. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 24.2

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

   9. Applied rewrites 24.2%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. sqrt-prod N/A

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

       6. pow1/2 N/A

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

   11. Applied rewrites 44.2%

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

   if 4.9999937e-319 < (*.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*)))) < 5.0000000000000001e215

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 97.6

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 97.6%

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

   6. Applied rewrites 36.4%

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

   7. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 40.5

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

   9. Applied rewrites 40.5%

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

       1. lift-*.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqrt-unprod N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

   11. Applied rewrites 92.7%

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

   if 5.0000000000000001e215 < (*.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 21.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 23.0

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 33.4%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 23.3%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\frac{\left(U* - U\right) \cdot n}{Om \cdot Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right)\\ t_2 := U \cdot \left(n \cdot 2\right)\\ t_3 := \left(\left(U* - U\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right) - \left(\frac{l\_m \cdot l\_m}{Om} \cdot 2 - t\right)\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{t\_1 \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, \frac{\left(\left(n \cdot n\right) \cdot U\right) \cdot \left(U* - U\right)}{Om \cdot Om}, \frac{U \cdot n}{Om} \cdot -4\right)} \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* -2.0 l_m) (/ l_m Om) t))
        (t_2 (* U (* n 2.0)))
        (t_3
         (*
          (-
           (* (- U* U) (* (pow (/ l_m Om) 2.0) n))
           (- (* (/ (* l_m l_m) Om) 2.0) t))
          t_2)))
   (if (<= t_3 5e-319)
     (* (sqrt U) (sqrt (* t_1 (* n 2.0))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_1 t_2))
       (*
        (sqrt
         (fma
          2.0
          (/ (* (* (* n n) U) (- U* U)) (* Om Om))
          (* (/ (* U n) Om) -4.0)))
        l_m)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma((-2.0 * l_m), (l_m / Om), t);
	double t_2 = U * (n * 2.0);
	double t_3 = (((U_42_ - U) * (pow((l_m / Om), 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * t_2;
	double tmp;
	if (t_3 <= 5e-319) {
		tmp = sqrt(U) * sqrt((t_1 * (n * 2.0)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * t_2));
	} else {
		tmp = sqrt(fma(2.0, ((((n * n) * U) * (U_42_ - U)) / (Om * Om)), (((U * n) / Om) * -4.0))) * l_m;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(-2.0 * l_m), Float64(l_m / Om), t)
	t_2 = Float64(U * Float64(n * 2.0))
	t_3 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l_m / Om) ^ 2.0) * n)) - Float64(Float64(Float64(Float64(l_m * l_m) / Om) * 2.0) - t)) * t_2)
	tmp = 0.0
	if (t_3 <= 5e-319)
		tmp = Float64(sqrt(U) * sqrt(Float64(t_1 * Float64(n * 2.0))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_1 * t_2));
	else
		tmp = Float64(sqrt(fma(2.0, Float64(Float64(Float64(Float64(n * n) * U) * Float64(U_42_ - U)) / Float64(Om * Om)), Float64(Float64(Float64(U * n) / Om) * -4.0))) * l_m);
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-319], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$1 * t$95$2), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(N[(N[(N[(n * n), $MachinePrecision] * U), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(U * n), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * l$95$m), $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*)))) < 4.9999937e-319

   1. Initial program 13.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 18.8

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 24.1%

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

   6. Applied rewrites 24.2%

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

   7. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 24.2

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

   9. Applied rewrites 24.2%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. sqrt-prod N/A

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

       6. pow1/2 N/A

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

   11. Applied rewrites 44.2%

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

   if 4.9999937e-319 < (*.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 65.9%

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

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 66.9

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 73.8%

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

   6. Applied rewrites 27.4%

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

   7. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.3

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

   9. Applied rewrites 27.3%

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

       1. lift-*.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqrt-unprod N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

   11. Applied rewrites 64.9%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 1.0

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 1.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 16.8

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

   9. Applied rewrites 16.8%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, \frac{\left(\left(n \cdot n\right) \cdot U\right) \cdot \left(U* - U\right)}{Om \cdot Om}, \frac{U \cdot n}{Om} \cdot -4\right)} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \left(\left(U* - U\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right) - \left(\frac{l\_m \cdot l\_m}{Om} \cdot 2 - t\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{n}{Om \cdot Om} \cdot U*\right) \cdot \left(l\_m \cdot l\_m\right)\right) \cdot \left(n \cdot 2\right)\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2
         (*
          (-
           (* (- U* U) (* (pow (/ l_m Om) 2.0) n))
           (- (* (/ (* l_m l_m) Om) 2.0) t))
          t_1)))
   (if (<= t_2 5e-319)
     (* (sqrt (* (* t n) 2.0)) (sqrt U))
     (if (<= t_2 INFINITY)
       (sqrt (* (fma (* -2.0 l_m) (/ l_m Om) t) t_1))
       (sqrt (* (* (* (* (/ n (* Om Om)) U*) (* l_m l_m)) (* n 2.0)) U))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (((U_42_ - U) * (pow((l_m / Om), 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * t_1;
	double tmp;
	if (t_2 <= 5e-319) {
		tmp = sqrt(((t * n) * 2.0)) * sqrt(U);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((fma((-2.0 * l_m), (l_m / Om), t) * t_1));
	} else {
		tmp = sqrt((((((n / (Om * Om)) * U_42_) * (l_m * l_m)) * (n * 2.0)) * U));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l_m / Om) ^ 2.0) * n)) - Float64(Float64(Float64(Float64(l_m * l_m) / Om) * 2.0) - t)) * t_1)
	tmp = 0.0
	if (t_2 <= 5e-319)
		tmp = Float64(sqrt(Float64(Float64(t * n) * 2.0)) * sqrt(U));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(fma(Float64(-2.0 * l_m), Float64(l_m / Om), t) * t_1));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(n / Float64(Om * Om)) * U_42_) * Float64(l_m * l_m)) * Float64(n * 2.0)) * U));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-319], N[(N[Sqrt[N[(N[(t * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * U$42$), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999937e-319

   1. Initial program 13.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 33.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.6

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

   7. Applied rewrites 41.6%

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

   if 4.9999937e-319 < (*.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 65.9%

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

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 66.9

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 73.8%

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

   6. Applied rewrites 27.4%

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

   7. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.3

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

   9. Applied rewrites 27.3%

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

       1. lift-*.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqrt-unprod N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

   11. Applied rewrites 64.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around inf

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 27.1%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{n}{Om \cdot Om} \cdot U*\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot 2\right)\right) \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \left(\left(U* - U\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right) - \left(\frac{l\_m \cdot l\_m}{Om} \cdot 2 - t\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot n\right) \cdot l\_m}{Om} \cdot \sqrt{U* \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2
         (*
          (-
           (* (- U* U) (* (pow (/ l_m Om) 2.0) n))
           (- (* (/ (* l_m l_m) Om) 2.0) t))
          t_1)))
   (if (<= t_2 5e-319)
     (* (sqrt (* (* t n) 2.0)) (sqrt U))
     (if (<= t_2 INFINITY)
       (sqrt (* (fma (* -2.0 l_m) (/ l_m Om) t) t_1))
       (* (/ (* (* (sqrt 2.0) n) l_m) Om) (sqrt (* U* U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (((U_42_ - U) * (pow((l_m / Om), 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * t_1;
	double tmp;
	if (t_2 <= 5e-319) {
		tmp = sqrt(((t * n) * 2.0)) * sqrt(U);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((fma((-2.0 * l_m), (l_m / Om), t) * t_1));
	} else {
		tmp = (((sqrt(2.0) * n) * l_m) / Om) * sqrt((U_42_ * U));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l_m / Om) ^ 2.0) * n)) - Float64(Float64(Float64(Float64(l_m * l_m) / Om) * 2.0) - t)) * t_1)
	tmp = 0.0
	if (t_2 <= 5e-319)
		tmp = Float64(sqrt(Float64(Float64(t * n) * 2.0)) * sqrt(U));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(fma(Float64(-2.0 * l_m), Float64(l_m / Om), t) * t_1));
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(2.0) * n) * l_m) / Om) * sqrt(Float64(U_42_ * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-319], N[(N[Sqrt[N[(N[(t * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999937e-319

   1. Initial program 13.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 33.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.6

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

   7. Applied rewrites 41.6%

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

   if 4.9999937e-319 < (*.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 65.9%

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

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 66.9

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 73.8%

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

   6. Applied rewrites 27.4%

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

   7. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.3

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

   9. Applied rewrites 27.3%

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

       1. lift-*.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqrt-unprod N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

   11. Applied rewrites 64.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 17.8

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

   5. Applied rewrites 17.8%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{U* \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \left(\left(U* - U\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right) - \left(\frac{l\_m \cdot l\_m}{Om} \cdot 2 - t\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(U* \cdot U\right) \cdot 2} \cdot \left(l\_m \cdot n\right)}{Om}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2
         (*
          (-
           (* (- U* U) (* (pow (/ l_m Om) 2.0) n))
           (- (* (/ (* l_m l_m) Om) 2.0) t))
          t_1)))
   (if (<= t_2 5e-319)
     (* (sqrt (* (* t n) 2.0)) (sqrt U))
     (if (<= t_2 INFINITY)
       (sqrt (* (fma (* -2.0 l_m) (/ l_m Om) t) t_1))
       (/ (* (sqrt (* (* U* U) 2.0)) (* l_m n)) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (((U_42_ - U) * (pow((l_m / Om), 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * t_1;
	double tmp;
	if (t_2 <= 5e-319) {
		tmp = sqrt(((t * n) * 2.0)) * sqrt(U);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((fma((-2.0 * l_m), (l_m / Om), t) * t_1));
	} else {
		tmp = (sqrt(((U_42_ * U) * 2.0)) * (l_m * n)) / Om;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l_m / Om) ^ 2.0) * n)) - Float64(Float64(Float64(Float64(l_m * l_m) / Om) * 2.0) - t)) * t_1)
	tmp = 0.0
	if (t_2 <= 5e-319)
		tmp = Float64(sqrt(Float64(Float64(t * n) * 2.0)) * sqrt(U));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(fma(Float64(-2.0 * l_m), Float64(l_m / Om), t) * t_1));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(U_42_ * U) * 2.0)) * Float64(l_m * n)) / 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[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-319], N[(N[Sqrt[N[(N[(t * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(N[(U$42$ * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $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*)))) < 4.9999937e-319

   1. Initial program 13.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 33.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.6

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

   7. Applied rewrites 41.6%

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

   if 4.9999937e-319 < (*.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 65.9%

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

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 66.9

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 73.8%

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

   6. Applied rewrites 27.4%

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

   7. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 27.3

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

   9. Applied rewrites 27.3%

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

       1. lift-*.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqrt-unprod N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

   11. Applied rewrites 64.9%

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

   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-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 0.2%

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

   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. 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 17.8

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

   7. Applied rewrites 17.8%

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

   9. Applied rewrites 17.7%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(U* \cdot U\right) \cdot 2} \cdot \left(\ell \cdot n\right)}{Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(U* - U\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right) - \left(\frac{l\_m \cdot l\_m}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(U* \cdot U\right) \cdot 2} \cdot \left(l\_m \cdot n\right)}{Om}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (-
           (* (- U* U) (* (pow (/ l_m Om) 2.0) n))
           (- (* (/ (* l_m l_m) Om) 2.0) t))
          (* U (* n 2.0)))))
   (if (<= t_1 5e-319)
     (* (sqrt (* (* t n) 2.0)) (sqrt U))
     (if (<= t_1 4e+288)
       (sqrt (* (* (* U n) t) 2.0))
       (/ (* (sqrt (* (* U* U) 2.0)) (* l_m n)) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (((U_42_ - U) * (pow((l_m / Om), 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * (U * (n * 2.0));
	double tmp;
	if (t_1 <= 5e-319) {
		tmp = sqrt(((t * n) * 2.0)) * sqrt(U);
	} else if (t_1 <= 4e+288) {
		tmp = sqrt((((U * n) * t) * 2.0));
	} else {
		tmp = (sqrt(((U_42_ * U) * 2.0)) * (l_m * n)) / Om;
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (((u_42 - u) * (((l_m / om) ** 2.0d0) * n)) - ((((l_m * l_m) / om) * 2.0d0) - t)) * (u * (n * 2.0d0))
    if (t_1 <= 5d-319) then
        tmp = sqrt(((t * n) * 2.0d0)) * sqrt(u)
    else if (t_1 <= 4d+288) then
        tmp = sqrt((((u * n) * t) * 2.0d0))
    else
        tmp = (sqrt(((u_42 * u) * 2.0d0)) * (l_m * n)) / om
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (((U_42_ - U) * (Math.pow((l_m / Om), 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * (U * (n * 2.0));
	double tmp;
	if (t_1 <= 5e-319) {
		tmp = Math.sqrt(((t * n) * 2.0)) * Math.sqrt(U);
	} else if (t_1 <= 4e+288) {
		tmp = Math.sqrt((((U * n) * t) * 2.0));
	} else {
		tmp = (Math.sqrt(((U_42_ * U) * 2.0)) * (l_m * n)) / Om;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (((U_42_ - U) * (math.pow((l_m / Om), 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * (U * (n * 2.0))
	tmp = 0
	if t_1 <= 5e-319:
		tmp = math.sqrt(((t * n) * 2.0)) * math.sqrt(U)
	elif t_1 <= 4e+288:
		tmp = math.sqrt((((U * n) * t) * 2.0))
	else:
		tmp = (math.sqrt(((U_42_ * U) * 2.0)) * (l_m * n)) / Om
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l_m / Om) ^ 2.0) * n)) - Float64(Float64(Float64(Float64(l_m * l_m) / Om) * 2.0) - t)) * Float64(U * Float64(n * 2.0)))
	tmp = 0.0
	if (t_1 <= 5e-319)
		tmp = Float64(sqrt(Float64(Float64(t * n) * 2.0)) * sqrt(U));
	elseif (t_1 <= 4e+288)
		tmp = sqrt(Float64(Float64(Float64(U * n) * t) * 2.0));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(U_42_ * U) * 2.0)) * Float64(l_m * n)) / Om);
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (((U_42_ - U) * (((l_m / Om) ^ 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * (U * (n * 2.0));
	tmp = 0.0;
	if (t_1 <= 5e-319)
		tmp = sqrt(((t * n) * 2.0)) * sqrt(U);
	elseif (t_1 <= 4e+288)
		tmp = sqrt((((U * n) * t) * 2.0));
	else
		tmp = (sqrt(((U_42_ * U) * 2.0)) * (l_m * n)) / Om;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-319], N[(N[Sqrt[N[(N[(t * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+288], N[Sqrt[N[(N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(N[(U$42$ * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $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*)))) < 4.9999937e-319

   1. Initial program 13.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 33.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.6

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

   7. Applied rewrites 41.6%

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

   if 4.9999937e-319 < (*.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*)))) < 4e288

   1. Initial program 97.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 82.6%

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

   if 4e288 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 19.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 12.1%

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

   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. 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 22.4

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

   7. Applied rewrites 22.4%

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

   9. Applied rewrites 21.7%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(U* \cdot U\right) \cdot 2} \cdot \left(\ell \cdot n\right)}{Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(U* - U\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right) - \left(\frac{l\_m \cdot l\_m}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 5 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (-
         (* (- U* U) (* (pow (/ l_m Om) 2.0) n))
         (- (* (/ (* l_m l_m) Om) 2.0) t))
        (* U (* n 2.0))))
      5e-156)
   (sqrt (* (* t U) (* n 2.0)))
   (sqrt (* (* (* U n) t) 2.0))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (sqrt(((((U_42_ - U) * (pow((l_m / Om), 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * (U * (n * 2.0)))) <= 5e-156) {
		tmp = sqrt(((t * U) * (n * 2.0)));
	} else {
		tmp = sqrt((((U * n) * t) * 2.0));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (sqrt(((((u_42 - u) * (((l_m / om) ** 2.0d0) * n)) - ((((l_m * l_m) / om) * 2.0d0) - t)) * (u * (n * 2.0d0)))) <= 5d-156) then
        tmp = sqrt(((t * u) * (n * 2.0d0)))
    else
        tmp = sqrt((((u * n) * t) * 2.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Math.sqrt(((((U_42_ - U) * (Math.pow((l_m / Om), 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * (U * (n * 2.0)))) <= 5e-156) {
		tmp = Math.sqrt(((t * U) * (n * 2.0)));
	} else {
		tmp = Math.sqrt((((U * n) * t) * 2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if math.sqrt(((((U_42_ - U) * (math.pow((l_m / Om), 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * (U * (n * 2.0)))) <= 5e-156:
		tmp = math.sqrt(((t * U) * (n * 2.0)))
	else:
		tmp = math.sqrt((((U * n) * t) * 2.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l_m / Om) ^ 2.0) * n)) - Float64(Float64(Float64(Float64(l_m * l_m) / Om) * 2.0) - t)) * Float64(U * Float64(n * 2.0)))) <= 5e-156)
		tmp = sqrt(Float64(Float64(t * U) * Float64(n * 2.0)));
	else
		tmp = sqrt(Float64(Float64(Float64(U * n) * t) * 2.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (sqrt(((((U_42_ - U) * (((l_m / Om) ^ 2.0) * n)) - ((((l_m * l_m) / Om) * 2.0) - t)) * (U * (n * 2.0)))) <= 5e-156)
		tmp = sqrt(((t * U) * (n * 2.0)));
	else
		tmp = sqrt((((U * n) * t) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e-156], N[Sqrt[N[(N[(t * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000007e-156

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 33.3

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

   5. Applied rewrites 33.3%

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

   7. Applied rewrites 35.2%

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

   if 5.00000000000000007e-156 < (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 53.3%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.8

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

   5. Applied rewrites 36.8%

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

   7. Applied rewrites 41.7%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 5 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right)\\ \mathbf{if}\;U \leq 7.4 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\left(t\_1 \cdot U\right) \cdot n} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{t\_1 \cdot \left(n \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* -2.0 l_m) (/ l_m Om) t)))
   (if (<= U 7.4e-290)
     (* (sqrt (* (* t_1 U) n)) (sqrt 2.0))
     (* (sqrt U) (sqrt (* t_1 (* n 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma((-2.0 * l_m), (l_m / Om), t);
	double tmp;
	if (U <= 7.4e-290) {
		tmp = sqrt(((t_1 * U) * n)) * sqrt(2.0);
	} else {
		tmp = sqrt(U) * sqrt((t_1 * (n * 2.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(-2.0 * l_m), Float64(l_m / Om), t)
	tmp = 0.0
	if (U <= 7.4e-290)
		tmp = Float64(sqrt(Float64(Float64(t_1 * U) * n)) * sqrt(2.0));
	else
		tmp = Float64(sqrt(U) * sqrt(Float64(t_1 * Float64(n * 2.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[U, 7.4e-290], N[(N[Sqrt[N[(N[(t$95$1 * U), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < 7.39999999999999954e-290

   1. Initial program 48.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 50.8

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 56.4%

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

   6. Applied rewrites 24.7%

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

   7. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 23.2

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

   9. Applied rewrites 23.2%

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

       1. lift-*.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqrt-unprod N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. sqrt-prod N/A

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

   11. Applied rewrites 51.4%

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

   if 7.39999999999999954e-290 < U

   1. Initial program 48.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 49.8

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 55.8%

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

   6. Applied rewrites 23.8%

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

   7. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 23.9

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

   9. Applied rewrites 23.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. sqrt-prod N/A

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

       6. pow1/2 N/A

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

   11. Applied rewrites 61.5%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 7.4 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) \cdot U\right) \cdot n} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(n \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right)\\ \mathbf{if}\;U \leq 7.2 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{t\_1 \cdot \left(n \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* -2.0 l_m) (/ l_m Om) t)))
   (if (<= U 7.2e-290)
     (sqrt (* t_1 (* U (* n 2.0))))
     (* (sqrt U) (sqrt (* t_1 (* n 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma((-2.0 * l_m), (l_m / Om), t);
	double tmp;
	if (U <= 7.2e-290) {
		tmp = sqrt((t_1 * (U * (n * 2.0))));
	} else {
		tmp = sqrt(U) * sqrt((t_1 * (n * 2.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(-2.0 * l_m), Float64(l_m / Om), t)
	tmp = 0.0
	if (U <= 7.2e-290)
		tmp = sqrt(Float64(t_1 * Float64(U * Float64(n * 2.0))));
	else
		tmp = Float64(sqrt(U) * sqrt(Float64(t_1 * Float64(n * 2.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[U, 7.2e-290], N[Sqrt[N[(t$95$1 * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < 7.19999999999999959e-290

   1. Initial program 48.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 50.8

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 56.4%

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

   6. Applied rewrites 24.7%

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

   7. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 23.2

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

   9. Applied rewrites 23.2%

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

       1. lift-*.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqrt-unprod N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

   11. Applied rewrites 48.5%

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

   if 7.19999999999999959e-290 < U

   1. Initial program 48.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 49.8

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 55.8%

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

   6. Applied rewrites 23.8%

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

   7. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 23.9

         \[\leadsto \sqrt{\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot U} \cdot \sqrt{2 \cdot n} \]

   9. Applied rewrites 23.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)} \cdot U} \cdot \sqrt{2 \cdot n} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot U} \cdot \sqrt{2 \cdot n}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot U}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{2 \cdot n} \cdot \color{blue}{\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot U}} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{2 \cdot n} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot U}} \]

       5. sqrt-prod N/A

          \[\leadsto \sqrt{2 \cdot n} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)} \cdot \sqrt{U}\right)} \]

       6. pow1/2 N/A

          \[\leadsto \sqrt{2 \cdot n} \cdot \left(\color{blue}{{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{U}\right) \]

   11. Applied rewrites 61.5%

       \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)} \cdot \sqrt{U}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 7.2 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) \cdot \left(n \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -5e-310)
   (sqrt (* (* (* (fma -2.0 (/ (* l_m l_m) Om) t) n) U) 2.0))
   (* (sqrt (* n 2.0)) (sqrt (* t U)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -5e-310) {
		tmp = sqrt((((fma(-2.0, ((l_m * l_m) / Om), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((t * U));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -5e-310)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(Float64(l_m * l_m) / Om), t) * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(t * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -5e-310], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.999999999999985e-310

   1. Initial program 52.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U\right) \cdot 2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U\right) \cdot 2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      13. lower-*.f64 48.0

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

   5. Applied rewrites 48.0%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}} \]

   if -4.999999999999985e-310 < n

   1. Initial program 44.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. lower-*.f64 47.3

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. lift--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t - 2 \cdot \frac{\ell \cdot \ell}{Om}}\right)} \]

   4. Applied rewrites 56.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right)\right)}} \]

   6. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\sqrt{\left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2 - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right) \cdot U} \cdot \sqrt{2 \cdot n}} \]

   7. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{2 \cdot n} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{2 \cdot n} \]

      2. lower-*.f64 43.7

         \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{2 \cdot n} \]

   9. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{2 \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq 7.2 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U 7.2e-290)
   (sqrt (* (* t U) (* n 2.0)))
   (* (sqrt (* (* t n) 2.0)) (sqrt U))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= 7.2e-290) {
		tmp = sqrt(((t * U) * (n * 2.0)));
	} else {
		tmp = sqrt(((t * n) * 2.0)) * sqrt(U);
	}
	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 <= 7.2d-290) then
        tmp = sqrt(((t * u) * (n * 2.0d0)))
    else
        tmp = sqrt(((t * n) * 2.0d0)) * sqrt(u)
    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 <= 7.2e-290) {
		tmp = Math.sqrt(((t * U) * (n * 2.0)));
	} else {
		tmp = Math.sqrt(((t * n) * 2.0)) * Math.sqrt(U);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= 7.2e-290:
		tmp = math.sqrt(((t * U) * (n * 2.0)))
	else:
		tmp = math.sqrt(((t * n) * 2.0)) * math.sqrt(U)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= 7.2e-290)
		tmp = sqrt(Float64(Float64(t * U) * Float64(n * 2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(t * n) * 2.0)) * sqrt(U));
	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 <= 7.2e-290)
		tmp = sqrt(((t * U) * (n * 2.0)));
	else
		tmp = sqrt(((t * n) * 2.0)) * sqrt(U);
	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, 7.2e-290], N[Sqrt[N[(N[(t * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(t * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 7.19999999999999959e-290

   1. Initial program 48.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 36.6

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 36.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]

   if 7.19999999999999959e-290 < U

   1. Initial program 48.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      6. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)} \cdot \sqrt{U}} \]

      7. pow1/2 N/A

         \[\leadsto \sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)} \cdot \color{blue}{{U}^{\frac{1}{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)} \cdot {U}^{\frac{1}{2}}} \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right) - \left(n \cdot \left(U - U*\right)\right) \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}} \]

   5. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot t\right)}} \cdot \sqrt{U} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot t\right)}} \cdot \sqrt{U} \]

      2. lower-*.f64 48.0

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot t\right)}} \cdot \sqrt{U} \]

   7. Applied rewrites 48.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot t\right)}} \cdot \sqrt{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 7.2 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -5e-310)
   (sqrt (* (* (* t n) U) 2.0))
   (* (sqrt (* n 2.0)) (sqrt (* t U)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -5e-310) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((n * 2.0)) * sqrt((t * U));
	}
	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 <= (-5d-310)) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((n * 2.0d0)) * sqrt((t * u))
    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 <= -5e-310) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((n * 2.0)) * Math.sqrt((t * U));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= -5e-310:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((n * 2.0)) * math.sqrt((t * U))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -5e-310)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(t * U)));
	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 <= -5e-310)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((n * 2.0)) * sqrt((t * U));
	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, -5e-310], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.999999999999985e-310

   1. Initial program 52.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 41.9

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 41.9%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]

   if -4.999999999999985e-310 < n

   1. Initial program 44.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. lower-*.f64 47.3

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. lift--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t - 2 \cdot \frac{\ell \cdot \ell}{Om}}\right)} \]

   4. Applied rewrites 56.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right)\right)}} \]

   6. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\sqrt{\left(t + \frac{\left(\ell \cdot \ell\right) \cdot -2 - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right) \cdot U} \cdot \sqrt{2 \cdot n}} \]

   7. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{2 \cdot n} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{2 \cdot n} \]

      2. lower-*.f64 43.7

         \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{2 \cdot n} \]

   9. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{2 \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* t U) (* n 2.0))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt(((t * U) * (n * 2.0)));
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt(((t * u) * (n * 2.0d0)))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt(((t * U) * (n * 2.0)));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt(((t * U) * (n * 2.0)))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(t * U) * Float64(n * 2.0)))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt(((t * U) * (n * 2.0)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(t * U), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.7%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

   3. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

   5. lower-*.f64 36.3

      \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

5. Applied rewrites 36.3%

   \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
6. Step-by-step derivation

7. Applied rewrites 37.4%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]

8. Final simplification 37.4%

   \[\leadsto \sqrt{\left(t \cdot U\right) \cdot \left(n \cdot 2\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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*))))))