Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 59.6%

Time: 12.0s

Alternatives: 21

Speedup: 2.3×

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: 59.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ \mathbf{if}\;l\_m \leq 0.7:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(\left(U - U*\right) \cdot \left(n \cdot \frac{l\_m}{Om}\right)\right) \cdot \frac{l\_m}{Om}\right)}\\ \mathbf{elif}\;l\_m \leq 1.6 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(\frac{\left(l\_m \cdot l\_m\right) \cdot U}{Om} \cdot \frac{\left(U - U*\right) \cdot n}{Om} - \mathsf{fma}\left(-2, t\_1, t\right) \cdot U\right)\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot n\right)}{Om} \cdot \frac{U - U*}{Om}, -4 \cdot \frac{U \cdot n}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om)))
   (if (<= l_m 0.7)
     (sqrt
      (*
       (* (* 2.0 n) U)
       (- (- t (* 2.0 t_1)) (* (* (- U U*) (* n (/ l_m Om))) (/ l_m Om)))))
     (if (<= l_m 1.6e+118)
       (sqrt
        (*
         (*
          -2.0
          (-
           (* (/ (* (* l_m l_m) U) Om) (/ (* (- U U*) n) Om))
           (* (fma -2.0 t_1 t) U)))
         n))
       (*
        l_m
        (sqrt
         (fma
          -2.0
          (* (/ (* U (* n n)) Om) (/ (- U U*) Om))
          (* -4.0 (/ (* U 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 = (l_m * l_m) / Om;
	double tmp;
	if (l_m <= 0.7) {
		tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) - (((U - U_42_) * (n * (l_m / Om))) * (l_m / Om)))));
	} else if (l_m <= 1.6e+118) {
		tmp = sqrt(((-2.0 * (((((l_m * l_m) * U) / Om) * (((U - U_42_) * n) / Om)) - (fma(-2.0, t_1, t) * U))) * n));
	} else {
		tmp = l_m * sqrt(fma(-2.0, (((U * (n * n)) / Om) * ((U - U_42_) / Om)), (-4.0 * ((U * n) / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	tmp = 0.0
	if (l_m <= 0.7)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(Float64(U - U_42_) * Float64(n * Float64(l_m / Om))) * Float64(l_m / Om)))));
	elseif (l_m <= 1.6e+118)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(Float64(Float64(Float64(l_m * l_m) * U) / Om) * Float64(Float64(Float64(U - U_42_) * n) / Om)) - Float64(fma(-2.0, t_1, t) * U))) * n));
	else
		tmp = Float64(l_m * sqrt(fma(-2.0, Float64(Float64(Float64(U * Float64(n * n)) / Om) * Float64(Float64(U - U_42_) / Om)), Float64(-4.0 * Float64(Float64(U * 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[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[l$95$m, 0.7], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.6e+118], N[Sqrt[N[(N[(-2.0 * N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(N[(U * N[(n * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(U * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 0.69999999999999996

   1. Initial program 54.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 56.8

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

   4. Applied rewrites 56.8%

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

   if 0.69999999999999996 < l < 1.60000000000000008e118

   1. Initial program 49.0%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 67.6%

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

   if 1.60000000000000008e118 < l

   1. Initial program 18.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 40.7

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 40.7%

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

   6. Applied rewrites 42.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-neg.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lift-neg.f64 N/A

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

      16. lower-*.f64 52.4

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

   8. Applied rewrites 52.4%

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

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om} + -2 \cdot \frac{U \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}} \]
   10. 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. associate-*r* N/A

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

       6. unpow2 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower--.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

   11. Applied rewrites 68.1%

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

Alternative 2: 57.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(n \cdot \left(U - U*\right)\right) \cdot \frac{-l\_m}{Om}, \frac{l\_m}{Om}, \mathsf{fma}\left(2, n, t\right)\right) \cdot U} \cdot \sqrt{n \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 1000000000:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(l\_m, \frac{\left(\frac{\left(U - U*\right) \cdot n}{Om} - -2\right) \cdot l\_m}{-Om}, t\right)}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot n\right)}{Om} \cdot \frac{U - U*}{Om}, -4 \cdot \frac{U \cdot n}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_3 0.0)
     (*
      (sqrt
       (* (fma (* (* n (- U U*)) (/ (- l_m) Om)) (/ l_m Om) (fma 2.0 n t)) U))
      (sqrt (* n 2.0)))
     (if (<= t_3 1000000000.0)
       (sqrt
        (* t_2 (fma l_m (/ (* (- (/ (* (- U U*) n) Om) -2.0) l_m) (- Om)) t)))
       (if (<= t_3 5e+307)
         (sqrt (* t_2 (fma -2.0 t_1 t)))
         (*
          l_m
          (sqrt
           (fma
            -2.0
            (* (/ (* U (* n n)) Om) (/ (- U U*) Om))
            (* -4.0 (/ (* U 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 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((fma(((n * (U - U_42_)) * (-l_m / Om)), (l_m / Om), fma(2.0, n, t)) * U)) * sqrt((n * 2.0));
	} else if (t_3 <= 1000000000.0) {
		tmp = sqrt((t_2 * fma(l_m, ((((((U - U_42_) * n) / Om) - -2.0) * l_m) / -Om), t)));
	} else if (t_3 <= 5e+307) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = l_m * sqrt(fma(-2.0, (((U * (n * n)) / Om) * ((U - U_42_) / Om)), (-4.0 * ((U * n) / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(n * Float64(U - U_42_)) * Float64(Float64(-l_m) / Om)), Float64(l_m / Om), fma(2.0, n, t)) * U)) * sqrt(Float64(n * 2.0)));
	elseif (t_3 <= 1000000000.0)
		tmp = sqrt(Float64(t_2 * fma(l_m, Float64(Float64(Float64(Float64(Float64(Float64(U - U_42_) * n) / Om) - -2.0) * l_m) / Float64(-Om)), t)));
	elseif (t_3 <= 5e+307)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = Float64(l_m * sqrt(fma(-2.0, Float64(Float64(Float64(U * Float64(n * n)) / Om) * Float64(Float64(U - U_42_) / Om)), Float64(-4.0 * Float64(Float64(U * 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[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[((-l$95$m) / Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + N[(2.0 * n + t), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1000000000.0], N[Sqrt[N[(t$95$2 * N[(l$95$m * N[(N[(N[(N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] - -2.0), $MachinePrecision] * l$95$m), $MachinePrecision] / (-Om)), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 5e+307], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(N[(U * N[(n * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(U * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 10.3%

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

   4. Applied rewrites 29.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 29.5

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 29.5

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

   6. Applied rewrites 29.5%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e9

   1. Initial program 95.7%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 87.8%

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

   7. Applied rewrites 93.7%

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

   if 1e9 < (*.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*)))) < 5e307

   1. Initial program 99.6%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 5e307 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 18.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 32.8

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 31.6%

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

   6. Applied rewrites 34.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-neg.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lift-neg.f64 N/A

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

      16. lower-*.f64 39.7

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

   8. Applied rewrites 39.7%

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

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om} + -2 \cdot \frac{U \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}} \]
   10. 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. associate-*r* N/A

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

       6. unpow2 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower--.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

   11. Applied rewrites 24.2%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(n \cdot \left(U - U*\right)\right) \cdot \frac{-\ell}{Om}, \frac{\ell}{Om}, \mathsf{fma}\left(2, n, t\right)\right) \cdot U} \cdot \sqrt{n \cdot 2}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 1000000000:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell, \frac{\left(\frac{\left(U - U*\right) \cdot n}{Om} - -2\right) \cdot \ell}{-Om}, t\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot n\right)}{Om} \cdot \frac{U - U*}{Om}, -4 \cdot \frac{U \cdot n}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{n \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 1000000000:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(l\_m, \frac{\left(\frac{\left(U - U*\right) \cdot n}{Om} - -2\right) \cdot l\_m}{-Om}, t\right)}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot n\right)}{Om} \cdot \frac{U - U*}{Om}, -4 \cdot \frac{U \cdot n}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_3 0.0)
     (* (sqrt (* U t)) (sqrt (* n 2.0)))
     (if (<= t_3 1000000000.0)
       (sqrt
        (* t_2 (fma l_m (/ (* (- (/ (* (- U U*) n) Om) -2.0) l_m) (- Om)) t)))
       (if (<= t_3 5e+307)
         (sqrt (* t_2 (fma -2.0 t_1 t)))
         (*
          l_m
          (sqrt
           (fma
            -2.0
            (* (/ (* U (* n n)) Om) (/ (- U U*) Om))
            (* -4.0 (/ (* U 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 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((U * t)) * sqrt((n * 2.0));
	} else if (t_3 <= 1000000000.0) {
		tmp = sqrt((t_2 * fma(l_m, ((((((U - U_42_) * n) / Om) - -2.0) * l_m) / -Om), t)));
	} else if (t_3 <= 5e+307) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = l_m * sqrt(fma(-2.0, (((U * (n * n)) / Om) * ((U - U_42_) / Om)), (-4.0 * ((U * n) / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(U * t)) * sqrt(Float64(n * 2.0)));
	elseif (t_3 <= 1000000000.0)
		tmp = sqrt(Float64(t_2 * fma(l_m, Float64(Float64(Float64(Float64(Float64(Float64(U - U_42_) * n) / Om) - -2.0) * l_m) / Float64(-Om)), t)));
	elseif (t_3 <= 5e+307)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = Float64(l_m * sqrt(fma(-2.0, Float64(Float64(Float64(U * Float64(n * n)) / Om) * Float64(Float64(U - U_42_) / Om)), Float64(-4.0 * Float64(Float64(U * 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[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1000000000.0], N[Sqrt[N[(t$95$2 * N[(l$95$m * N[(N[(N[(N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] - -2.0), $MachinePrecision] * l$95$m), $MachinePrecision] / (-Om)), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 5e+307], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(N[(U * N[(n * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(U * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 10.3%

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

   4. Applied rewrites 29.3%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 27.1

         \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   7. Applied rewrites 27.1%

      \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e9

   1. Initial program 95.7%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 87.8%

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

   7. Applied rewrites 93.7%

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

   if 1e9 < (*.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*)))) < 5e307

   1. Initial program 99.6%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 5e307 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 18.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 32.8

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 31.6%

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

   6. Applied rewrites 34.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-neg.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lift-neg.f64 N/A

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

      16. lower-*.f64 39.7

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

   8. Applied rewrites 39.7%

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

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om} + -2 \cdot \frac{U \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}} \]
   10. 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. associate-*r* N/A

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

       6. unpow2 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower--.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

   11. Applied rewrites 24.2%

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

Alternative 4: 59.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot \left(U - U*\right)\right) \cdot \frac{-l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(t\_1, \frac{l\_m}{Om}, \mathsf{fma}\left(2, n, t\right)\right) \cdot U} \cdot \sqrt{n \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(t\_1, \frac{l\_m}{Om}, \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot n\right)}{Om} \cdot \frac{U - U*}{Om}, -4 \cdot \frac{U \cdot n}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (- U U*)) (/ (- l_m) Om)))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_3 0.0)
     (* (sqrt (* (fma t_1 (/ l_m Om) (fma 2.0 n t)) U)) (sqrt (* n 2.0)))
     (if (<= t_3 5e+307)
       (sqrt (* t_2 (fma t_1 (/ l_m Om) (fma (* (/ l_m Om) l_m) -2.0 t))))
       (*
        l_m
        (sqrt
         (fma
          -2.0
          (* (/ (* U (* n n)) Om) (/ (- U U*) Om))
          (* -4.0 (/ (* U 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 = (n * (U - U_42_)) * (-l_m / Om);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((fma(t_1, (l_m / Om), fma(2.0, n, t)) * U)) * sqrt((n * 2.0));
	} else if (t_3 <= 5e+307) {
		tmp = sqrt((t_2 * fma(t_1, (l_m / Om), fma(((l_m / Om) * l_m), -2.0, t))));
	} else {
		tmp = l_m * sqrt(fma(-2.0, (((U * (n * n)) / Om) * ((U - U_42_) / Om)), (-4.0 * ((U * n) / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * Float64(U - U_42_)) * Float64(Float64(-l_m) / Om))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(fma(t_1, Float64(l_m / Om), fma(2.0, n, t)) * U)) * sqrt(Float64(n * 2.0)));
	elseif (t_3 <= 5e+307)
		tmp = sqrt(Float64(t_2 * fma(t_1, Float64(l_m / Om), fma(Float64(Float64(l_m / Om) * l_m), -2.0, t))));
	else
		tmp = Float64(l_m * sqrt(fma(-2.0, Float64(Float64(Float64(U * Float64(n * n)) / Om) * Float64(Float64(U - U_42_) / Om)), Float64(-4.0 * Float64(Float64(U * 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[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[((-l$95$m) / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(t$95$1 * N[(l$95$m / Om), $MachinePrecision] + N[(2.0 * n + t), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+307], N[Sqrt[N[(t$95$2 * N[(t$95$1 * N[(l$95$m / Om), $MachinePrecision] + N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(N[(U * N[(n * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(U * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.3%

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

   4. Applied rewrites 29.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 29.5

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 29.5

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

   6. Applied rewrites 29.5%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5e307

   1. Initial program 97.7%

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

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 97.7

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 92.6%

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

   6. Applied rewrites 92.7%

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

   if 5e307 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 18.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 32.8

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 31.6%

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

   6. Applied rewrites 34.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-neg.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lift-neg.f64 N/A

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

      16. lower-*.f64 39.7

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

   8. Applied rewrites 39.7%

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

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om} + -2 \cdot \frac{U \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}} \]
   10. 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. associate-*r* N/A

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

       6. unpow2 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower--.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

   11. Applied rewrites 24.2%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(n \cdot \left(U - U*\right)\right) \cdot \frac{-\ell}{Om}, \frac{\ell}{Om}, \mathsf{fma}\left(2, n, t\right)\right) \cdot U} \cdot \sqrt{n \cdot 2}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(n \cdot \left(U - U*\right)\right) \cdot \frac{-\ell}{Om}, \frac{\ell}{Om}, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot n\right)}{Om} \cdot \frac{U - U*}{Om}, -4 \cdot \frac{U \cdot n}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{n \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 10^{+154}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(\frac{l\_m}{Om} \cdot l\_m\right) \cdot U*\right) \cdot n\right) \cdot \frac{n}{Om}\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (*
           t_2
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_3 0.0)
     (* (sqrt (* U t)) (sqrt (* n 2.0)))
     (if (<= t_3 1e+154)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (sqrt (* (* (* (* (* (/ l_m Om) l_m) U*) n) (/ n Om)) (* 2.0 U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((U * t)) * sqrt((n * 2.0));
	} else if (t_3 <= 1e+154) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = sqrt(((((((l_m / Om) * l_m) * U_42_) * n) * (n / Om)) * (2.0 * U)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(U * t)) * sqrt(Float64(n * 2.0)));
	elseif (t_3 <= 1e+154)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(l_m / Om) * l_m) * U_42_) * n) * Float64(n / Om)) * Float64(2.0 * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+154], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * U$42$), $MachinePrecision] * n), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.6%

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 28.8

         \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   7. Applied rewrites 28.8%

      \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.00000000000000004e154

   1. Initial program 97.7%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if 1.00000000000000004e154 < (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 18.0%

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

   3. Taylor expanded in U* around inf

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

      1. count-2-rev N/A

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

      2. associate-/l* N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-*.f64 26.1

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

   5. Applied rewrites 26.1%

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

   7. Applied rewrites 28.5%

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

Alternative 6: 52.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{n \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 10^{+154}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(U \cdot U*\right) \cdot \left(\left(n \cdot l\_m\right) \cdot \left(n \cdot l\_m\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (*
           t_2
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_3 0.0)
     (* (sqrt (* U t)) (sqrt (* n 2.0)))
     (if (<= t_3 1e+154)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (sqrt (* 2.0 (/ (* (* U U*) (* (* n l_m) (* n l_m))) (* Om Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((U * t)) * sqrt((n * 2.0));
	} else if (t_3 <= 1e+154) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = sqrt((2.0 * (((U * U_42_) * ((n * l_m) * (n * l_m))) / (Om * Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(U * t)) * sqrt(Float64(n * 2.0)));
	elseif (t_3 <= 1e+154)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(U * U_42_) * Float64(Float64(n * l_m) * Float64(n * l_m))) / Float64(Om * Om))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+154], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(N[(U * U$42$), $MachinePrecision] * N[(N[(n * l$95$m), $MachinePrecision] * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.6%

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 28.8

         \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   7. Applied rewrites 28.8%

      \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.00000000000000004e154

   1. Initial program 97.7%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if 1.00000000000000004e154 < (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 18.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 32.5

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 31.3%

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

   6. Applied rewrites 34.4%

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

   7. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 24.6

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

   9. Applied rewrites 24.6%

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

   11. Applied rewrites 28.4%

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

Alternative 7: 52.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{n \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{n \cdot n}{Om} \cdot U*\right) \cdot \left(\frac{l\_m}{Om} \cdot l\_m\right)\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_3 0.0)
     (* (sqrt (* U t)) (sqrt (* n 2.0)))
     (if (<= t_3 5e+307)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (sqrt (* (* (* (/ (* n n) Om) U*) (* (/ l_m Om) l_m)) (* 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 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((U * t)) * sqrt((n * 2.0));
	} else if (t_3 <= 5e+307) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = sqrt((((((n * n) / Om) * U_42_) * ((l_m / Om) * l_m)) * (2.0 * U)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(U * t)) * sqrt(Float64(n * 2.0)));
	elseif (t_3 <= 5e+307)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(n * n) / Om) * U_42_) * Float64(Float64(l_m / Om) * l_m)) * Float64(2.0 * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+307], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(n * n), $MachinePrecision] / Om), $MachinePrecision] * U$42$), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * 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*)))) < 0.0

   1. Initial program 10.3%

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

   4. Applied rewrites 29.3%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 27.1

         \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   7. Applied rewrites 27.1%

      \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5e307

   1. Initial program 97.7%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if 5e307 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 18.6%

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

   3. Taylor expanded in U* around inf

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

      1. count-2-rev N/A

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

      2. associate-/l* N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-*.f64 26.9

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

   5. Applied rewrites 26.9%

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

   7. Applied rewrites 29.1%

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

Alternative 8: 50.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 10.3%

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

   4. Applied rewrites 29.3%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 27.1

         \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   7. Applied rewrites 27.1%

      \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

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

   1. Initial program 66.3%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 58.1

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

   5. Applied rewrites 58.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around inf

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

      1. count-2-rev N/A

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

      2. associate-/l* N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-*.f64 37.3

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

   5. Applied rewrites 37.3%

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

   7. Applied rewrites 34.4%

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

Alternative 9: 50.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{n \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(-2, t\_1, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(U \cdot U*\right) \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \left(n \cdot n\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (*
          t_2
          (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_3 0.0)
     (* (sqrt (* U t)) (sqrt (* n 2.0)))
     (if (<= t_3 5e+307)
       (sqrt (* t_2 (fma -2.0 t_1 t)))
       (sqrt (* 2.0 (/ (* (* U U*) (* (* l_m l_m) (* n n))) (* Om Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((U * t)) * sqrt((n * 2.0));
	} else if (t_3 <= 5e+307) {
		tmp = sqrt((t_2 * fma(-2.0, t_1, t)));
	} else {
		tmp = sqrt((2.0 * (((U * U_42_) * ((l_m * l_m) * (n * n))) / (Om * Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(U * t)) * sqrt(Float64(n * 2.0)));
	elseif (t_3 <= 5e+307)
		tmp = sqrt(Float64(t_2 * fma(-2.0, t_1, t)));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(U * U_42_) * Float64(Float64(l_m * l_m) * Float64(n * n))) / Float64(Om * Om))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+307], N[Sqrt[N[(t$95$2 * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(N[(U * U$42$), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.3%

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

   4. Applied rewrites 29.3%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 27.1

         \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   7. Applied rewrites 27.1%

      \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5e307

   1. Initial program 97.7%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if 5e307 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 18.6%

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

   3. Taylor expanded in U* around inf

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

      1. count-2-rev N/A

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

      2. associate-/l* N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-*.f64 26.9

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

   5. Applied rewrites 26.9%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 25.4%

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

Alternative 10: 49.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.3%

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

   4. Applied rewrites 29.3%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 27.1

         \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   7. Applied rewrites 27.1%

      \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

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

   1. Initial program 66.3%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 58.1

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

   5. Applied rewrites 58.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 21.5

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

   5. Applied rewrites 21.5%

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

Alternative 11: 47.8% accurate, 0.5× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (fma -2.0 t_1 t))
        (t_3 (* (* 2.0 n) U))
        (t_4
         (*
          t_3
          (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_4 0.0)
     (* (sqrt (* U t)) (sqrt (* n 2.0)))
     (if (<= t_4 1e+306) (sqrt (* t_3 t_2)) (sqrt (* (* (* t_2 n) U) 2.0))))))

C

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * t$95$1 + t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e+306], N[Sqrt[N[(t$95$3 * t$95$2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(t$95$2 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.3%

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

   4. Applied rewrites 29.3%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 27.1

         \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   7. Applied rewrites 27.1%

      \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.00000000000000002e306

   1. Initial program 97.7%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if 1.00000000000000002e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 19.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      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 20.3

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

   5. Applied rewrites 20.3%

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

Alternative 12: 59.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.7 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(\left(U - U*\right) \cdot \left(n \cdot \frac{l\_m}{Om}\right)\right) \cdot \frac{l\_m}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot n\right)}{Om} \cdot \frac{U - U*}{Om}, -4 \cdot \frac{U \cdot n}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.7e+118)
   (sqrt
    (*
     (* (* 2.0 n) U)
     (-
      (- t (* 2.0 (/ (* l_m l_m) Om)))
      (* (* (- U U*) (* n (/ l_m Om))) (/ l_m Om)))))
   (*
    l_m
    (sqrt
     (fma
      -2.0
      (* (/ (* U (* n n)) Om) (/ (- U U*) Om))
      (* -4.0 (/ (* U n) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.7e+118) {
		tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - (((U - U_42_) * (n * (l_m / Om))) * (l_m / Om)))));
	} else {
		tmp = l_m * sqrt(fma(-2.0, (((U * (n * n)) / Om) * ((U - U_42_) / Om)), (-4.0 * ((U * n) / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.7e+118)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(Float64(U - U_42_) * Float64(n * Float64(l_m / Om))) * Float64(l_m / Om)))));
	else
		tmp = Float64(l_m * sqrt(fma(-2.0, Float64(Float64(Float64(U * Float64(n * n)) / Om) * Float64(Float64(U - U_42_) / Om)), Float64(-4.0 * Float64(Float64(U * n) / Om)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.7e+118], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(N[(U * N[(n * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(U * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.7e118

   1. Initial program 54.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 56.3

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

   4. Applied rewrites 56.3%

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

   if 2.7e118 < l

   1. Initial program 18.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 40.7

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 40.7%

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

   6. Applied rewrites 42.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-neg.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lift-neg.f64 N/A

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

      16. lower-*.f64 52.4

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

   8. Applied rewrites 52.4%

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

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om} + -2 \cdot \frac{U \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}} \]
   10. 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. associate-*r* N/A

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

       6. unpow2 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower--.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

   11. Applied rewrites 68.1%

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

Alternative 13: 57.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;l\_m \leq 1.7 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\frac{\frac{U* \cdot \left(\left(l\_m \cdot l\_m\right) \cdot n\right)}{-Om}}{-Om} + t\right)}\\ \mathbf{elif}\;l\_m \leq 1.7 \cdot 10^{+251}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m, \frac{\left(\frac{\left(U - U*\right) \cdot n}{Om} - -2\right) \cdot l\_m}{-Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{\mathsf{fma}\left(-4, \frac{U \cdot n}{Om}, -2 \cdot \frac{\left(U \cdot \left(n \cdot n\right)\right) \cdot \left(U - U*\right)}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<= l_m 1.7e-107)
     (sqrt (* t_1 (+ (/ (/ (* U* (* (* l_m l_m) n)) (- Om)) (- Om)) t)))
     (if (<= l_m 1.7e+251)
       (sqrt
        (* t_1 (fma l_m (/ (* (- (/ (* (- U U*) n) Om) -2.0) l_m) (- Om)) t)))
       (*
        l_m
        (sqrt
         (fma
          -4.0
          (/ (* U n) Om)
          (* -2.0 (/ (* (* U (* n n)) (- U U*)) (* Om Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (l_m <= 1.7e-107) {
		tmp = sqrt((t_1 * ((((U_42_ * ((l_m * l_m) * n)) / -Om) / -Om) + t)));
	} else if (l_m <= 1.7e+251) {
		tmp = sqrt((t_1 * fma(l_m, ((((((U - U_42_) * n) / Om) - -2.0) * l_m) / -Om), t)));
	} else {
		tmp = l_m * sqrt(fma(-4.0, ((U * n) / Om), (-2.0 * (((U * (n * n)) * (U - U_42_)) / (Om * Om)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (l_m <= 1.7e-107)
		tmp = sqrt(Float64(t_1 * Float64(Float64(Float64(Float64(U_42_ * Float64(Float64(l_m * l_m) * n)) / Float64(-Om)) / Float64(-Om)) + t)));
	elseif (l_m <= 1.7e+251)
		tmp = sqrt(Float64(t_1 * fma(l_m, Float64(Float64(Float64(Float64(Float64(Float64(U - U_42_) * n) / Om) - -2.0) * l_m) / Float64(-Om)), t)));
	else
		tmp = Float64(l_m * sqrt(fma(-4.0, Float64(Float64(U * n) / Om), Float64(-2.0 * Float64(Float64(Float64(U * Float64(n * n)) * Float64(U - U_42_)) / Float64(Om * Om))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[l$95$m, 1.7e-107], N[Sqrt[N[(t$95$1 * N[(N[(N[(N[(U$42$ * N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / (-Om)), $MachinePrecision] / (-Om)), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.7e+251], N[Sqrt[N[(t$95$1 * N[(l$95$m * N[(N[(N[(N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] - -2.0), $MachinePrecision] * l$95$m), $MachinePrecision] / (-Om)), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-4.0 * N[(N[(U * n), $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 * N[(N[(N[(U * N[(n * n), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.69999999999999997e-107

   1. Initial program 52.4%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 49.0%

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

   6. Taylor expanded in U* around inf

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

   8. Applied rewrites 51.1%

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

   if 1.69999999999999997e-107 < l < 1.70000000000000006e251

   1. Initial program 42.9%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 48.6%

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

   7. Applied rewrites 59.5%

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

   if 1.70000000000000006e251 < l

   1. Initial program 12.7%

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

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 13.3

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 13.3%

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

   6. Applied rewrites 13.0%

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

   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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 77.8

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

   9. Applied rewrites 77.8%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{-Om}}{-Om} + t\right)}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+251}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell, \frac{\left(\frac{\left(U - U*\right) \cdot n}{Om} - -2\right) \cdot \ell}{-Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{\mathsf{fma}\left(-4, \frac{U \cdot n}{Om}, -2 \cdot \frac{\left(U \cdot \left(n \cdot n\right)\right) \cdot \left(U - U*\right)}{Om \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U* \leq -4.5 \cdot 10^{-188} \lor \neg \left(U* \leq 5.5 \cdot 10^{-88}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\frac{U* \cdot \left(\left(l\_m \cdot l\_m\right) \cdot n\right)}{-Om}}{-Om} + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= U* -4.5e-188) (not (<= U* 5.5e-88)))
   (sqrt
    (* (* (* 2.0 n) U) (+ (/ (/ (* U* (* (* l_m l_m) n)) (- Om)) (- Om)) t)))
   (sqrt (* (* (* (fma -2.0 (/ (* l_m l_m) Om) t) n) U) 2.0))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -4.5e-188) || !(U_42_ <= 5.5e-88)) {
		tmp = sqrt((((2.0 * n) * U) * ((((U_42_ * ((l_m * l_m) * n)) / -Om) / -Om) + t)));
	} else {
		tmp = sqrt((((fma(-2.0, ((l_m * l_m) / Om), t) * n) * U) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((U_42_ <= -4.5e-188) || !(U_42_ <= 5.5e-88))
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(Float64(Float64(U_42_ * Float64(Float64(l_m * l_m) * n)) / Float64(-Om)) / Float64(-Om)) + t)));
	else
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(Float64(l_m * l_m) / Om), t) * n) * U) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[U$42$, -4.5e-188], N[Not[LessEqual[U$42$, 5.5e-88]], $MachinePrecision]], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(N[(N[(U$42$ * N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / (-Om)), $MachinePrecision] / (-Om)), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if U* < -4.49999999999999993e-188 or 5.49999999999999971e-88 < U*

   1. Initial program 50.4%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in U* around inf

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

   8. Applied rewrites 52.6%

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

   if -4.49999999999999993e-188 < U* < 5.49999999999999971e-88

   1. Initial program 41.8%

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

   3. Taylor expanded in 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. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -4.5 \cdot 10^{-188} \lor \neg \left(U* \leq 5.5 \cdot 10^{-88}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{-Om}}{-Om} + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;l\_m \leq 5.8 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\frac{\frac{U* \cdot \left(\left(l\_m \cdot l\_m\right) \cdot n\right)}{-Om}}{-Om} + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m, \frac{\left(\frac{\left(U - U*\right) \cdot n}{Om} - -2\right) \cdot l\_m}{-Om}, t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<= l_m 5.8e-89)
     (sqrt (* t_1 (+ (/ (/ (* U* (* (* l_m l_m) n)) (- Om)) (- Om)) t)))
     (sqrt
      (* t_1 (fma l_m (/ (* (- (/ (* (- U U*) n) Om) -2.0) l_m) (- Om)) t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (l_m <= 5.8e-89) {
		tmp = sqrt((t_1 * ((((U_42_ * ((l_m * l_m) * n)) / -Om) / -Om) + t)));
	} else {
		tmp = sqrt((t_1 * fma(l_m, ((((((U - U_42_) * n) / Om) - -2.0) * l_m) / -Om), t)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (l_m <= 5.8e-89)
		tmp = sqrt(Float64(t_1 * Float64(Float64(Float64(Float64(U_42_ * Float64(Float64(l_m * l_m) * n)) / Float64(-Om)) / Float64(-Om)) + t)));
	else
		tmp = sqrt(Float64(t_1 * fma(l_m, Float64(Float64(Float64(Float64(Float64(Float64(U - U_42_) * n) / Om) - -2.0) * l_m) / Float64(-Om)), t)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[l$95$m, 5.8e-89], N[Sqrt[N[(t$95$1 * N[(N[(N[(N[(U$42$ * N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / (-Om)), $MachinePrecision] / (-Om)), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(l$95$m * N[(N[(N[(N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] - -2.0), $MachinePrecision] * l$95$m), $MachinePrecision] / (-Om)), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 5.79999999999999984e-89

   1. Initial program 52.9%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 49.6%

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

   6. Taylor expanded in U* around inf

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

   8. Applied rewrites 51.7%

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

   if 5.79999999999999984e-89 < l

   1. Initial program 37.4%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 45.8%

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

   7. Applied rewrites 56.2%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{-Om}}{-Om} + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell, \frac{\left(\frac{\left(U - U*\right) \cdot n}{Om} - -2\right) \cdot \ell}{-Om}, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 48.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;l\_m \leq 3.4 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\frac{\frac{U* \cdot \left(\left(l\_m \cdot l\_m\right) \cdot n\right)}{-Om}}{-Om} + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \frac{\left(l\_m \cdot l\_m\right) \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)}{-Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<= l_m 3.4e+36)
     (sqrt (* t_1 (+ (/ (/ (* U* (* (* l_m l_m) n)) (- Om)) (- Om)) t)))
     (sqrt (* t_1 (/ (* (* l_m l_m) (+ 2.0 (/ (* n (- U U*)) Om))) (- Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (l_m <= 3.4e+36) {
		tmp = sqrt((t_1 * ((((U_42_ * ((l_m * l_m) * n)) / -Om) / -Om) + t)));
	} else {
		tmp = sqrt((t_1 * (((l_m * l_m) * (2.0 + ((n * (U - U_42_)) / Om))) / -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 = (2.0d0 * n) * u
    if (l_m <= 3.4d+36) then
        tmp = sqrt((t_1 * ((((u_42 * ((l_m * l_m) * n)) / -om) / -om) + t)))
    else
        tmp = sqrt((t_1 * (((l_m * l_m) * (2.0d0 + ((n * (u - u_42)) / om))) / -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 = (2.0 * n) * U;
	double tmp;
	if (l_m <= 3.4e+36) {
		tmp = Math.sqrt((t_1 * ((((U_42_ * ((l_m * l_m) * n)) / -Om) / -Om) + t)));
	} else {
		tmp = Math.sqrt((t_1 * (((l_m * l_m) * (2.0 + ((n * (U - U_42_)) / Om))) / -Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (2.0 * n) * U
	tmp = 0
	if l_m <= 3.4e+36:
		tmp = math.sqrt((t_1 * ((((U_42_ * ((l_m * l_m) * n)) / -Om) / -Om) + t)))
	else:
		tmp = math.sqrt((t_1 * (((l_m * l_m) * (2.0 + ((n * (U - U_42_)) / Om))) / -Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (l_m <= 3.4e+36)
		tmp = sqrt(Float64(t_1 * Float64(Float64(Float64(Float64(U_42_ * Float64(Float64(l_m * l_m) * n)) / Float64(-Om)) / Float64(-Om)) + t)));
	else
		tmp = sqrt(Float64(t_1 * Float64(Float64(Float64(l_m * l_m) * Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om))) / Float64(-Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (2.0 * n) * U;
	tmp = 0.0;
	if (l_m <= 3.4e+36)
		tmp = sqrt((t_1 * ((((U_42_ * ((l_m * l_m) * n)) / -Om) / -Om) + t)));
	else
		tmp = sqrt((t_1 * (((l_m * l_m) * (2.0 + ((n * (U - U_42_)) / Om))) / -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[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[l$95$m, 3.4e+36], N[Sqrt[N[(t$95$1 * N[(N[(N[(N[(U$42$ * N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / (-Om)), $MachinePrecision] / (-Om)), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-Om)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.3999999999999998e36

   1. Initial program 54.2%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in U* around inf

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

   8. Applied rewrites 52.6%

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

   if 3.3999999999999998e36 < l

   1. Initial program 22.8%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 36.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 34.0%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{-Om}}{-Om} + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)}{-Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 10^{+120}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot t} \cdot \sqrt{U \cdot n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 1e+120)
   (sqrt (* (* (* (fma -2.0 (/ (* l_m l_m) Om) t) n) U) 2.0))
   (* (sqrt (* 2.0 t)) (sqrt (* U n)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= 1e+120) {
		tmp = sqrt((((fma(-2.0, ((l_m * l_m) / Om), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt((2.0 * t)) * sqrt((U * n));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 1e+120)
		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(2.0 * t)) * sqrt(Float64(U * n)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, 1e+120], 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[(2.0 * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.9999999999999998e119

   1. Initial program 47.2%

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

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

      8. fp-cancel-sign-sub-inv N/A

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

      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 41.6

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

   5. Applied rewrites 41.6%

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

   if 9.9999999999999998e119 < t

   1. Initial program 54.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 17.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

   7. Applied rewrites 22.3%

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

   8. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot t}} \cdot \sqrt{U \cdot n} \]
   9. Step-by-step derivation

      1. lower-*.f64 67.2

         \[\leadsto \sqrt{\color{blue}{2 \cdot t}} \cdot \sqrt{U \cdot n} \]

   10. Applied rewrites 67.2%

       \[\leadsto \sqrt{\color{blue}{2 \cdot t}} \cdot \sqrt{U \cdot n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 39.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot t} \cdot \sqrt{U \cdot n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 9e-285)
   (sqrt (* (* (* n t) U) 2.0))
   (* (sqrt (* 2.0 t)) (sqrt (* U n)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= 9e-285) {
		tmp = sqrt((((n * t) * U) * 2.0));
	} else {
		tmp = sqrt((2.0 * t)) * sqrt((U * n));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 9d-285) then
        tmp = sqrt((((n * t) * u) * 2.0d0))
    else
        tmp = sqrt((2.0d0 * t)) * sqrt((u * n))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= 9e-285) {
		tmp = Math.sqrt((((n * t) * U) * 2.0));
	} else {
		tmp = Math.sqrt((2.0 * t)) * Math.sqrt((U * n));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= 9e-285:
		tmp = math.sqrt((((n * t) * U) * 2.0))
	else:
		tmp = math.sqrt((2.0 * t)) * math.sqrt((U * n))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 9e-285)
		tmp = sqrt(Float64(Float64(Float64(n * t) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(2.0 * t)) * sqrt(Float64(U * n)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= 9e-285)
		tmp = sqrt((((n * t) * U) * 2.0));
	else
		tmp = sqrt((2.0 * t)) * sqrt((U * n));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, 9e-285], N[Sqrt[N[(N[(N[(n * t), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.0000000000000005e-285

   1. Initial program 47.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 32.3

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

   5. Applied rewrites 32.3%

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

   if 9.0000000000000005e-285 < t

   1. Initial program 50.1%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 28.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

   7. Applied rewrites 26.9%

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

   8. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot t}} \cdot \sqrt{U \cdot n} \]
   9. Step-by-step derivation

      1. lower-*.f64 48.8

         \[\leadsto \sqrt{\color{blue}{2 \cdot t}} \cdot \sqrt{U \cdot n} \]

   10. Applied rewrites 48.8%

       \[\leadsto \sqrt{\color{blue}{2 \cdot t}} \cdot \sqrt{U \cdot n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 39.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 5.5 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot t} \cdot \sqrt{n \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 5.5e-302)
   (sqrt (* (* (* U n) t) 2.0))
   (* (sqrt (* U t)) (sqrt (* n 2.0)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 5.5e-302) {
		tmp = sqrt((((U * n) * t) * 2.0));
	} else {
		tmp = sqrt((U * t)) * sqrt((n * 2.0));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 5.5d-302) then
        tmp = sqrt((((u * n) * t) * 2.0d0))
    else
        tmp = sqrt((u * t)) * sqrt((n * 2.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 5.5e-302) {
		tmp = Math.sqrt((((U * n) * t) * 2.0));
	} else {
		tmp = Math.sqrt((U * t)) * Math.sqrt((n * 2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 5.5e-302:
		tmp = math.sqrt((((U * n) * t) * 2.0))
	else:
		tmp = math.sqrt((U * t)) * math.sqrt((n * 2.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 5.5e-302)
		tmp = sqrt(Float64(Float64(Float64(U * n) * t) * 2.0));
	else
		tmp = Float64(sqrt(Float64(U * t)) * sqrt(Float64(n * 2.0)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= 5.5e-302)
		tmp = sqrt((((U * n) * t) * 2.0));
	else
		tmp = sqrt((U * t)) * sqrt((n * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 5.5e-302], N[Sqrt[N[(N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 5.5000000000000001e-302

   1. Initial program 49.1%

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

   3. Taylor expanded in t around inf

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

      1. *-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 35.2

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

   5. Applied rewrites 35.2%

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

   7. Applied rewrites 37.8%

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

   if 5.5000000000000001e-302 < n

   1. Initial program 47.5%

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

   4. Applied rewrites 45.1%

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

   5. Taylor expanded in n around 0

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

      1. lower-*.f64 39.4

         \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]

   7. Applied rewrites 39.4%

      \[\leadsto \sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{n \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 36.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (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_) {
	return sqrt((((U * n) * t) * 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((((u * n) * t) * 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((((U * n) * t) * 2.0));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((((U * n) * t) * 2.0))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(Float64(U * n) * t) * 2.0))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((((U * n) * t) * 2.0));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
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 33.9

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

5. Applied rewrites 33.9%

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

7. Applied rewrites 35.7%

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

Alternative 21: 35.9% accurate, 7.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{n \cdot \left(t \cdot \left(U + U\right)\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* n (* t (+ U U)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt((n * (t * (U + U))));
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((n * (t * (u + u))))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt((n * (t * (U + U))));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((n * (t * (U + U))))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(n * Float64(t * Float64(U + U))))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((n * (t * (U + U))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(n * N[(t * N[(U + U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
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 33.9

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

5. Applied rewrites 33.9%

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

7. Applied rewrites 32.5%

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

9. Applied rewrites 32.5%

   \[\leadsto \sqrt{n \cdot \left(t \cdot \left(U + \color{blue}{U}\right)\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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*))))))