Toniolo and Linder, Equation (13)

Percentage Accurate: 49.9% → 63.2%

Time: 17.4s

Alternatives: 18

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 63.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-145}:\\ \;\;\;\;n \cdot \sqrt{\mathsf{fma}\left(-2, \left(U \cdot U\right) \cdot \frac{l\_m \cdot l\_m}{Om \cdot Om}, \frac{\left(2 \cdot U\right) \cdot \mathsf{fma}\left(-2, t\_1, t\right)}{n}\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{n \cdot \mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}, -4 \cdot \frac{U}{Om}\right)}\\ \end{array} \end{array} \]

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
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_2 2e-145)
     (*
      n
      (sqrt
       (fma
        -2.0
        (* (* U U) (/ (* l_m l_m) (* Om Om)))
        (/ (* (* 2.0 U) (fma -2.0 t_1 t)) n))))
     (if (<= t_2 5e+152)
       t_2
       (*
        l_m
        (sqrt
         (*
          n
          (fma
           -2.0
           (/ (* U (* n (- U U*))) (* Om Om))
           (* -4.0 (/ U 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 = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 2e-145) {
		tmp = n * sqrt(fma(-2.0, ((U * U) * ((l_m * l_m) / (Om * Om))), (((2.0 * U) * fma(-2.0, t_1, t)) / n)));
	} else if (t_2 <= 5e+152) {
		tmp = t_2;
	} else {
		tmp = l_m * sqrt((n * fma(-2.0, ((U * (n * (U - U_42_))) / (Om * Om)), (-4.0 * (U / 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 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 2e-145)
		tmp = Float64(n * sqrt(fma(-2.0, Float64(Float64(U * U) * Float64(Float64(l_m * l_m) / Float64(Om * Om))), Float64(Float64(Float64(2.0 * U) * fma(-2.0, t_1, t)) / n))));
	elseif (t_2 <= 5e+152)
		tmp = t_2;
	else
		tmp = Float64(l_m * sqrt(Float64(n * fma(-2.0, Float64(Float64(U * Float64(n * Float64(U - U_42_))) / Float64(Om * Om)), Float64(-4.0 * Float64(U / 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[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2e-145], N[(n * N[Sqrt[N[(-2.0 * N[(N[(U * U), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * U), $MachinePrecision] * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+152], t$95$2, N[(l$95$m * N[Sqrt[N[(n * N[(-2.0 * N[(N[(U * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.99999999999999983e-145

   1. Initial program 13.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   5. Simplified 18.2%

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

   6. Taylor expanded in U* around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lower-/.f64 N/A

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

   8. Simplified 47.0%

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

   if 1.99999999999999983e-145 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5e152

   1. Initial program 98.1%

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

   if 5e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 28.0%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   5. Simplified 22.3%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 22.2

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

   8. Simplified 22.2%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-145}:\\ \;\;\;\;n \cdot \sqrt{\mathsf{fma}\left(-2, \left(U \cdot U\right) \cdot \frac{\ell \cdot \ell}{Om \cdot Om}, \frac{\left(2 \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}{n}\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{n \cdot \mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}, -4 \cdot \frac{U}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := t - 2 \cdot t\_1\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \sqrt{t\_3 \cdot \left(t\_2 + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_4 \leq 2 \cdot 10^{-145}:\\ \;\;\;\;n \cdot \sqrt{\mathsf{fma}\left(-2, \left(U \cdot U\right) \cdot \frac{l\_m \cdot l\_m}{Om \cdot Om}, \frac{\left(2 \cdot U\right) \cdot \mathsf{fma}\left(-2, t\_1, t\right)}{n}\right)}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{t\_3 \cdot \left(t\_2 + \left(l\_m \cdot l\_m\right) \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{n \cdot \mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}, -4 \cdot \frac{U}{Om}\right)}\\ \end{array} \end{array} \]

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 (- t (* 2.0 t_1)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (sqrt (* t_3 (+ t_2 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_4 2e-145)
     (*
      n
      (sqrt
       (fma
        -2.0
        (* (* U U) (/ (* l_m l_m) (* Om Om)))
        (/ (* (* 2.0 U) (fma -2.0 t_1 t)) n))))
     (if (<= t_4 5e+152)
       (sqrt (* t_3 (+ t_2 (* (* l_m l_m) (/ (* n (- U* U)) (* Om Om))))))
       (*
        l_m
        (sqrt
         (*
          n
          (fma
           -2.0
           (/ (* U (* n (- U U*))) (* Om Om))
           (* -4.0 (/ U 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 = t - (2.0 * t_1);
	double t_3 = (2.0 * n) * U;
	double t_4 = sqrt((t_3 * (t_2 + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_4 <= 2e-145) {
		tmp = n * sqrt(fma(-2.0, ((U * U) * ((l_m * l_m) / (Om * Om))), (((2.0 * U) * fma(-2.0, t_1, t)) / n)));
	} else if (t_4 <= 5e+152) {
		tmp = sqrt((t_3 * (t_2 + ((l_m * l_m) * ((n * (U_42_ - U)) / (Om * Om))))));
	} else {
		tmp = l_m * sqrt((n * fma(-2.0, ((U * (n * (U - U_42_))) / (Om * Om)), (-4.0 * (U / 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(t - Float64(2.0 * t_1))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = sqrt(Float64(t_3 * Float64(t_2 + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_4 <= 2e-145)
		tmp = Float64(n * sqrt(fma(-2.0, Float64(Float64(U * U) * Float64(Float64(l_m * l_m) / Float64(Om * Om))), Float64(Float64(Float64(2.0 * U) * fma(-2.0, t_1, t)) / n))));
	elseif (t_4 <= 5e+152)
		tmp = sqrt(Float64(t_3 * Float64(t_2 + Float64(Float64(l_m * l_m) * Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om))))));
	else
		tmp = Float64(l_m * sqrt(Float64(n * fma(-2.0, Float64(Float64(U * Float64(n * Float64(U - U_42_))) / Float64(Om * Om)), Float64(-4.0 * Float64(U / 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[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(t$95$2 + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 2e-145], N[(n * N[Sqrt[N[(-2.0 * N[(N[(U * U), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * U), $MachinePrecision] * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+152], N[Sqrt[N[(t$95$3 * N[(t$95$2 + N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(n * N[(-2.0 * N[(N[(U * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.99999999999999983e-145

   1. Initial program 13.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   5. Simplified 18.2%

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

   6. Taylor expanded in U* around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lower-/.f64 N/A

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

   8. Simplified 47.0%

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

   if 1.99999999999999983e-145 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5e152

   1. Initial program 98.1%

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

   3. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 82.4

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

   5. Simplified 82.4%

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

   if 5e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 28.0%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   5. Simplified 22.3%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 22.2

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

   8. Simplified 22.2%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-145}:\\ \;\;\;\;n \cdot \sqrt{\mathsf{fma}\left(-2, \left(U \cdot U\right) \cdot \frac{\ell \cdot \ell}{Om \cdot Om}, \frac{\left(2 \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}{n}\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{n \cdot \mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}, -4 \cdot \frac{U}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := t - 2 \cdot t\_1\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \sqrt{t\_3 \cdot \left(t\_2 + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_4 \leq 2 \cdot 10^{-145}:\\ \;\;\;\;n \cdot \sqrt{U \cdot \left(2 \cdot \mathsf{fma}\left(U*, \frac{l\_m \cdot l\_m}{Om \cdot Om}, \frac{\mathsf{fma}\left(-2, t\_1, t\right)}{n}\right)\right)}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{t\_3 \cdot \left(t\_2 + \left(l\_m \cdot l\_m\right) \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{n \cdot \mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}, -4 \cdot \frac{U}{Om}\right)}\\ \end{array} \end{array} \]

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 (- t (* 2.0 t_1)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (sqrt (* t_3 (+ t_2 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_4 2e-145)
     (*
      n
      (sqrt
       (*
        U
        (* 2.0 (fma U* (/ (* l_m l_m) (* Om Om)) (/ (fma -2.0 t_1 t) n))))))
     (if (<= t_4 5e+152)
       (sqrt (* t_3 (+ t_2 (* (* l_m l_m) (/ (* n (- U* U)) (* Om Om))))))
       (*
        l_m
        (sqrt
         (*
          n
          (fma
           -2.0
           (/ (* U (* n (- U U*))) (* Om Om))
           (* -4.0 (/ U 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 = t - (2.0 * t_1);
	double t_3 = (2.0 * n) * U;
	double t_4 = sqrt((t_3 * (t_2 + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_4 <= 2e-145) {
		tmp = n * sqrt((U * (2.0 * fma(U_42_, ((l_m * l_m) / (Om * Om)), (fma(-2.0, t_1, t) / n)))));
	} else if (t_4 <= 5e+152) {
		tmp = sqrt((t_3 * (t_2 + ((l_m * l_m) * ((n * (U_42_ - U)) / (Om * Om))))));
	} else {
		tmp = l_m * sqrt((n * fma(-2.0, ((U * (n * (U - U_42_))) / (Om * Om)), (-4.0 * (U / 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(t - Float64(2.0 * t_1))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = sqrt(Float64(t_3 * Float64(t_2 + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_4 <= 2e-145)
		tmp = Float64(n * sqrt(Float64(U * Float64(2.0 * fma(U_42_, Float64(Float64(l_m * l_m) / Float64(Om * Om)), Float64(fma(-2.0, t_1, t) / n))))));
	elseif (t_4 <= 5e+152)
		tmp = sqrt(Float64(t_3 * Float64(t_2 + Float64(Float64(l_m * l_m) * Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om))))));
	else
		tmp = Float64(l_m * sqrt(Float64(n * fma(-2.0, Float64(Float64(U * Float64(n * Float64(U - U_42_))) / Float64(Om * Om)), Float64(-4.0 * Float64(U / 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[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(t$95$2 + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 2e-145], N[(n * N[Sqrt[N[(U * N[(2.0 * N[(U$42$ * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+152], N[Sqrt[N[(t$95$3 * N[(t$95$2 + N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(n * N[(-2.0 * N[(N[(U * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.99999999999999983e-145

   1. Initial program 13.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   5. Simplified 18.2%

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

   6. Taylor expanded in U around 0

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

      1. lower-*.f64 N/A

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

   8. Simplified 39.1%

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

   if 1.99999999999999983e-145 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5e152

   1. Initial program 98.1%

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

   3. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 82.4

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

   5. Simplified 82.4%

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

   if 5e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 28.0%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   5. Simplified 22.3%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 22.2

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

   8. Simplified 22.2%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-145}:\\ \;\;\;\;n \cdot \sqrt{U \cdot \left(2 \cdot \mathsf{fma}\left(U*, \frac{\ell \cdot \ell}{Om \cdot Om}, \frac{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}{n}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U* - U\right)}{Om \cdot Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{n \cdot \mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}, -4 \cdot \frac{U}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-145}:\\ \;\;\;\;n \cdot \sqrt{U \cdot \left(2 \cdot \mathsf{fma}\left(U*, \frac{l\_m \cdot l\_m}{Om \cdot Om}, \frac{\mathsf{fma}\left(-2, t\_1, t\right)}{n}\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(t\_1, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{n \cdot \mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}, -4 \cdot \frac{U}{Om}\right)}\\ \end{array} \end{array} \]

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
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_2 2e-145)
     (*
      n
      (sqrt
       (*
        U
        (* 2.0 (fma U* (/ (* l_m l_m) (* Om Om)) (/ (fma -2.0 t_1 t) n))))))
     (if (<= t_2 5e+152)
       (sqrt (* 2.0 (* (* n U) (fma t_1 -2.0 t))))
       (*
        l_m
        (sqrt
         (*
          n
          (fma
           -2.0
           (/ (* U (* n (- U U*))) (* Om Om))
           (* -4.0 (/ U 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 = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 2e-145) {
		tmp = n * sqrt((U * (2.0 * fma(U_42_, ((l_m * l_m) / (Om * Om)), (fma(-2.0, t_1, t) / n)))));
	} else if (t_2 <= 5e+152) {
		tmp = sqrt((2.0 * ((n * U) * fma(t_1, -2.0, t))));
	} else {
		tmp = l_m * sqrt((n * fma(-2.0, ((U * (n * (U - U_42_))) / (Om * Om)), (-4.0 * (U / 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 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 2e-145)
		tmp = Float64(n * sqrt(Float64(U * Float64(2.0 * fma(U_42_, Float64(Float64(l_m * l_m) / Float64(Om * Om)), Float64(fma(-2.0, t_1, t) / n))))));
	elseif (t_2 <= 5e+152)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(t_1, -2.0, t))));
	else
		tmp = Float64(l_m * sqrt(Float64(n * fma(-2.0, Float64(Float64(U * Float64(n * Float64(U - U_42_))) / Float64(Om * Om)), Float64(-4.0 * Float64(U / 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[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2e-145], N[(n * N[Sqrt[N[(U * N[(2.0 * N[(U$42$ * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+152], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(n * N[(-2.0 * N[(N[(U * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.99999999999999983e-145

   1. Initial program 13.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   5. Simplified 18.2%

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

   6. Taylor expanded in U around 0

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

      1. lower-*.f64 N/A

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

   8. Simplified 39.1%

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

   if 1.99999999999999983e-145 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5e152

   1. Initial program 98.1%

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

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 77.7

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

   5. Simplified 77.7%

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

   if 5e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 28.0%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   5. Simplified 22.3%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 22.2

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

   8. Simplified 22.2%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-145}:\\ \;\;\;\;n \cdot \sqrt{U \cdot \left(2 \cdot \mathsf{fma}\left(U*, \frac{\ell \cdot \ell}{Om \cdot Om}, \frac{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)}{n}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{n \cdot \mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}, -4 \cdot \frac{U}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;n \cdot \sqrt{\mathsf{fma}\left(-2, \frac{\left(l\_m \cdot l\_m\right) \cdot \left(U \cdot U\right)}{Om \cdot Om}, \frac{-4 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{n \cdot Om}\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(t\_1, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{n \cdot \mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}, -4 \cdot \frac{U}{Om}\right)}\\ \end{array} \end{array} \]

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
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_2 0.0)
     (*
      n
      (sqrt
       (fma
        -2.0
        (/ (* (* l_m l_m) (* U U)) (* Om Om))
        (/ (* -4.0 (* U (* l_m l_m))) (* n Om)))))
     (if (<= t_2 5e+152)
       (sqrt (* 2.0 (* (* n U) (fma t_1 -2.0 t))))
       (*
        l_m
        (sqrt
         (*
          n
          (fma
           -2.0
           (/ (* U (* n (- U U*))) (* Om Om))
           (* -4.0 (/ U 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 = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = n * sqrt(fma(-2.0, (((l_m * l_m) * (U * U)) / (Om * Om)), ((-4.0 * (U * (l_m * l_m))) / (n * Om))));
	} else if (t_2 <= 5e+152) {
		tmp = sqrt((2.0 * ((n * U) * fma(t_1, -2.0, t))));
	} else {
		tmp = l_m * sqrt((n * fma(-2.0, ((U * (n * (U - U_42_))) / (Om * Om)), (-4.0 * (U / 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 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(n * sqrt(fma(-2.0, Float64(Float64(Float64(l_m * l_m) * Float64(U * U)) / Float64(Om * Om)), Float64(Float64(-4.0 * Float64(U * Float64(l_m * l_m))) / Float64(n * Om)))));
	elseif (t_2 <= 5e+152)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(t_1, -2.0, t))));
	else
		tmp = Float64(l_m * sqrt(Float64(n * fma(-2.0, Float64(Float64(U * Float64(n * Float64(U - U_42_))) / Float64(Om * Om)), Float64(-4.0 * Float64(U / 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[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(n * N[Sqrt[N[(-2.0 * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(U * U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+152], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(n * N[(-2.0 * N[(N[(U * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   5. Simplified 16.1%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   8. Simplified 30.5%

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

   9. Taylor expanded in U* around 0

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

       1. lower-sqrt.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. associate-*r/ N/A

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

       14. lower-/.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 N/A

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

       19. *-commutative N/A

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

       20. lower-*.f64 30.2

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

   11. Simplified 30.2%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5e152

   1. Initial program 98.1%

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

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 77.9

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

   5. Simplified 77.9%

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

   if 5e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 28.0%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   5. Simplified 22.3%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 22.2

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

   8. Simplified 22.2%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;n \cdot \sqrt{\mathsf{fma}\left(-2, \frac{\left(\ell \cdot \ell\right) \cdot \left(U \cdot U\right)}{Om \cdot Om}, \frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{n \cdot Om}\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{n \cdot \mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}, -4 \cdot \frac{U}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(t\_1, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{n \cdot \mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}, -4 \cdot \frac{U}{Om}\right)}\\ \end{array} \end{array} \]

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 (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 5e-290)
     (sqrt (* n (fma 2.0 (* U t) (/ (* -4.0 (* U (* l_m l_m))) Om))))
     (if (<= t_2 4e+305)
       (sqrt (* 2.0 (* (* n U) (fma t_1 -2.0 t))))
       (*
        l_m
        (sqrt
         (*
          n
          (fma
           -2.0
           (/ (* U (* n (- U U*))) (* Om Om))
           (* -4.0 (/ U 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) * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 5e-290) {
		tmp = sqrt((n * fma(2.0, (U * t), ((-4.0 * (U * (l_m * l_m))) / Om))));
	} else if (t_2 <= 4e+305) {
		tmp = sqrt((2.0 * ((n * U) * fma(t_1, -2.0, t))));
	} else {
		tmp = l_m * sqrt((n * fma(-2.0, ((U * (n * (U - U_42_))) / (Om * Om)), (-4.0 * (U / 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(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 5e-290)
		tmp = sqrt(Float64(n * fma(2.0, Float64(U * t), Float64(Float64(-4.0 * Float64(U * Float64(l_m * l_m))) / Om))));
	elseif (t_2 <= 4e+305)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(t_1, -2.0, t))));
	else
		tmp = Float64(l_m * sqrt(Float64(n * fma(-2.0, Float64(Float64(U * Float64(n * Float64(U - U_42_))) / Float64(Om * Om)), Float64(-4.0 * Float64(U / 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[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-290], N[Sqrt[N[(n * N[(2.0 * N[(U * t), $MachinePrecision] + N[(N[(-4.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 4e+305], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(n * N[(-2.0 * N[(N[(U * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(U / Om), $MachinePrecision]), $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*)))) < 5.0000000000000001e-290

   1. Initial program 12.6%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

   5. Simplified 26.0%

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

   6. Taylor expanded in n around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 28.4

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

   8. Simplified 28.4%

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

   if 5.0000000000000001e-290 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999999999998e305

   1. Initial program 98.1%

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

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 77.7

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

   5. Simplified 77.7%

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

   if 3.9999999999999998e305 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 28.8%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   5. Simplified 23.0%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 22.8

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

   8. Simplified 22.8%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{n \cdot \mathsf{fma}\left(-2, \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om \cdot Om}, -4 \cdot \frac{U}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(t\_1, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \left(\mathsf{fma}\left(U - U*, \frac{n}{Om \cdot Om}, \frac{2}{Om}\right) \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 5e-290)
     (sqrt (* n (fma 2.0 (* U t) (/ (* -4.0 (* U (* l_m l_m))) Om))))
     (if (<= t_2 4e+305)
       (sqrt (* 2.0 (* (* n U) (fma t_1 -2.0 t))))
       (sqrt
        (*
         (* U -2.0)
         (* (fma (- U U*) (/ n (* Om Om)) (/ 2.0 Om)) (* n (* l_m l_m)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 5e-290) {
		tmp = sqrt((n * fma(2.0, (U * t), ((-4.0 * (U * (l_m * l_m))) / Om))));
	} else if (t_2 <= 4e+305) {
		tmp = sqrt((2.0 * ((n * U) * fma(t_1, -2.0, t))));
	} else {
		tmp = sqrt(((U * -2.0) * (fma((U - U_42_), (n / (Om * Om)), (2.0 / Om)) * (n * (l_m * l_m)))));
	}
	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(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 5e-290)
		tmp = sqrt(Float64(n * fma(2.0, Float64(U * t), Float64(Float64(-4.0 * Float64(U * Float64(l_m * l_m))) / Om))));
	elseif (t_2 <= 4e+305)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(t_1, -2.0, t))));
	else
		tmp = sqrt(Float64(Float64(U * -2.0) * Float64(fma(Float64(U - U_42_), Float64(n / Float64(Om * Om)), Float64(2.0 / Om)) * Float64(n * Float64(l_m * l_m)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-290], N[Sqrt[N[(n * N[(2.0 * N[(U * t), $MachinePrecision] + N[(N[(-4.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 4e+305], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * -2.0), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(n * N[(l$95$m * l$95$m), $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*)))) < 5.0000000000000001e-290

   1. Initial program 12.6%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

   5. Simplified 26.0%

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

   6. Taylor expanded in n around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 28.4

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

   8. Simplified 28.4%

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

   if 5.0000000000000001e-290 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999999999998e305

   1. Initial program 98.1%

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

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 77.7

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

   5. Simplified 77.7%

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

   if 3.9999999999999998e305 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 28.8%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. associate-*r/ N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 35.6

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

   5. Simplified 35.6%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \left(\mathsf{fma}\left(U - U*, \frac{n}{Om \cdot Om}, \frac{2}{Om}\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(t\_1, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \frac{l\_m \cdot \left(n \cdot \sqrt{2}\right)}{Om}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_2 2e-145)
     (sqrt (* n (fma 2.0 (* U t) (/ (* -4.0 (* U (* l_m l_m))) Om))))
     (if (<= t_2 INFINITY)
       (sqrt (* 2.0 (* (* n U) (fma t_1 -2.0 t))))
       (* (sqrt (* U U*)) (/ (* l_m (* n (sqrt 2.0))) Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 2e-145) {
		tmp = sqrt((n * fma(2.0, (U * t), ((-4.0 * (U * (l_m * l_m))) / Om))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * ((n * U) * fma(t_1, -2.0, t))));
	} else {
		tmp = sqrt((U * U_42_)) * ((l_m * (n * sqrt(2.0))) / Om);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 2e-145)
		tmp = sqrt(Float64(n * fma(2.0, Float64(U * t), Float64(Float64(-4.0 * Float64(U * Float64(l_m * l_m))) / Om))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(t_1, -2.0, t))));
	else
		tmp = Float64(sqrt(Float64(U * U_42_)) * Float64(Float64(l_m * Float64(n * sqrt(2.0))) / Om));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = 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 * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2e-145], N[Sqrt[N[(n * N[(2.0 * N[(U * t), $MachinePrecision] + N[(N[(-4.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 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*))))) < 1.99999999999999983e-145

   1. Initial program 13.9%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

   5. Simplified 28.6%

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

   6. Taylor expanded in n around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 28.7

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

   8. Simplified 28.7%

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

   if 1.99999999999999983e-145 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 68.3%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.4

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

   5. Simplified 50.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-sqrt.f64 24.8

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

   5. Simplified 24.8%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(t\_1, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\sqrt{U \cdot U*} \cdot \frac{l\_m \cdot \sqrt{2}}{Om}\right)\\ \end{array} \end{array} \]

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
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_2 2e-145)
     (sqrt (* n (fma 2.0 (* U t) (/ (* -4.0 (* U (* l_m l_m))) Om))))
     (if (<= t_2 INFINITY)
       (sqrt (* 2.0 (* (* n U) (fma t_1 -2.0 t))))
       (* n (* (sqrt (* U U*)) (/ (* l_m (sqrt 2.0)) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 2e-145) {
		tmp = sqrt((n * fma(2.0, (U * t), ((-4.0 * (U * (l_m * l_m))) / Om))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * ((n * U) * fma(t_1, -2.0, t))));
	} else {
		tmp = n * (sqrt((U * U_42_)) * ((l_m * sqrt(2.0)) / Om));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 2e-145)
		tmp = sqrt(Float64(n * fma(2.0, Float64(U * t), Float64(Float64(-4.0 * Float64(U * Float64(l_m * l_m))) / Om))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(t_1, -2.0, t))));
	else
		tmp = Float64(n * Float64(sqrt(Float64(U * U_42_)) * Float64(Float64(l_m * sqrt(2.0)) / Om)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = 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 * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2e-145], N[Sqrt[N[(n * N[(2.0 * N[(U * t), $MachinePrecision] + N[(N[(-4.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(n * N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $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*))))) < 1.99999999999999983e-145

   1. Initial program 13.9%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

   5. Simplified 28.6%

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

   6. Taylor expanded in n around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 28.7

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

   8. Simplified 28.7%

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

   if 1.99999999999999983e-145 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 68.3%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.4

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

   5. Simplified 50.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   5. Simplified 0.1%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   8. Simplified 0.6%

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

   9. Taylor expanded in U* around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 20.3

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

   11. Simplified 20.3%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\sqrt{U \cdot U*} \cdot \frac{\ell \cdot \sqrt{2}}{Om}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.8% accurate, 0.4× speedup

Math

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

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
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_2 0.0)
     (sqrt (* (* 2.0 U) (* n t)))
     (if (<= t_2 INFINITY)
       (sqrt (* 2.0 (* (* n U) (fma t_1 -2.0 t))))
       (* n (* (sqrt (* U U*)) (/ (* l_m (sqrt 2.0)) Om)))))))

C

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = 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 * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(n * N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $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 11.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 26.8

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

   5. Simplified 26.8%

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

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

   1. Initial program 68.5%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 50.7

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

   5. Simplified 50.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   5. Simplified 0.1%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   8. Simplified 0.6%

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

   9. Taylor expanded in U* around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 20.3

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

   11. Simplified 20.3%

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

4. Final simplification 42.2%

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

Alternative 11: 51.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(t\_1, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \frac{\left(n \cdot \left(l\_m \cdot l\_m\right)\right) \cdot \mathsf{fma}\left(2, Om, n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 5e-290)
     (sqrt (* n (fma 2.0 (* U t) (/ (* -4.0 (* U (* l_m l_m))) Om))))
     (if (<= t_2 4e+305)
       (sqrt (* 2.0 (* (* n U) (fma t_1 -2.0 t))))
       (sqrt
        (*
         -2.0
         (*
          U
          (/
           (* (* n (* l_m l_m)) (fma 2.0 Om (* n (- U U*))))
           (* Om Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 5e-290) {
		tmp = sqrt((n * fma(2.0, (U * t), ((-4.0 * (U * (l_m * l_m))) / Om))));
	} else if (t_2 <= 4e+305) {
		tmp = sqrt((2.0 * ((n * U) * fma(t_1, -2.0, t))));
	} else {
		tmp = sqrt((-2.0 * (U * (((n * (l_m * l_m)) * fma(2.0, Om, (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(l_m * l_m) / Om)
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 5e-290)
		tmp = sqrt(Float64(n * fma(2.0, Float64(U * t), Float64(Float64(-4.0 * Float64(U * Float64(l_m * l_m))) / Om))));
	elseif (t_2 <= 4e+305)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(t_1, -2.0, t))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(U * Float64(Float64(Float64(n * Float64(l_m * l_m)) * fma(2.0, Om, Float64(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[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-290], N[Sqrt[N[(n * N[(2.0 * N[(U * t), $MachinePrecision] + N[(N[(-4.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 4e+305], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(U * N[(N[(N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * Om + N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000001e-290

   1. Initial program 12.6%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

   5. Simplified 26.0%

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

   6. Taylor expanded in n around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{n \cdot \left(2 \cdot \left(U \cdot t\right) - 4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{n \cdot \color{blue}{\left(2 \cdot \left(U \cdot t\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right) + \color{blue}{-4} \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \sqrt{n \cdot \color{blue}{\mathsf{fma}\left(2, U \cdot t, -4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, \color{blue}{U \cdot t}, -4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \color{blue}{\frac{-4 \cdot \left(U \cdot {\ell}^{2}\right)}{Om}}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \color{blue}{\frac{-4 \cdot \left(U \cdot {\ell}^{2}\right)}{Om}}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{\color{blue}{-4 \cdot \left(U \cdot {\ell}^{2}\right)}}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \color{blue}{\left(U \cdot {\ell}^{2}\right)}}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{Om}\right)} \]

      12. lower-*.f64 28.4

          \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{Om}\right)} \]

   8. Simplified 28.4%

      \[\leadsto \color{blue}{\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}} \]

   if 5.0000000000000001e-290 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999999999998e305

   1. Initial program 98.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + t\right)}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot 2}\right)\right) + t\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot \left(\mathsf{neg}\left(2\right)\right)} + t\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\frac{{\ell}^{2}}{Om} \cdot \color{blue}{-2} + t\right)\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{\ell}^{2}}{Om}}, -2, t\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)\right)} \]

      13. lower-*.f64 77.7

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)\right)} \]

   5. Simplified 77.7%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}} \]

   if 3.9999999999999998e305 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 28.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \color{blue}{\frac{{\ell}^{2}}{Om}}, \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      9. associate-/l* N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{{\ell}^{2} \cdot \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{{\ell}^{2} \cdot \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{n \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{n \cdot \left(U - U*\right)}}{{Om}^{2}}\right)\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}\right)\right)} \]

      16. unpow2 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}\right)\right)} \]

      17. lower-*.f64 23.5

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}\right)\right)} \]

   5. Simplified 23.5%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right)\right)}} \]

   6. Taylor expanded in Om around 0

      \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\frac{2 \cdot \left(Om \cdot {\ell}^{2}\right) + {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\frac{2 \cdot \left(Om \cdot {\ell}^{2}\right) + {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(2, Om \cdot {\ell}^{2}, {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}}{{Om}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{\mathsf{fma}\left(2, \color{blue}{Om \cdot {\ell}^{2}}, {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)} \]

      4. unpow2 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{\mathsf{fma}\left(2, Om \cdot \color{blue}{\left(\ell \cdot \ell\right)}, {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{\mathsf{fma}\left(2, Om \cdot \color{blue}{\left(\ell \cdot \ell\right)}, {\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{\mathsf{fma}\left(2, Om \cdot \left(\ell \cdot \ell\right), \color{blue}{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)}{{Om}^{2}}\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{\mathsf{fma}\left(2, Om \cdot \left(\ell \cdot \ell\right), \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{\mathsf{fma}\left(2, Om \cdot \left(\ell \cdot \ell\right), \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{\mathsf{fma}\left(2, Om \cdot \left(\ell \cdot \ell\right), \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(n \cdot \left(U - U*\right)\right)}\right)}{{Om}^{2}}\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{\mathsf{fma}\left(2, Om \cdot \left(\ell \cdot \ell\right), \left(\ell \cdot \ell\right) \cdot \left(n \cdot \color{blue}{\left(U - U*\right)}\right)\right)}{{Om}^{2}}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{\mathsf{fma}\left(2, Om \cdot \left(\ell \cdot \ell\right), \left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{\color{blue}{Om \cdot Om}}\right)} \]

      12. lower-*.f64 25.0

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{\mathsf{fma}\left(2, Om \cdot \left(\ell \cdot \ell\right), \left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{\color{blue}{Om \cdot Om}}\right)} \]

   8. Simplified 25.0%

      \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(2, Om \cdot \left(\ell \cdot \ell\right), \left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om \cdot Om}}\right)} \]

   9. Taylor expanded in l around 0

      \[\leadsto \sqrt{-2 \cdot \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot Om + n \cdot \left(U - U*\right)\right)\right)\right)}{{Om}^{2}}}} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(U \cdot \frac{{\ell}^{2} \cdot \left(n \cdot \left(2 \cdot Om + n \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)}} \]

       2. lower-*.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(U \cdot \frac{{\ell}^{2} \cdot \left(n \cdot \left(2 \cdot Om + n \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)}} \]

       3. lower-/.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(U \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(n \cdot \left(2 \cdot Om + n \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}}\right)} \]

       4. associate-*r* N/A

          \[\leadsto \sqrt{-2 \cdot \left(U \cdot \frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot Om + n \cdot \left(U - U*\right)\right)}}{{Om}^{2}}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(U \cdot \frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot Om + n \cdot \left(U - U*\right)\right)}}{{Om}^{2}}\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(U \cdot \frac{\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot \left(2 \cdot Om + n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)} \]

       7. unpow2 N/A

          \[\leadsto \sqrt{-2 \cdot \left(U \cdot \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot Om + n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(U \cdot \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot Om + n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(U \cdot \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(2, Om, n \cdot \left(U - U*\right)\right)}}{{Om}^{2}}\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \sqrt{-2 \cdot \left(U \cdot \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(2, Om, \color{blue}{n \cdot \left(U - U*\right)}\right)}{{Om}^{2}}\right)} \]

       11. lower--.f64 N/A

           \[\leadsto \sqrt{-2 \cdot \left(U \cdot \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(2, Om, n \cdot \color{blue}{\left(U - U*\right)}\right)}{{Om}^{2}}\right)} \]

       12. unpow2 N/A

           \[\leadsto \sqrt{-2 \cdot \left(U \cdot \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(2, Om, n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}}\right)} \]

       13. lower-*.f64 33.9

           \[\leadsto \sqrt{-2 \cdot \left(U \cdot \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(2, Om, n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}}\right)} \]

   11. Simplified 33.9%

       \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(U \cdot \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(2, Om, n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \mathsf{fma}\left(2, Om, n \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_1 2e-137)
     (sqrt (* (* 2.0 U) (* n t)))
     (if (<= t_1 5e+152)
       (* (sqrt 2.0) (sqrt (* t (* n U))))
       (sqrt (/ (* -4.0 (* n (* U (* l_m l_m)))) Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 2e-137) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else if (t_1 <= 5e+152) {
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	} else {
		tmp = sqrt(((-4.0 * (n * (U * (l_m * l_m)))) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) + ((n * ((l_m / om) ** 2.0d0)) * (u_42 - u)))))
    if (t_1 <= 2d-137) then
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    else if (t_1 <= 5d+152) then
        tmp = sqrt(2.0d0) * sqrt((t * (n * u)))
    else
        tmp = sqrt((((-4.0d0) * (n * (u * (l_m * l_m)))) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 2e-137) {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	} else if (t_1 <= 5e+152) {
		tmp = Math.sqrt(2.0) * Math.sqrt((t * (n * U)));
	} else {
		tmp = Math.sqrt(((-4.0 * (n * (U * (l_m * l_m)))) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)))))
	tmp = 0
	if t_1 <= 2e-137:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	elif t_1 <= 5e+152:
		tmp = math.sqrt(2.0) * math.sqrt((t * (n * U)))
	else:
		tmp = math.sqrt(((-4.0 * (n * (U * (l_m * l_m)))) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 2e-137)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	elseif (t_1 <= 5e+152)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(t * Float64(n * U))));
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(n * Float64(U * Float64(l_m * l_m)))) / Om));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U)))));
	tmp = 0.0;
	if (t_1 <= 2e-137)
		tmp = sqrt(((2.0 * U) * (n * t)));
	elseif (t_1 <= 5e+152)
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	else
		tmp = sqrt(((-4.0 * (n * (U * (l_m * l_m)))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = 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 * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 2e-137], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e+152], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-4.0 * N[(n * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $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*))))) < 1.99999999999999996e-137

   1. Initial program 18.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 32.1

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 32.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 1.99999999999999996e-137 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5e152

   1. Initial program 98.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      6. sub-neg N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \cdot \sqrt{2} \]

      7. +-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + t\right)}} \cdot \sqrt{2} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot 2}\right)\right) + t\right)} \cdot \sqrt{2} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot \left(\mathsf{neg}\left(2\right)\right)} + t\right)} \cdot \sqrt{2} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{{\ell}^{2}}{Om} \cdot \color{blue}{-2} + t\right)} \cdot \sqrt{2} \]

      11. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)}} \cdot \sqrt{2} \]

      12. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{\ell}^{2}}{Om}}, -2, t\right)} \cdot \sqrt{2} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)} \cdot \sqrt{2} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)} \cdot \sqrt{2} \]

      15. lower-sqrt.f64 76.9

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)} \cdot \color{blue}{\sqrt{2}} \]

   5. Simplified 76.9%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)} \cdot \sqrt{2}} \]

   6. Taylor expanded in U around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}\right)} \cdot \sqrt{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)}\right)} \cdot \sqrt{2} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right)\right)} \cdot \sqrt{2} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)} \cdot \sqrt{2} \]

      8. lower-*.f64 67.0

         \[\leadsto \sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)} \cdot \sqrt{2} \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \cdot \sqrt{2} \]

   9. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot t\right)}} \cdot \sqrt{2} \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot t\right)}} \cdot \sqrt{2} \]

       2. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot t}} \cdot \sqrt{2} \]

       3. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot t}} \cdot \sqrt{2} \]

       4. lower-*.f64 65.6

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right)} \cdot t} \cdot \sqrt{2} \]

   11. Simplified 65.6%

       \[\leadsto \color{blue}{\sqrt{\left(U \cdot n\right) \cdot t}} \cdot \sqrt{2} \]

   if 5e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 28.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \color{blue}{\frac{{\ell}^{2}}{Om}}, \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      9. associate-/l* N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{{\ell}^{2} \cdot \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{{\ell}^{2} \cdot \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{n \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{n \cdot \left(U - U*\right)}}{{Om}^{2}}\right)\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}\right)\right)} \]

      16. unpow2 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}\right)\right)} \]

      17. lower-*.f64 22.8

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}\right)\right)} \]

   5. Simplified 22.8%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right)\right)}} \]

   6. Taylor expanded in Om around inf

      \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\frac{2 \cdot {\ell}^{2}}{Om}}\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\frac{2 \cdot {\ell}^{2}}{Om}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{\color{blue}{2 \cdot {\ell}^{2}}}{Om}\right)} \]

      4. unpow2 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om}\right)} \]

      5. lower-*.f64 16.0

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{Om}\right)} \]

   8. Simplified 16.0%

      \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{Om}}\right)} \]

   9. Taylor expanded in U around 0

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}} \]

       2. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}} \]

       3. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}}{Om}} \]

       4. associate-*r* N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(U \cdot {\ell}^{2}\right) \cdot n\right)}}{Om}} \]

       5. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(U \cdot {\ell}^{2}\right) \cdot n\right)}}{Om}} \]

       6. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \left(\color{blue}{\left(U \cdot {\ell}^{2}\right)} \cdot n\right)}{Om}} \]

       7. unpow2 N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \left(\left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot n\right)}{Om}} \]

       8. lower-*.f64 17.3

          \[\leadsto \sqrt{\frac{-4 \cdot \left(\left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot n\right)}{Om}} \]

   11. Simplified 17.3%

       \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot \left(\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n\right)}{Om}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(t\_1, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(\left(l\_m \cdot l\_m\right) \cdot U*\right) \cdot \left(n \cdot n\right)\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

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 (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 5e-290)
     (sqrt (* n (fma 2.0 (* U t) (/ (* -4.0 (* U (* l_m l_m))) Om))))
     (if (<= t_2 4e+305)
       (sqrt (* 2.0 (* (* n U) (fma t_1 -2.0 t))))
       (sqrt (/ (* 2.0 (* U (* (* (* l_m l_m) U*) (* 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) * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 5e-290) {
		tmp = sqrt((n * fma(2.0, (U * t), ((-4.0 * (U * (l_m * l_m))) / Om))));
	} else if (t_2 <= 4e+305) {
		tmp = sqrt((2.0 * ((n * U) * fma(t_1, -2.0, t))));
	} else {
		tmp = sqrt(((2.0 * (U * (((l_m * l_m) * U_42_) * (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(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 5e-290)
		tmp = sqrt(Float64(n * fma(2.0, Float64(U * t), Float64(Float64(-4.0 * Float64(U * Float64(l_m * l_m))) / Om))));
	elseif (t_2 <= 4e+305)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(t_1, -2.0, t))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * Float64(Float64(Float64(l_m * l_m) * U_42_) * 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[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-290], N[Sqrt[N[(n * N[(2.0 * N[(U * t), $MachinePrecision] + N[(N[(-4.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 4e+305], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(U * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * U$42$), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000001e-290

   1. Initial program 12.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around -inf

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -1 \cdot \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}} \]

      2. mul-1-neg N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)}} \]

      3. unsub-neg N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) - \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) - \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)} - \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)} - \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right) - \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)} - \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) - \color{blue}{\frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}} \]

   5. Simplified 26.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) - \frac{\mathsf{fma}\left(2, \left(\left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U - U*\right)\right) \cdot \frac{U}{Om}, 4 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om}}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right) - 4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right) - 4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{n \cdot \left(2 \cdot \left(U \cdot t\right) - 4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{n \cdot \color{blue}{\left(2 \cdot \left(U \cdot t\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right) + \color{blue}{-4} \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \sqrt{n \cdot \color{blue}{\mathsf{fma}\left(2, U \cdot t, -4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, \color{blue}{U \cdot t}, -4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \color{blue}{\frac{-4 \cdot \left(U \cdot {\ell}^{2}\right)}{Om}}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \color{blue}{\frac{-4 \cdot \left(U \cdot {\ell}^{2}\right)}{Om}}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{\color{blue}{-4 \cdot \left(U \cdot {\ell}^{2}\right)}}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \color{blue}{\left(U \cdot {\ell}^{2}\right)}}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{Om}\right)} \]

      12. lower-*.f64 28.4

          \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{Om}\right)} \]

   8. Simplified 28.4%

      \[\leadsto \color{blue}{\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}} \]

   if 5.0000000000000001e-290 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999999999998e305

   1. Initial program 98.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + t\right)}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot 2}\right)\right) + t\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot \left(\mathsf{neg}\left(2\right)\right)} + t\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\frac{{\ell}^{2}}{Om} \cdot \color{blue}{-2} + t\right)\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{\ell}^{2}}{Om}}, -2, t\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)\right)} \]

      13. lower-*.f64 77.7

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)\right)} \]

   5. Simplified 77.7%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}} \]

   if 3.9999999999999998e305 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 28.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      14. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

   5. Simplified 23.7%

      \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \frac{\left(2 \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)}{n}\right)}} \]

   6. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}{{Om}^{2}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}{{Om}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}}{{Om}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{2 \cdot \color{blue}{\left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}}{{Om}^{2}}} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\frac{2 \cdot \left(U \cdot \color{blue}{\left(\left(U* \cdot {\ell}^{2}\right) \cdot {n}^{2}\right)}\right)}{{Om}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{2 \cdot \left(U \cdot \color{blue}{\left(\left(U* \cdot {\ell}^{2}\right) \cdot {n}^{2}\right)}\right)}{{Om}^{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{2 \cdot \left(U \cdot \left(\color{blue}{\left(U* \cdot {\ell}^{2}\right)} \cdot {n}^{2}\right)\right)}{{Om}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\frac{2 \cdot \left(U \cdot \left(\left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot {n}^{2}\right)\right)}{{Om}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{2 \cdot \left(U \cdot \left(\left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot {n}^{2}\right)\right)}{{Om}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\frac{2 \cdot \left(U \cdot \left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\left(n \cdot n\right)}\right)\right)}{{Om}^{2}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{2 \cdot \left(U \cdot \left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\left(n \cdot n\right)}\right)\right)}{{Om}^{2}}} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\frac{2 \cdot \left(U \cdot \left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)\right)\right)}{\color{blue}{Om \cdot Om}}} \]

      13. lower-*.f64 32.6

          \[\leadsto \sqrt{\frac{2 \cdot \left(U \cdot \left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)\right)\right)}{\color{blue}{Om \cdot Om}}} \]

   8. Simplified 32.6%

      \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)\right)\right)}{Om \cdot Om}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(U \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot U*\right) \cdot \left(n \cdot n\right)\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(t\_1, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \frac{\left(l\_m \cdot l\_m\right) \cdot \left(n \cdot n\right)}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 5e-290)
     (sqrt (* n (fma 2.0 (* U t) (/ (* -4.0 (* U (* l_m l_m))) Om))))
     (if (<= t_2 4e+305)
       (sqrt (* 2.0 (* (* n U) (fma t_1 -2.0 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) * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 5e-290) {
		tmp = sqrt((n * fma(2.0, (U * t), ((-4.0 * (U * (l_m * l_m))) / Om))));
	} else if (t_2 <= 4e+305) {
		tmp = sqrt((2.0 * ((n * U) * fma(t_1, -2.0, 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(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 5e-290)
		tmp = sqrt(Float64(n * fma(2.0, Float64(U * t), Float64(Float64(-4.0 * Float64(U * Float64(l_m * l_m))) / Om))));
	elseif (t_2 <= 4e+305)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(t_1, -2.0, t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(U * U_42_) * Float64(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[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-290], N[Sqrt[N[(n * N[(2.0 * N[(U * t), $MachinePrecision] + N[(N[(-4.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 4e+305], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(U * U$42$), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000001e-290

   1. Initial program 12.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around -inf

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -1 \cdot \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}} \]

      2. mul-1-neg N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)}} \]

      3. unsub-neg N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) - \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}} \]

      4. lower--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) - \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)} - \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)} - \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right) - \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)} - \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) - \color{blue}{\frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}} \]

   5. Simplified 26.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) - \frac{\mathsf{fma}\left(2, \left(\left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U - U*\right)\right) \cdot \frac{U}{Om}, 4 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{Om}}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right) - 4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right) - 4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{n \cdot \left(2 \cdot \left(U \cdot t\right) - 4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{n \cdot \color{blue}{\left(2 \cdot \left(U \cdot t\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{n \cdot \left(2 \cdot \left(U \cdot t\right) + \color{blue}{-4} \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \sqrt{n \cdot \color{blue}{\mathsf{fma}\left(2, U \cdot t, -4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, \color{blue}{U \cdot t}, -4 \cdot \frac{U \cdot {\ell}^{2}}{Om}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \color{blue}{\frac{-4 \cdot \left(U \cdot {\ell}^{2}\right)}{Om}}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \color{blue}{\frac{-4 \cdot \left(U \cdot {\ell}^{2}\right)}{Om}}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{\color{blue}{-4 \cdot \left(U \cdot {\ell}^{2}\right)}}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \color{blue}{\left(U \cdot {\ell}^{2}\right)}}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{Om}\right)} \]

      12. lower-*.f64 28.4

          \[\leadsto \sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{Om}\right)} \]

   8. Simplified 28.4%

      \[\leadsto \color{blue}{\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}} \]

   if 5.0000000000000001e-290 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999999999998e305

   1. Initial program 98.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + t\right)}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot 2}\right)\right) + t\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot \left(\mathsf{neg}\left(2\right)\right)} + t\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\frac{{\ell}^{2}}{Om} \cdot \color{blue}{-2} + t\right)\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{\ell}^{2}}{Om}}, -2, t\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)\right)} \]

      13. lower-*.f64 77.7

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)\right)} \]

   5. Simplified 77.7%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}} \]

   if 3.9999999999999998e305 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 28.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. 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. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \frac{\color{blue}{\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}}{{Om}^{2}}} \]

      3. associate-/l* N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot \frac{{\ell}^{2} \cdot {n}^{2}}{{Om}^{2}}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot \frac{{\ell}^{2} \cdot {n}^{2}}{{Om}^{2}}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot U*\right)} \cdot \frac{{\ell}^{2} \cdot {n}^{2}}{{Om}^{2}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \color{blue}{\frac{{\ell}^{2} \cdot {n}^{2}}{{Om}^{2}}}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \frac{\color{blue}{{\ell}^{2} \cdot {n}^{2}}}{{Om}^{2}}\right)} \]

      8. unpow2 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot {n}^{2}}{{Om}^{2}}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot {n}^{2}}{{Om}^{2}}\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(n \cdot n\right)}}{{Om}^{2}}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(n \cdot n\right)}}{{Om}^{2}}\right)} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)}{\color{blue}{Om \cdot Om}}\right)} \]

      13. lower-*.f64 31.2

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)}{\color{blue}{Om \cdot Om}}\right)} \]

   5. Simplified 31.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot U*\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)}{Om \cdot Om}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{n \cdot \mathsf{fma}\left(2, U \cdot t, \frac{-4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)}{Om \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(t\_1, -2, t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om)))
   (if (<=
        (sqrt
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))
        0.0)
     (sqrt (* (* 2.0 U) (* n t)))
     (sqrt (* 2.0 (* (* n U) (fma t_1 -2.0 t)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double tmp;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))))) <= 0.0) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = sqrt((2.0 * ((n * U) * fma(t_1, -2.0, t))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))) <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(t_1, -2.0, t))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t$95$1 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 11.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 26.8

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 26.8%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 55.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. 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. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + t\right)}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot 2}\right)\right) + t\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot \left(\mathsf{neg}\left(2\right)\right)} + t\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\frac{{\ell}^{2}}{Om} \cdot \color{blue}{-2} + t\right)\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)}\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{\ell}^{2}}{Om}}, -2, t\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)\right)} \]

      13. lower-*.f64 41.5

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)\right)} \]

   5. Simplified 41.5%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (+
         (- t (* 2.0 (/ (* l_m l_m) Om)))
         (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))
      2e-137)
   (sqrt (* (* 2.0 U) (* n t)))
   (* (sqrt 2.0) (sqrt (* t (* n U))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))))) <= 2e-137) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) + ((n * ((l_m / om) ** 2.0d0)) * (u_42 - u))))) <= 2d-137) then
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    else
        tmp = sqrt(2.0d0) * sqrt((t * (n * u)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U))))) <= 2e-137) {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	} else {
		tmp = Math.sqrt(2.0) * Math.sqrt((t * (n * U)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U))))) <= 2e-137:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	else:
		tmp = math.sqrt(2.0) * math.sqrt((t * (n * U)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))) <= 2e-137)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	else
		tmp = Float64(sqrt(2.0) * sqrt(Float64(t * Float64(n * U))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U))))) <= 2e-137)
		tmp = sqrt(((2.0 * U) * (n * t)));
	else
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[N[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 * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-137], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.99999999999999996e-137

   1. Initial program 18.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 32.1

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 32.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 1.99999999999999996e-137 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 54.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      6. sub-neg N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \cdot \sqrt{2} \]

      7. +-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + t\right)}} \cdot \sqrt{2} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot 2}\right)\right) + t\right)} \cdot \sqrt{2} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot \left(\mathsf{neg}\left(2\right)\right)} + t\right)} \cdot \sqrt{2} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{{\ell}^{2}}{Om} \cdot \color{blue}{-2} + t\right)} \cdot \sqrt{2} \]

      11. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)}} \cdot \sqrt{2} \]

      12. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{\ell}^{2}}{Om}}, -2, t\right)} \cdot \sqrt{2} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)} \cdot \sqrt{2} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)} \cdot \sqrt{2} \]

      15. lower-sqrt.f64 40.6

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)} \cdot \color{blue}{\sqrt{2}} \]

   5. Simplified 40.6%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)} \cdot \sqrt{2}} \]

   6. Taylor expanded in U around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}\right)} \cdot \sqrt{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)}\right)} \cdot \sqrt{2} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right)\right)} \cdot \sqrt{2} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)} \cdot \sqrt{2} \]

      8. lower-*.f64 38.9

         \[\leadsto \sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)} \cdot \sqrt{2} \]

   8. Simplified 38.9%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \cdot \sqrt{2} \]

   9. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot t\right)}} \cdot \sqrt{2} \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot t\right)}} \cdot \sqrt{2} \]

       2. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot t}} \cdot \sqrt{2} \]

       3. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot t}} \cdot \sqrt{2} \]

       4. lower-*.f64 31.4

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right)} \cdot t} \cdot \sqrt{2} \]

   11. Simplified 31.4%

       \[\leadsto \color{blue}{\sqrt{\left(U \cdot n\right) \cdot t}} \cdot \sqrt{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 39.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(2 \cdot U\right) \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5e+39)
   (* (sqrt 2.0) (sqrt (* t (* n U))))
   (sqrt (* -2.0 (/ (* (* 2.0 U) (* n (* l_m l_m))) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 5e+39) {
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	} else {
		tmp = sqrt((-2.0 * (((2.0 * U) * (n * (l_m * l_m))) / Om)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 5d+39) then
        tmp = sqrt(2.0d0) * sqrt((t * (n * u)))
    else
        tmp = sqrt(((-2.0d0) * (((2.0d0 * u) * (n * (l_m * l_m))) / om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 5e+39) {
		tmp = Math.sqrt(2.0) * Math.sqrt((t * (n * U)));
	} else {
		tmp = Math.sqrt((-2.0 * (((2.0 * U) * (n * (l_m * l_m))) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 5e+39:
		tmp = math.sqrt(2.0) * math.sqrt((t * (n * U)))
	else:
		tmp = math.sqrt((-2.0 * (((2.0 * U) * (n * (l_m * l_m))) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 5e+39)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(t * Float64(n * U))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(Float64(2.0 * U) * Float64(n * Float64(l_m * l_m))) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 5e+39)
		tmp = sqrt(2.0) * sqrt((t * (n * U)));
	else
		tmp = sqrt((-2.0 * (((2.0 * U) * (n * (l_m * l_m))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 5e+39], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.00000000000000015e39

   1. Initial program 53.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \cdot \sqrt{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right)} \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot \sqrt{2} \]

      6. sub-neg N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \cdot \sqrt{2} \]

      7. +-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + t\right)}} \cdot \sqrt{2} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot 2}\right)\right) + t\right)} \cdot \sqrt{2} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot \left(\mathsf{neg}\left(2\right)\right)} + t\right)} \cdot \sqrt{2} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{{\ell}^{2}}{Om} \cdot \color{blue}{-2} + t\right)} \cdot \sqrt{2} \]

      11. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)}} \cdot \sqrt{2} \]

      12. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{\ell}^{2}}{Om}}, -2, t\right)} \cdot \sqrt{2} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)} \cdot \sqrt{2} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)} \cdot \sqrt{2} \]

      15. lower-sqrt.f64 39.6

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)} \cdot \color{blue}{\sqrt{2}} \]

   5. Simplified 39.6%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)} \cdot \sqrt{2}} \]

   6. Taylor expanded in U around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}\right)} \cdot \sqrt{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)}\right)} \cdot \sqrt{2} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right)\right)} \cdot \sqrt{2} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)} \cdot \sqrt{2} \]

      8. lower-*.f64 40.0

         \[\leadsto \sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)} \cdot \sqrt{2} \]

   8. Simplified 40.0%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \cdot \sqrt{2} \]

   9. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot t\right)}} \cdot \sqrt{2} \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot t\right)}} \cdot \sqrt{2} \]

       2. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot t}} \cdot \sqrt{2} \]

       3. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot t}} \cdot \sqrt{2} \]

       4. lower-*.f64 33.3

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right)} \cdot t} \cdot \sqrt{2} \]

   11. Simplified 33.3%

       \[\leadsto \color{blue}{\sqrt{\left(U \cdot n\right) \cdot t}} \cdot \sqrt{2} \]

   if 5.00000000000000015e39 < l

   1. Initial program 33.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \color{blue}{\frac{{\ell}^{2}}{Om}}, \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      9. associate-/l* N/A

         \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{{\ell}^{2} \cdot \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{{\ell}^{2} \cdot \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{n \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{\color{blue}{n \cdot \left(U - U*\right)}}{{Om}^{2}}\right)\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}\right)\right)} \]

      16. unpow2 N/A

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}\right)\right)} \]

      17. lower-*.f64 26.8

          \[\leadsto \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}\right)\right)} \]

   5. Simplified 26.8%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right)\right)}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\frac{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\frac{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{-2 \cdot \frac{\color{blue}{\left(2 \cdot U\right) \cdot \left({\ell}^{2} \cdot n\right)}}{Om}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \frac{\color{blue}{\left(2 \cdot U\right) \cdot \left({\ell}^{2} \cdot n\right)}}{Om}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \frac{\color{blue}{\left(2 \cdot U\right)} \cdot \left({\ell}^{2} \cdot n\right)}{Om}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \frac{\left(2 \cdot U\right) \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}}{Om}} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{-2 \cdot \frac{\left(2 \cdot U\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)}{Om}} \]

      8. lower-*.f64 27.1

         \[\leadsto \sqrt{-2 \cdot \frac{\left(2 \cdot U\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)}{Om}} \]

   8. Simplified 27.1%

      \[\leadsto \sqrt{-2 \cdot \color{blue}{\frac{\left(2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{t \cdot \left(n \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* 2.0 U) (* n t))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt(((2.0 * U) * (n * t)));
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt(((2.0d0 * u) * (n * t)))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt(((2.0 * U) * (n * t)));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt(((2.0 * U) * (n * t)))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(2.0 * U) * Float64(n * t)))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt(((2.0 * U) * (n * t)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.8%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

   4. lower-*.f64 28.9

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

5. Simplified 28.9%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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*))))))