Toniolo and Linder, Equation (13)

Percentage Accurate: 50.7% → 66.5%

Time: 20.3s

Alternatives: 26

Speedup: 2.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(U* - U, n \cdot \frac{l\_m}{Om}, l\_m \cdot -2\right), t\right)\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(t\_1 \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{t\_1 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (/ l_m Om) (fma (- U* U) (* n (/ l_m Om)) (* l_m -2.0)) t))
        (t_2 (* U (* 2.0 n)))
        (t_3
         (*
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
          t_2)))
   (if (<= t_3 0.0)
     (sqrt (* U (* t_1 (* 2.0 n))))
     (if (<= t_3 5e+299)
       (sqrt (* t_1 t_2))
       (*
        (sqrt (* (* n U) (fma n (/ (- U* U) (* Om Om)) (/ -2.0 Om))))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma((l_m / Om), fma((U_42_ - U), (n * (l_m / Om)), (l_m * -2.0)), t);
	double t_2 = U * (2.0 * n);
	double t_3 = ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_2;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((U * (t_1 * (2.0 * n))));
	} else if (t_3 <= 5e+299) {
		tmp = sqrt((t_1 * t_2));
	} else {
		tmp = sqrt(((n * U) * fma(n, ((U_42_ - U) / (Om * Om)), (-2.0 / Om)))) * (l_m * sqrt(2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(l_m / Om), fma(Float64(U_42_ - U), Float64(n * Float64(l_m / Om)), Float64(l_m * -2.0)), t)
	t_2 = Float64(U * Float64(2.0 * n))
	t_3 = Float64(Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))) * t_2)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(U * Float64(t_1 * Float64(2.0 * n))));
	elseif (t_3 <= 5e+299)
		tmp = sqrt(Float64(t_1 * t_2));
	else
		tmp = Float64(sqrt(Float64(Float64(n * U) * fma(n, Float64(Float64(U_42_ - U) / Float64(Om * Om)), Float64(-2.0 / Om)))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(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] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(U * N[(t$95$1 * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 5e+299], N[Sqrt[N[(t$95$1 * t$95$2), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 17.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 25.3%

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

   6. Applied rewrites 47.1%

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

   7. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 47.1

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

   9. Applied rewrites 47.1%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000003e299

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

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

   if 5.0000000000000003e299 < (*.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 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 36.3%

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

   6. Applied rewrites 49.4%

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

   7. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 27.4%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right) \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(U* - U, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right) \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(U* - U, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 51.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \sqrt{\left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot t\_1}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+150}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(U - U*\right) \cdot \left(\left(n \cdot l\_m\right) \cdot \left(n \cdot l\_m\right)\right)\right) \cdot \left(U \cdot -2\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (sqrt
          (*
           (+
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
           t_1))))
   (if (<= t_2 0.0)
     (sqrt (* (* n t) (* 2.0 U)))
     (if (<= t_2 1e+150)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (sqrt
        (/ (* (* (- U U*) (* (* n l_m) (* n l_m))) (* U -2.0)) (* Om Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = sqrt((((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_1));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((n * t) * (2.0 * U)));
	} else if (t_2 <= 1e+150) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = sqrt(((((U - U_42_) * ((n * l_m) * (n * l_m))) * (U * -2.0)) / (Om * Om)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = sqrt(Float64(Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))) * t_1))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * U)));
	elseif (t_2 <= 1e+150)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U - U_42_) * Float64(Float64(n * l_m) * Float64(n * l_m))) * Float64(U * -2.0)) / Float64(Om * Om)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(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] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+150], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(U - U$42$), $MachinePrecision] * N[(N[(n * l$95$m), $MachinePrecision] * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om * 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 20.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 40.8

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

   5. Applied rewrites 40.8%

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

   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*))))) < 9.99999999999999981e149

   1. Initial program 99.6%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if 9.99999999999999981e149 < (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 31.4%

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in Om around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 33.3

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

   7. Applied rewrites 33.3%

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

   8. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 35.0

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

   10. Applied rewrites 35.0%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)} \leq 10^{+150}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(U - U*\right) \cdot \left(\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)\right) \cdot \left(U \cdot -2\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \sqrt{\left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot t\_1}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+150}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{\left(l\_m \cdot \left(U \cdot -2\right)\right) \cdot \left(l\_m \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (sqrt
          (*
           (+
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
           t_1))))
   (if (<= t_2 0.0)
     (sqrt (* (* n t) (* 2.0 U)))
     (if (<= t_2 1e+150)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (sqrt
        (* n (/ (* (* l_m (* U -2.0)) (* l_m (* n (- U U*)))) (* Om Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = sqrt((((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_1));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((n * t) * (2.0 * U)));
	} else if (t_2 <= 1e+150) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = sqrt((n * (((l_m * (U * -2.0)) * (l_m * (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(U * Float64(2.0 * n))
	t_2 = sqrt(Float64(Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))) * t_1))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * U)));
	elseif (t_2 <= 1e+150)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = sqrt(Float64(n * Float64(Float64(Float64(l_m * Float64(U * -2.0)) * Float64(l_m * 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[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(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] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+150], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(N[(l$95$m * N[(U * -2.0), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 40.8

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

   5. Applied rewrites 40.8%

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

   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*))))) < 9.99999999999999981e149

   1. Initial program 99.6%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if 9.99999999999999981e149 < (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 31.4%

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in Om around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.9

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

   7. Applied rewrites 33.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 35.0

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

   9. Applied rewrites 35.0%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)} \leq 10^{+150}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{\left(\ell \cdot \left(U \cdot -2\right)\right) \cdot \left(\ell \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \sqrt{\left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot t\_1}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+150}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(-2 \cdot \left(U \cdot \frac{l\_m \cdot \left(l\_m \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om \cdot Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (sqrt
          (*
           (+
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
           t_1))))
   (if (<= t_2 0.0)
     (sqrt (* (* n t) (* 2.0 U)))
     (if (<= t_2 1e+150)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (sqrt
        (* n (* -2.0 (* U (/ (* l_m (* l_m (* n (- U U*)))) (* Om Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = sqrt((((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_1));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((n * t) * (2.0 * U)));
	} else if (t_2 <= 1e+150) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = sqrt((n * (-2.0 * (U * ((l_m * (l_m * (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(U * Float64(2.0 * n))
	t_2 = sqrt(Float64(Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))) * t_1))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * U)));
	elseif (t_2 <= 1e+150)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = sqrt(Float64(n * Float64(-2.0 * Float64(U * Float64(Float64(l_m * Float64(l_m * 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[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(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] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+150], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(-2.0 * N[(U * N[(N[(l$95$m * N[(l$95$m * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * 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 20.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 40.8

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

   5. Applied rewrites 40.8%

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

   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*))))) < 9.99999999999999981e149

   1. Initial program 99.6%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if 9.99999999999999981e149 < (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 31.4%

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in Om around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.9

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

   7. Applied rewrites 33.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 33.9

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 35.0

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

   9. Applied rewrites 35.0%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)} \leq 10^{+150}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(-2 \cdot \left(U \cdot \frac{\ell \cdot \left(\ell \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om \cdot Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \sqrt{\left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot t\_1}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+150}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(l\_m \cdot l\_m\right) \cdot U*\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (sqrt
          (*
           (+
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
           t_1))))
   (if (<= t_2 0.0)
     (sqrt (* (* n t) (* 2.0 U)))
     (if (<= t_2 1e+150)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (sqrt (* n (/ (* (* 2.0 U) (* n (* (* l_m l_m) U*))) (* Om Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = sqrt((((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_1));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((n * t) * (2.0 * U)));
	} else if (t_2 <= 1e+150) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = sqrt((n * (((2.0 * U) * (n * ((l_m * l_m) * U_42_))) / (Om * Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = sqrt(Float64(Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))) * t_1))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * U)));
	elseif (t_2 <= 1e+150)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = sqrt(Float64(n * Float64(Float64(Float64(2.0 * U) * Float64(n * Float64(Float64(l_m * l_m) * 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[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(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] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+150], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(N[(l$95$m * l$95$m), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 40.8

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

   5. Applied rewrites 40.8%

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

   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*))))) < 9.99999999999999981e149

   1. Initial program 99.6%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if 9.99999999999999981e149 < (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 31.4%

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in Om around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.9

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

   7. Applied rewrites 33.9%

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

   8. Taylor expanded in U around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.2

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

   10. Applied rewrites 34.2%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)} \leq 10^{+150}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \frac{\left(2 \cdot U\right) \cdot \left(n \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.8% accurate, 0.4× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 17.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 25.3%

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

   6. Applied rewrites 47.1%

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

   7. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 47.1

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

   9. Applied rewrites 47.1%

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

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

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 79.6%

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

   6. Applied rewrites 79.6%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 2.3%

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

   6. Applied rewrites 43.8%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 51.5%

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

4. Final simplification 70.4%

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

Alternative 7: 65.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 17.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 25.3%

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

   6. Applied rewrites 47.1%

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

   7. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 47.1

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

   9. Applied rewrites 47.1%

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

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

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 79.6%

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

   6. Applied rewrites 73.2%

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

   7. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 73.2

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

   9. Applied rewrites 73.2%

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

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

   11. Applied rewrites 76.7%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 2.3%

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

   6. Applied rewrites 43.8%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 51.5%

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

4. Final simplification 68.4%

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

Alternative 8: 63.3% accurate, 0.4× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 17.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 25.3%

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

   6. Applied rewrites 47.1%

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

   7. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 42.5

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

   9. Applied rewrites 42.5%

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

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

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 79.6%

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

   6. Applied rewrites 73.2%

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

   7. Taylor expanded in U around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 74.9

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

   9. Applied rewrites 74.9%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 2.3%

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

   6. Applied rewrites 43.8%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 51.5%

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

4. Final simplification 66.4%

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

Alternative 9: 62.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 17.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 25.3%

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

   6. Applied rewrites 47.1%

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

   7. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 42.5

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

   9. Applied rewrites 42.5%

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

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

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 79.6%

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

   6. Applied rewrites 73.2%

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

   7. Taylor expanded in U around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 74.9

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

   9. Applied rewrites 74.9%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 2.3%

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

   6. Applied rewrites 43.8%

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

   7. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 43.8

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

   9. Applied rewrites 43.8%

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

   10. Taylor expanded in t around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

   12. Applied rewrites 48.9%

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

4. Final simplification 66.0%

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

Alternative 10: 57.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \frac{U* \cdot \left(n \cdot l\_m\right)}{Om}, t\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \frac{2 \cdot \left(\left(n \cdot l\_m\right) \cdot \left(l\_m \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
          t_1)))
   (if (<= t_2 0.0)
     (sqrt (* U (* (* 2.0 n) (fma (/ l_m Om) (/ (* U* (* n l_m)) Om) t))))
     (if (<= t_2 5e+299)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (sqrt
        (*
         U
         (/
          (* 2.0 (* (* n l_m) (* l_m (fma n (/ (- U* U) Om) -2.0))))
          Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((U * ((2.0 * n) * fma((l_m / Om), ((U_42_ * (n * l_m)) / Om), t))));
	} else if (t_2 <= 5e+299) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = sqrt((U * ((2.0 * ((n * l_m) * (l_m * fma(n, ((U_42_ - U) / Om), -2.0)))) / Om)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))) * t_1)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(Float64(l_m / Om), Float64(Float64(U_42_ * Float64(n * l_m)) / Om), t))));
	elseif (t_2 <= 5e+299)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * Float64(Float64(n * l_m) * Float64(l_m * fma(n, Float64(Float64(U_42_ - U) / Om), -2.0)))) / Om)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+299], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(U * N[(N[(2.0 * N[(N[(n * l$95$m), $MachinePrecision] * N[(l$95$m * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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*)))) < 0.0

   1. Initial program 17.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 25.3%

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

   6. Applied rewrites 47.1%

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

   7. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 42.5

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

   9. Applied rewrites 42.5%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000003e299

   1. Initial program 99.6%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if 5.0000000000000003e299 < (*.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 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 36.3%

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

   6. Applied rewrites 49.4%

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

   7. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower--.f64 49.4

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

   9. Applied rewrites 49.4%

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

   10. Taylor expanded in t around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

   12. Applied rewrites 48.7%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right) \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \frac{U* \cdot \left(n \cdot \ell\right)}{Om}, t\right)\right)}\\ \mathbf{elif}\;\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right) \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \frac{2 \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \frac{U* \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
          t_1)))
   (if (<= t_2 0.0)
     (sqrt (* (* n t) (* 2.0 U)))
     (if (<= t_2 5e+299)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (sqrt (* t_1 (/ (* U* (* n (* l_m l_m))) (* Om Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((n * t) * (2.0 * U)));
	} else if (t_2 <= 5e+299) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = sqrt((t_1 * ((U_42_ * (n * (l_m * l_m))) / (Om * Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))) * t_1)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * U)));
	elseif (t_2 <= 5e+299)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = sqrt(Float64(t_1 * Float64(Float64(U_42_ * Float64(n * Float64(l_m * l_m))) / Float64(Om * Om))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+299], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(N[(U$42$ * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 17.2%

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

   3. Taylor expanded in 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 35.0

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

   5. Applied rewrites 35.0%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000003e299

   1. Initial program 99.6%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if 5.0000000000000003e299 < (*.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 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 U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 34.8

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

   5. Applied rewrites 34.8%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right) \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \frac{\left(\left(l\_m \cdot l\_m\right) \cdot U*\right) \cdot \left(n \cdot n\right)}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
          t_1)))
   (if (<= t_2 0.0)
     (sqrt (* (* n t) (* 2.0 U)))
     (if (<= t_2 5e+299)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) 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 = U * (2.0 * n);
	double t_2 = ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((n * t) * (2.0 * U)));
	} else if (t_2 <= 5e+299) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), 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(U * Float64(2.0 * n))
	t_2 = Float64(Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))) * t_1)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * U)));
	elseif (t_2 <= 5e+299)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(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[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+299], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * U$42$), $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*)))) < 0.0

   1. Initial program 17.2%

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

   3. Taylor expanded in 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 35.0

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

   5. Applied rewrites 35.0%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000003e299

   1. Initial program 99.6%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if 5.0000000000000003e299 < (*.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 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   4. Applied rewrites 36.3%

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

   5. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.0

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

   7. Applied rewrites 34.0%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right) \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \frac{\left(\left(\ell \cdot \ell\right) \cdot U*\right) \cdot \left(n \cdot n\right)}{Om \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(l\_m \cdot \left(U \cdot \left(n \cdot l\_m\right)\right)\right) \cdot -4}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
          t_1)))
   (if (<= t_2 0.0)
     (sqrt (* (* n t) (* 2.0 U)))
     (if (<= t_2 5e+299)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (sqrt (/ (* (* l_m (* U (* n l_m))) -4.0) 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 = U * (2.0 * n);
	double t_2 = ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((n * t) * (2.0 * U)));
	} else if (t_2 <= 5e+299) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = sqrt((((l_m * (U * (n * l_m))) * -4.0) / Om));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))) * t_1)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * U)));
	elseif (t_2 <= 5e+299)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = sqrt(Float64(Float64(Float64(l_m * Float64(U * Float64(n * l_m))) * -4.0) / 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[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+299], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(l$95$m * N[(U * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 17.2%

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

   3. Taylor expanded in 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 35.0

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

   5. Applied rewrites 35.0%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000003e299

   1. Initial program 99.6%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if 5.0000000000000003e299 < (*.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 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 Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 24.6

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

   5. Applied rewrites 24.6%

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

   6. Taylor expanded in t around 0

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   7. 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. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 21.9

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

   8. Applied rewrites 21.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 29.2

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

   10. Applied rewrites 29.2%

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

       1. lift-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 30.9

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

   12. Applied rewrites 30.9%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right) \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right) \cdot -4}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -4\right) \cdot \frac{n \cdot \left(l\_m \cdot l\_m\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
          t_1)))
   (if (<= t_2 0.0)
     (sqrt (* (* n t) (* 2.0 U)))
     (if (<= t_2 5e+299)
       (sqrt (* t t_1))
       (sqrt (* (* U -4.0) (/ (* n (* l_m l_m)) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((n * t) * (2.0 * U)));
	} else if (t_2 <= 5e+299) {
		tmp = sqrt((t * t_1));
	} else {
		tmp = sqrt(((U * -4.0) * ((n * (l_m * l_m)) / Om)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = u * (2.0d0 * n)
    t_2 = ((t - (2.0d0 * ((l_m * l_m) / om))) + ((n * ((l_m / om) ** 2.0d0)) * (u_42 - u))) * t_1
    if (t_2 <= 0.0d0) then
        tmp = sqrt(((n * t) * (2.0d0 * u)))
    else if (t_2 <= 5d+299) then
        tmp = sqrt((t * t_1))
    else
        tmp = sqrt(((u * (-4.0d0)) * ((n * (l_m * l_m)) / om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt(((n * t) * (2.0 * U)));
	} else if (t_2 <= 5e+299) {
		tmp = Math.sqrt((t * t_1));
	} else {
		tmp = Math.sqrt(((U * -4.0) * ((n * (l_m * l_m)) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (2.0 * n)
	t_2 = ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_1
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt(((n * t) * (2.0 * U)))
	elif t_2 <= 5e+299:
		tmp = math.sqrt((t * t_1))
	else:
		tmp = math.sqrt(((U * -4.0) * ((n * (l_m * l_m)) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))) * t_1)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * U)));
	elseif (t_2 <= 5e+299)
		tmp = sqrt(Float64(t * t_1));
	else
		tmp = sqrt(Float64(Float64(U * -4.0) * Float64(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_)
	t_1 = U * (2.0 * n);
	t_2 = ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U))) * t_1;
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt(((n * t) * (2.0 * U)));
	elseif (t_2 <= 5e+299)
		tmp = sqrt((t * t_1));
	else
		tmp = sqrt(((U * -4.0) * ((n * (l_m * l_m)) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+299], N[Sqrt[N[(t * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * -4.0), $MachinePrecision] * N[(N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / 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*)))) < 0.0

   1. Initial program 17.2%

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

   3. Taylor expanded in 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 35.0

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

   5. Applied rewrites 35.0%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000003e299

   1. Initial program 99.6%

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

   3. Taylor expanded in 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 67.8

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

   5. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 75.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 75.1

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

   7. Applied rewrites 75.1%

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

   if 5.0000000000000003e299 < (*.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 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 Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 24.6

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

   5. Applied rewrites 24.6%

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

   6. Taylor expanded in t around 0

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   7. 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. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 21.9

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

   8. Applied rewrites 21.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 21.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 21.9

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

   10. Applied rewrites 21.9%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right) \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -4\right) \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = u * (2.0d0 * n)
    if ((((t - (2.0d0 * ((l_m * l_m) / om))) + ((n * ((l_m / om) ** 2.0d0)) * (u_42 - u))) * t_1) <= 0.0d0) then
        tmp = sqrt(((n * t) * (2.0d0 * u)))
    else
        tmp = sqrt((t * t_1))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double tmp;
	if ((((t - (2.0 * ((l_m * l_m) / Om))) + ((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_1) <= 0.0) {
		tmp = Math.sqrt(((n * t) * (2.0 * U)));
	} else {
		tmp = Math.sqrt((t * t_1));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (2.0 * n)
	tmp = 0
	if (((t - (2.0 * ((l_m * l_m) / Om))) + ((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U))) * t_1) <= 0.0:
		tmp = math.sqrt(((n * t) * (2.0 * U)))
	else:
		tmp = math.sqrt((t * t_1))
	return tmp

Julia

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

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U * (2.0 * n);
	tmp = 0.0;
	if ((((t - (2.0 * ((l_m * l_m) / Om))) + ((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U))) * t_1) <= 0.0)
		tmp = sqrt(((n * t) * (2.0 * U)));
	else
		tmp = sqrt((t * t_1));
	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[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(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] * t$95$1), $MachinePrecision], 0.0], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t * t$95$1), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 17.2%

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

   3. Taylor expanded in 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 35.0

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

   5. Applied rewrites 35.0%

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

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

   1. Initial program 64.0%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 38.2

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Applied rewrites 38.2%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot U\right)} \cdot n\right) \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(U \cdot 2\right)} \cdot n\right) \cdot t} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(2 \cdot n\right)}\right) \cdot t} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      9. lower-*.f64 41.7

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      12. lower-*.f64 41.7

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(2 \cdot n\right)}\right) \cdot t} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(n \cdot 2\right)}\right) \cdot t} \]

      15. lower-*.f64 41.7

          \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(n \cdot 2\right)}\right) \cdot t} \]

   7. Applied rewrites 41.7%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot 2\right)\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(2 \cdot n\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -4.1 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(l\_m \cdot \left(U \cdot \left(n \cdot l\_m\right)\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{elif}\;Om \leq 2.75 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \frac{l\_m \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, l\_m \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right), t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om -4.1e-82)
   (sqrt (fma 2.0 (* U (* n t)) (/ (* (* l_m (* U (* n l_m))) -4.0) Om)))
   (if (<= Om 2.75e-73)
     (sqrt
      (* U (* (* 2.0 n) (fma (/ l_m Om) (/ (* l_m (* n (- U* U))) Om) t))))
     (sqrt
      (*
       U
       (*
        (* 2.0 n)
        (fma (/ l_m Om) (* l_m (fma n (/ (- U* U) Om) -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 tmp;
	if (Om <= -4.1e-82) {
		tmp = sqrt(fma(2.0, (U * (n * t)), (((l_m * (U * (n * l_m))) * -4.0) / Om)));
	} else if (Om <= 2.75e-73) {
		tmp = sqrt((U * ((2.0 * n) * fma((l_m / Om), ((l_m * (n * (U_42_ - U))) / Om), t))));
	} else {
		tmp = sqrt((U * ((2.0 * n) * fma((l_m / Om), (l_m * fma(n, ((U_42_ - U) / Om), -2.0)), t))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Om <= -4.1e-82)
		tmp = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(Float64(l_m * Float64(U * Float64(n * l_m))) * -4.0) / Om)));
	elseif (Om <= 2.75e-73)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(Float64(l_m / Om), Float64(Float64(l_m * Float64(n * Float64(U_42_ - U))) / Om), t))));
	else
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(Float64(l_m / Om), Float64(l_m * fma(n, Float64(Float64(U_42_ - U) / Om), -2.0)), t))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[Om, -4.1e-82], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m * N[(U * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 2.75e-73], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(l$95$m * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < -4.09999999999999996e-82

   1. Initial program 57.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. lower-*.f64 48.6

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Applied rewrites 48.6%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right)} \cdot -4}{Om}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right)} \cdot U\right) \cdot -4}{Om}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U\right) \cdot -4}{Om}\right)} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot U\right) \cdot -4}{Om}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot U\right)\right) \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right)}\right) \cdot -4}{Om}\right)} \]

      11. lower-*.f64 56.9

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot U\right)\right) \cdot -4}{Om}\right)} \]

   7. Applied rewrites 56.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

   if -4.09999999999999996e-82 < Om < 2.75000000000000003e-73

   1. Initial program 58.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\color{blue}{\left(\frac{\ell}{Om}\right)}}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      10. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   6. Applied rewrites 73.1%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(-\left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in U around 0

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\color{blue}{U* + -1 \cdot U}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(U* + \color{blue}{\left(\mathsf{neg}\left(U\right)\right)}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\color{blue}{U* - U}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      3. lower--.f64 73.1

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\color{blue}{U* - U}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   9. Applied rewrites 73.1%

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\color{blue}{U* - U}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   10. Taylor expanded in n around inf

       \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       2. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\color{blue}{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}}{Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       3. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot \color{blue}{\left(n \cdot \left(U* - U\right)\right)}}{Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       4. lower--.f64 71.8

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot \left(n \cdot \color{blue}{\left(U* - U\right)}\right)}{Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   12. Applied rewrites 71.8%

       \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   if 2.75000000000000003e-73 < Om

   1. Initial program 54.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\color{blue}{\left(\frac{\ell}{Om}\right)}}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      10. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

   4. Applied rewrites 58.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   6. Applied rewrites 64.2%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(-\left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\ell \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\ell \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om} - 2\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{Om} + \left(\mathsf{neg}\left(2\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      3. associate-/l* N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot \left(\color{blue}{n \cdot \frac{U* - U}{Om}} + \left(\mathsf{neg}\left(2\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot \left(n \cdot \frac{U* - U}{Om} + \color{blue}{-2}\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      5. lower-fma.f64 N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot \color{blue}{\mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot \mathsf{fma}\left(n, \color{blue}{\frac{U* - U}{Om}}, -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      7. lower--.f64 62.1

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot \mathsf{fma}\left(n, \frac{\color{blue}{U* - U}}{Om}, -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   9. Applied rewrites 62.1%

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\ell \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4.1 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{elif}\;Om \leq 2.75 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot \mathsf{fma}\left(n, \frac{U* - U}{Om}, -2\right), t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(l\_m \cdot \left(U \cdot \left(n \cdot l\_m\right)\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{if}\;Om \leq -4.1 \cdot 10^{-82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 4.3 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \frac{l\_m \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (fma 2.0 (* U (* n t)) (/ (* (* l_m (* U (* n l_m))) -4.0) Om)))))
   (if (<= Om -4.1e-82)
     t_1
     (if (<= Om 4.3e+48)
       (sqrt
        (* U (* (* 2.0 n) (fma (/ l_m Om) (/ (* l_m (* n (- U* U))) Om) t))))
       t_1))))

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(fma(2.0, (U * (n * t)), (((l_m * (U * (n * l_m))) * -4.0) / Om)));
	double tmp;
	if (Om <= -4.1e-82) {
		tmp = t_1;
	} else if (Om <= 4.3e+48) {
		tmp = sqrt((U * ((2.0 * n) * fma((l_m / Om), ((l_m * (n * (U_42_ - U))) / Om), t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(Float64(l_m * Float64(U * Float64(n * l_m))) * -4.0) / Om)))
	tmp = 0.0
	if (Om <= -4.1e-82)
		tmp = t_1;
	elseif (Om <= 4.3e+48)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(Float64(l_m / Om), Float64(Float64(l_m * Float64(n * Float64(U_42_ - U))) / Om), t))));
	else
		tmp = t_1;
	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[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m * N[(U * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -4.1e-82], t$95$1, If[LessEqual[Om, 4.3e+48], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(l$95$m * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -4.09999999999999996e-82 or 4.29999999999999978e48 < Om

   1. Initial program 58.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. lower-*.f64 51.9

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Applied rewrites 51.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right)} \cdot -4}{Om}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right)} \cdot U\right) \cdot -4}{Om}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U\right) \cdot -4}{Om}\right)} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot U\right) \cdot -4}{Om}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot U\right)\right) \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right)}\right) \cdot -4}{Om}\right)} \]

      11. lower-*.f64 60.4

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot U\right)\right) \cdot -4}{Om}\right)} \]

   7. Applied rewrites 60.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

   if -4.09999999999999996e-82 < Om < 4.29999999999999978e48

   1. Initial program 53.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\color{blue}{\left(\frac{\ell}{Om}\right)}}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      10. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

   4. Applied rewrites 55.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   6. Applied rewrites 70.2%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(-\left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in U around 0

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\color{blue}{U* + -1 \cdot U}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(U* + \color{blue}{\left(\mathsf{neg}\left(U\right)\right)}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\color{blue}{U* - U}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      3. lower--.f64 70.2

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\color{blue}{U* - U}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   9. Applied rewrites 70.2%

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\color{blue}{U* - U}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   10. Taylor expanded in n around inf

       \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       2. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\color{blue}{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}}{Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       3. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot \color{blue}{\left(n \cdot \left(U* - U\right)\right)}}{Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       4. lower--.f64 66.2

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot \left(n \cdot \color{blue}{\left(U* - U\right)}\right)}{Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   12. Applied rewrites 66.2%

       \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4.1 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{elif}\;Om \leq 4.3 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right) \cdot -4}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(l\_m \cdot \left(U \cdot \left(n \cdot l\_m\right)\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{if}\;Om \leq -4.1 \cdot 10^{-82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 4.3 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, l\_m \cdot \frac{n \cdot \left(U* - U\right)}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (fma 2.0 (* U (* n t)) (/ (* (* l_m (* U (* n l_m))) -4.0) Om)))))
   (if (<= Om -4.1e-82)
     t_1
     (if (<= Om 4.3e+48)
       (sqrt
        (* U (* (* 2.0 n) (fma (/ l_m Om) (* l_m (/ (* n (- U* U)) Om)) t))))
       t_1))))

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(fma(2.0, (U * (n * t)), (((l_m * (U * (n * l_m))) * -4.0) / Om)));
	double tmp;
	if (Om <= -4.1e-82) {
		tmp = t_1;
	} else if (Om <= 4.3e+48) {
		tmp = sqrt((U * ((2.0 * n) * fma((l_m / Om), (l_m * ((n * (U_42_ - U)) / Om)), t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(Float64(l_m * Float64(U * Float64(n * l_m))) * -4.0) / Om)))
	tmp = 0.0
	if (Om <= -4.1e-82)
		tmp = t_1;
	elseif (Om <= 4.3e+48)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(Float64(l_m / Om), Float64(l_m * Float64(Float64(n * Float64(U_42_ - U)) / Om)), t))));
	else
		tmp = t_1;
	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[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m * N[(U * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -4.1e-82], t$95$1, If[LessEqual[Om, 4.3e+48], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -4.09999999999999996e-82 or 4.29999999999999978e48 < Om

   1. Initial program 58.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. lower-*.f64 51.9

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Applied rewrites 51.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right)} \cdot -4}{Om}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right)} \cdot U\right) \cdot -4}{Om}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U\right) \cdot -4}{Om}\right)} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot U\right) \cdot -4}{Om}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot U\right)\right) \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right)}\right) \cdot -4}{Om}\right)} \]

      11. lower-*.f64 60.4

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot U\right)\right) \cdot -4}{Om}\right)} \]

   7. Applied rewrites 60.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

   if -4.09999999999999996e-82 < Om < 4.29999999999999978e48

   1. Initial program 53.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\color{blue}{\left(\frac{\ell}{Om}\right)}}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      10. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

   4. Applied rewrites 55.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   6. Applied rewrites 70.2%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(-\left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in n around inf

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot \left(n \cdot \left(U* - U\right)\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\ell \cdot \frac{n \cdot \left(U* - U\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\ell \cdot \frac{n \cdot \left(U* - U\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot \color{blue}{\frac{n \cdot \left(U* - U\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot \frac{\color{blue}{n \cdot \left(U* - U\right)}}{Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      5. lower--.f64 63.3

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot \frac{n \cdot \color{blue}{\left(U* - U\right)}}{Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   9. Applied rewrites 63.3%

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\ell \cdot \frac{n \cdot \left(U* - U\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4.1 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{elif}\;Om \leq 4.3 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot \frac{n \cdot \left(U* - U\right)}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right) \cdot -4}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 60.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 3.2 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om}, \mathsf{fma}\left(U* - U, n \cdot \frac{l\_m}{Om}, l\_m \cdot -2\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(l\_m, \frac{\mathsf{fma}\left(U*, \frac{n \cdot l\_m}{Om}, l\_m \cdot -2\right)}{Om}, t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 3.2e+54)
   (sqrt
    (*
     U
     (*
      (fma (/ l_m Om) (fma (- U* U) (* n (/ l_m Om)) (* l_m -2.0)) t)
      (* 2.0 n))))
   (sqrt
    (*
     2.0
     (* (* n U) (fma l_m (/ (fma U* (/ (* n l_m) Om) (* l_m -2.0)) Om) t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 3.2e+54) {
		tmp = sqrt((U * (fma((l_m / Om), fma((U_42_ - U), (n * (l_m / Om)), (l_m * -2.0)), t) * (2.0 * n))));
	} else {
		tmp = sqrt((2.0 * ((n * U) * fma(l_m, (fma(U_42_, ((n * l_m) / Om), (l_m * -2.0)) / Om), t))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 3.2e+54)
		tmp = sqrt(Float64(U * Float64(fma(Float64(l_m / Om), fma(Float64(U_42_ - U), Float64(n * Float64(l_m / Om)), Float64(l_m * -2.0)), t) * Float64(2.0 * n))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * fma(l_m, Float64(fma(U_42_, Float64(Float64(n * l_m) / Om), Float64(l_m * -2.0)) / Om), t))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 3.2e+54], N[Sqrt[N[(U * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(l$95$m * N[(N[(U$42$ * N[(N[(n * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + N[(l$95$m * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 3.2e54

   1. Initial program 54.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\color{blue}{\left(\frac{\ell}{Om}\right)}}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      10. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

   4. Applied rewrites 57.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   6. Applied rewrites 64.4%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(-\left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in U around 0

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\color{blue}{U* + -1 \cdot U}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(U* + \color{blue}{\left(\mathsf{neg}\left(U\right)\right)}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\color{blue}{U* - U}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      3. lower--.f64 64.4

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\color{blue}{U* - U}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   9. Applied rewrites 64.4%

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\color{blue}{U* - U}, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   if 3.2e54 < n

   1. Initial program 64.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\color{blue}{\left(\frac{\ell}{Om}\right)}}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      10. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

   4. Applied rewrites 67.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   6. Applied rewrites 66.1%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(-\left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in U around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{Om}\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{Om}\right)\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{Om}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{Om}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(U \cdot n\right)} \cdot \left(t + \frac{\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{Om}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(\frac{\ell \cdot \left(-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}{Om} + t\right)}\right)} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\color{blue}{\ell \cdot \frac{-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}}{Om}} + t\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\ell, \frac{-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}}{Om}, t\right)}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\ell, \color{blue}{\frac{-2 \cdot \ell + \frac{U* \cdot \left(\ell \cdot n\right)}{Om}}{Om}}, t\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\ell, \frac{\color{blue}{\frac{U* \cdot \left(\ell \cdot n\right)}{Om} + -2 \cdot \ell}}{Om}, t\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\ell, \frac{\color{blue}{U* \cdot \frac{\ell \cdot n}{Om}} + -2 \cdot \ell}{Om}, t\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\ell, \frac{\color{blue}{\mathsf{fma}\left(U*, \frac{\ell \cdot n}{Om}, -2 \cdot \ell\right)}}{Om}, t\right)\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\ell, \frac{\mathsf{fma}\left(U*, \color{blue}{\frac{\ell \cdot n}{Om}}, -2 \cdot \ell\right)}{Om}, t\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\ell, \frac{\mathsf{fma}\left(U*, \frac{\color{blue}{\ell \cdot n}}{Om}, -2 \cdot \ell\right)}{Om}, t\right)\right)} \]

      14. lower-*.f64 74.9

          \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\ell, \frac{\mathsf{fma}\left(U*, \frac{\ell \cdot n}{Om}, \color{blue}{-2 \cdot \ell}\right)}{Om}, t\right)\right)} \]

   9. Applied rewrites 74.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(\ell, \frac{\mathsf{fma}\left(U*, \frac{\ell \cdot n}{Om}, -2 \cdot \ell\right)}{Om}, t\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.2 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(U* - U, n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \mathsf{fma}\left(\ell, \frac{\mathsf{fma}\left(U*, \frac{n \cdot \ell}{Om}, \ell \cdot -2\right)}{Om}, t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 53.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(l\_m \cdot \left(U \cdot \left(n \cdot l\_m\right)\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{if}\;Om \leq -4.1 \cdot 10^{-82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 8.8 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \frac{U* \cdot \left(n \cdot l\_m\right)}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (fma 2.0 (* U (* n t)) (/ (* (* l_m (* U (* n l_m))) -4.0) Om)))))
   (if (<= Om -4.1e-82)
     t_1
     (if (<= Om 8.8e+48)
       (sqrt (* U (* (* 2.0 n) (fma (/ l_m Om) (/ (* U* (* n l_m)) Om) t))))
       t_1))))

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(fma(2.0, (U * (n * t)), (((l_m * (U * (n * l_m))) * -4.0) / Om)));
	double tmp;
	if (Om <= -4.1e-82) {
		tmp = t_1;
	} else if (Om <= 8.8e+48) {
		tmp = sqrt((U * ((2.0 * n) * fma((l_m / Om), ((U_42_ * (n * l_m)) / Om), t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(Float64(l_m * Float64(U * Float64(n * l_m))) * -4.0) / Om)))
	tmp = 0.0
	if (Om <= -4.1e-82)
		tmp = t_1;
	elseif (Om <= 8.8e+48)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(Float64(l_m / Om), Float64(Float64(U_42_ * Float64(n * l_m)) / Om), t))));
	else
		tmp = t_1;
	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[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m * N[(U * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -4.1e-82], t$95$1, If[LessEqual[Om, 8.8e+48], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U$42$ * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -4.09999999999999996e-82 or 8.7999999999999997e48 < Om

   1. Initial program 58.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. lower-*.f64 51.9

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Applied rewrites 51.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right)} \cdot -4}{Om}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right)} \cdot U\right) \cdot -4}{Om}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U\right) \cdot -4}{Om}\right)} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot U\right) \cdot -4}{Om}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot U\right)\right) \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right)}\right) \cdot -4}{Om}\right)} \]

      11. lower-*.f64 60.4

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot U\right)\right) \cdot -4}{Om}\right)} \]

   7. Applied rewrites 60.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

   if -4.09999999999999996e-82 < Om < 8.7999999999999997e48

   1. Initial program 53.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\color{blue}{\left(\frac{\ell}{Om}\right)}}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      10. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

   4. Applied rewrites 55.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   6. Applied rewrites 70.2%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(-\left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{U* \cdot \left(\ell \cdot n\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{U* \cdot \left(\ell \cdot n\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\color{blue}{U* \cdot \left(\ell \cdot n\right)}}{Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

      3. lower-*.f64 63.3

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{U* \cdot \color{blue}{\left(\ell \cdot n\right)}}{Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   9. Applied rewrites 63.3%

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{U* \cdot \left(\ell \cdot n\right)}{Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -4.1 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{elif}\;Om \leq 8.8 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \frac{U* \cdot \left(n \cdot \ell\right)}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right) \cdot -4}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 49.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(l\_m \cdot \left(U \cdot \left(n \cdot l\_m\right)\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{if}\;Om \leq -1.6 \cdot 10^{-277}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 3.1 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{\left(U \cdot -2\right) \cdot \left(n \cdot \left(l\_m \cdot \left(l\_m \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}}{Om}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (fma 2.0 (* U (* n t)) (/ (* (* l_m (* U (* n l_m))) -4.0) Om)))))
   (if (<= Om -1.6e-277)
     t_1
     (if (<= Om 3.1e-189)
       (/ (sqrt (* (* U -2.0) (* n (* l_m (* l_m (* n (- U U*))))))) Om)
       t_1))))

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(fma(2.0, (U * (n * t)), (((l_m * (U * (n * l_m))) * -4.0) / Om)));
	double tmp;
	if (Om <= -1.6e-277) {
		tmp = t_1;
	} else if (Om <= 3.1e-189) {
		tmp = sqrt(((U * -2.0) * (n * (l_m * (l_m * (n * (U - U_42_))))))) / Om;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(Float64(l_m * Float64(U * Float64(n * l_m))) * -4.0) / Om)))
	tmp = 0.0
	if (Om <= -1.6e-277)
		tmp = t_1;
	elseif (Om <= 3.1e-189)
		tmp = Float64(sqrt(Float64(Float64(U * -2.0) * Float64(n * Float64(l_m * Float64(l_m * Float64(n * Float64(U - U_42_))))))) / Om);
	else
		tmp = t_1;
	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[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m * N[(U * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -1.6e-277], t$95$1, If[LessEqual[Om, 3.1e-189], N[(N[Sqrt[N[(N[(U * -2.0), $MachinePrecision] * N[(n * N[(l$95$m * N[(l$95$m * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Om), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.5999999999999999e-277 or 3.1e-189 < Om

   1. Initial program 57.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. lower-*.f64 50.2

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Applied rewrites 50.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right)} \cdot -4}{Om}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right)} \cdot U\right) \cdot -4}{Om}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U\right) \cdot -4}{Om}\right)} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot U\right) \cdot -4}{Om}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot U\right)\right) \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right)}\right) \cdot -4}{Om}\right)} \]

      11. lower-*.f64 55.5

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot U\right)\right) \cdot -4}{Om}\right)} \]

   7. Applied rewrites 55.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right)} \cdot -4}{Om}\right)} \]

   if -1.5999999999999999e-277 < Om < 3.1e-189

   1. Initial program 52.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 28.6%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om \cdot Om} \cdot \left(U - U*\right), -n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right) \cdot \left(U \cdot 2\right)\right) \cdot n}} \]

   5. Taylor expanded in Om around 0

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)} \cdot n} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}{{Om}^{2}}} \cdot n} \]

      2. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}{{Om}^{2}}} \cdot n} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}}{{Om}^{2}} \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}}{{Om}^{2}} \cdot n} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-2 \cdot U\right)} \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}} \cdot n} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(-2 \cdot U\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}}{{Om}^{2}} \cdot n} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{\left(-2 \cdot U\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}} \cdot n} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(-2 \cdot U\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}} \cdot n} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(n \cdot \left(U - U*\right)\right)}\right)}{{Om}^{2}} \cdot n} \]

      10. lower--.f64 N/A

          \[\leadsto \sqrt{\frac{\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \color{blue}{\left(U - U*\right)}\right)\right)}{{Om}^{2}} \cdot n} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{\color{blue}{Om \cdot Om}} \cdot n} \]

      12. lower-*.f64 44.9

          \[\leadsto \sqrt{\frac{\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{\color{blue}{Om \cdot Om}} \cdot n} \]

   7. Applied rewrites 44.9%

      \[\leadsto \sqrt{\color{blue}{\frac{\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om \cdot Om}} \cdot n} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-2 \cdot U\right)} \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om \cdot Om} \cdot n} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(-2 \cdot U\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om \cdot Om} \cdot n} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \color{blue}{\left(U - U*\right)}\right)\right)}{Om \cdot Om} \cdot n} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(n \cdot \left(U - U*\right)\right)}\right)}{Om \cdot Om} \cdot n} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}}{Om \cdot Om} \cdot n} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}}{Om \cdot Om} \cdot n} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{\color{blue}{Om \cdot Om}} \cdot n} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right) \cdot n}{Om \cdot Om}}} \]

      9. sqrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right) \cdot n}}{\sqrt{Om \cdot Om}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right) \cdot n}}{\sqrt{\color{blue}{Om \cdot Om}}} \]

      11. pow2 N/A

          \[\leadsto \frac{\sqrt{\left(\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right) \cdot n}}{\sqrt{\color{blue}{{Om}^{2}}}} \]

      12. sqrt-pow1 N/A

          \[\leadsto \frac{\sqrt{\left(\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right) \cdot n}}{\color{blue}{{Om}^{\left(\frac{2}{2}\right)}}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sqrt{\left(\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right) \cdot n}}{{Om}^{\color{blue}{1}}} \]

      14. unpow1 N/A

          \[\leadsto \frac{\sqrt{\left(\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right) \cdot n}}{\color{blue}{Om}} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right) \cdot n}}{Om}} \]

   9. Applied rewrites 61.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(U \cdot -2\right) \cdot \left(\left(\ell \cdot \left(\ell \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right) \cdot n\right)}}{Om}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.6 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{elif}\;Om \leq 3.1 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{\left(U \cdot -2\right) \cdot \left(n \cdot \left(\ell \cdot \left(\ell \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}}{Om}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right) \cdot -4}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 50.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.2 \cdot 10^{+229}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, l\_m \cdot -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{l\_m} \cdot \sqrt{\frac{n \cdot \left(-4 \cdot \left(U \cdot l\_m\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.2e+229)
   (sqrt (* U (* (* 2.0 n) (fma (/ l_m Om) (* l_m -2.0) t))))
   (* (sqrt l_m) (sqrt (/ (* n (* -4.0 (* U 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 <= 2.2e+229) {
		tmp = sqrt((U * ((2.0 * n) * fma((l_m / Om), (l_m * -2.0), t))));
	} else {
		tmp = sqrt(l_m) * sqrt(((n * (-4.0 * (U * 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 <= 2.2e+229)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(Float64(l_m / Om), Float64(l_m * -2.0), t))));
	else
		tmp = Float64(sqrt(l_m) * sqrt(Float64(Float64(n * Float64(-4.0 * Float64(U * l_m))) / Om)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.2e+229], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[l$95$m], $MachinePrecision] * N[Sqrt[N[(N[(n * N[(-4.0 * N[(U * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.20000000000000004e229

   1. Initial program 59.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\color{blue}{\left(\frac{\ell}{Om}\right)}}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      10. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

   4. Applied rewrites 62.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   6. Applied rewrites 66.9%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(-\left(U - U*\right), n \cdot \frac{\ell}{Om}, \ell \cdot -2\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   7. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{-2 \cdot \ell}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
   8. Step-by-step derivation

      1. lower-*.f64 51.9

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{-2 \cdot \ell}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   9. Applied rewrites 51.9%

      \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{-2 \cdot \ell}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   if 2.20000000000000004e229 < l

   1. Initial program 14.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. lower-*.f64 16.6

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Applied rewrites 16.6%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   7. 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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}}{Om}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right)}{Om}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right)}{Om}} \]

      7. lower-*.f64 16.6

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right)}{Om}} \]

   8. Applied rewrites 16.6%

      \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}}} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)}\right)}{Om}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\left(n \cdot \ell\right)}\right)\right)}{Om}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\left(n \cdot \ell\right)}\right)\right)}{Om}} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)}}{Om}} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(\color{blue}{\left(\ell \cdot U\right)} \cdot \left(n \cdot \ell\right)\right)}{Om}} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)}}{Om}} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(\ell \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right)}\right)}{Om}} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(\ell \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right)}\right)}{Om}} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right)}}{Om}} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{\color{blue}{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right) \cdot -4}}{Om}} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right)} \cdot -4}{Om}} \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{\frac{\color{blue}{\ell \cdot \left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot -4\right)}}{Om}} \]

      13. associate-/l* N/A

          \[\leadsto \sqrt{\color{blue}{\ell \cdot \frac{\left(\left(n \cdot \ell\right) \cdot U\right) \cdot -4}{Om}}} \]

      14. sqrt-prod N/A

          \[\leadsto \color{blue}{\sqrt{\ell} \cdot \sqrt{\frac{\left(\left(n \cdot \ell\right) \cdot U\right) \cdot -4}{Om}}} \]

      15. pow1/2 N/A

          \[\leadsto \color{blue}{{\ell}^{\frac{1}{2}}} \cdot \sqrt{\frac{\left(\left(n \cdot \ell\right) \cdot U\right) \cdot -4}{Om}} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{{\ell}^{\frac{1}{2}} \cdot \sqrt{\frac{\left(\left(n \cdot \ell\right) \cdot U\right) \cdot -4}{Om}}} \]

   10. Applied rewrites 41.1%

       \[\leadsto \color{blue}{\sqrt{\ell} \cdot \sqrt{\frac{n \cdot \left(\left(U \cdot \ell\right) \cdot -4\right)}{Om}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+229}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \ell \cdot -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\ell} \cdot \sqrt{\frac{n \cdot \left(-4 \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 49.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5.5 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(l\_m \cdot \left(U \cdot \left(n \cdot l\_m\right)\right)\right) \cdot -4}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5.5e+147)
   (sqrt (* 2.0 (* U (* n (fma -2.0 (/ (* l_m l_m) Om) t)))))
   (sqrt (/ (* (* l_m (* U (* n l_m))) -4.0) Om))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 5.5e+147) {
		tmp = sqrt((2.0 * (U * (n * fma(-2.0, ((l_m * l_m) / Om), t)))));
	} else {
		tmp = sqrt((((l_m * (U * (n * l_m))) * -4.0) / Om));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 5.5e+147)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(-2.0, Float64(Float64(l_m * l_m) / Om), t)))));
	else
		tmp = sqrt(Float64(Float64(Float64(l_m * Float64(U * Float64(n * l_m))) * -4.0) / Om));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 5.5e+147], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(l$95$m * N[(U * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.4999999999999997e147

   1. Initial program 61.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\color{blue}{\left(\frac{\ell}{Om}\right)}}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U - U*\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      10. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

   4. Applied rewrites 63.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}\right)\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right)\right)\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)\right)} \]

      12. lower-*.f64 51.0

          \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)\right)} \]

   7. Applied rewrites 51.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)\right)}} \]

   if 5.4999999999999997e147 < l

   1. Initial program 16.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. lower-*.f64 14.0

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Applied rewrites 14.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   7. 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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}}{Om}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right)}{Om}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right)}{Om}} \]

      7. lower-*.f64 14.0

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right)}{Om}} \]

   8. Applied rewrites 14.0%

      \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right)}{Om}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(\ell \cdot \ell\right)\right)}\right)}{Om}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)}{Om}} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \ell\right)}\right)}{Om}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot \ell\right)\right)}{Om}} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(U \cdot \left(n \cdot \ell\right)\right) \cdot \ell\right)}}{Om}} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(\color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right)} \cdot \ell\right)}{Om}} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right)}}{Om}} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(\left(n \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot U\right)}\right)}{Om}} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}}{Om}} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \left(\left(n \cdot \ell\right) \cdot \color{blue}{\left(U \cdot \ell\right)}\right)}{Om}} \]

      12. lower-*.f64 38.5

          \[\leadsto \sqrt{\frac{-4 \cdot \left(\left(n \cdot \ell\right) \cdot \color{blue}{\left(U \cdot \ell\right)}\right)}{Om}} \]

   10. Applied rewrites 38.5%

       \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right)}}{Om}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot \left(U \cdot \ell\right)\right)}{Om}} \]

       2. associate-*r* N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \ell\right)}}{Om}} \]

       3. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \ell\right)}}{Om}} \]

       4. *-commutative N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \left(\color{blue}{\left(U \cdot \left(n \cdot \ell\right)\right)} \cdot \ell\right)}{Om}} \]

       5. lower-*.f64 38.5

          \[\leadsto \sqrt{\frac{-4 \cdot \left(\color{blue}{\left(U \cdot \left(n \cdot \ell\right)\right)} \cdot \ell\right)}{Om}} \]

   12. Applied rewrites 38.5%

       \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(U \cdot \left(n \cdot \ell\right)\right) \cdot \ell\right)}}{Om}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right) \cdot -4}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 45.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.65 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(l\_m \cdot \left(U \cdot \left(n \cdot l\_m\right)\right)\right) \cdot -4}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.65e+70)
   (sqrt (* t (* U (* 2.0 n))))
   (sqrt (/ (* (* l_m (* U (* n l_m))) -4.0) Om))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.65e+70) {
		tmp = sqrt((t * (U * (2.0 * n))));
	} else {
		tmp = sqrt((((l_m * (U * (n * l_m))) * -4.0) / 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 <= 2.65d+70) then
        tmp = sqrt((t * (u * (2.0d0 * n))))
    else
        tmp = sqrt((((l_m * (u * (n * l_m))) * (-4.0d0)) / 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 <= 2.65e+70) {
		tmp = Math.sqrt((t * (U * (2.0 * n))));
	} else {
		tmp = Math.sqrt((((l_m * (U * (n * l_m))) * -4.0) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.65e+70:
		tmp = math.sqrt((t * (U * (2.0 * n))))
	else:
		tmp = math.sqrt((((l_m * (U * (n * l_m))) * -4.0) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.65e+70)
		tmp = sqrt(Float64(t * Float64(U * Float64(2.0 * n))));
	else
		tmp = sqrt(Float64(Float64(Float64(l_m * Float64(U * Float64(n * l_m))) * -4.0) / 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 <= 2.65e+70)
		tmp = sqrt((t * (U * (2.0 * n))));
	else
		tmp = sqrt((((l_m * (U * (n * l_m))) * -4.0) / 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, 2.65e+70], N[Sqrt[N[(t * N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(l$95$m * N[(U * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.65e70

   1. Initial program 63.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 43.2

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Applied rewrites 43.2%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot U\right)} \cdot n\right) \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(U \cdot 2\right)} \cdot n\right) \cdot t} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(2 \cdot n\right)}\right) \cdot t} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      9. lower-*.f64 43.5

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      12. lower-*.f64 43.5

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(2 \cdot n\right)}\right) \cdot t} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(n \cdot 2\right)}\right) \cdot t} \]

      15. lower-*.f64 43.5

          \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(n \cdot 2\right)}\right) \cdot t} \]

   7. Applied rewrites 43.5%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot 2\right)\right) \cdot t}} \]

   if 2.65e70 < l

   1. Initial program 29.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. lower-*.f64 34.8

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Applied rewrites 34.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   7. 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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}}{Om}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right)}{Om}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right)}{Om}} \]

      7. lower-*.f64 26.9

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right)}{Om}} \]

   8. Applied rewrites 26.9%

      \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right)}{Om}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \color{blue}{\left(n \cdot \left(\ell \cdot \ell\right)\right)}\right)}{Om}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)}{Om}} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \ell\right)}\right)}{Om}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot \ell\right)\right)}{Om}} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(U \cdot \left(n \cdot \ell\right)\right) \cdot \ell\right)}}{Om}} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(\color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right)} \cdot \ell\right)}{Om}} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right)}}{Om}} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(\left(n \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot U\right)}\right)}{Om}} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot U\right)\right)}}{Om}} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \left(\left(n \cdot \ell\right) \cdot \color{blue}{\left(U \cdot \ell\right)}\right)}{Om}} \]

      12. lower-*.f64 39.1

          \[\leadsto \sqrt{\frac{-4 \cdot \left(\left(n \cdot \ell\right) \cdot \color{blue}{\left(U \cdot \ell\right)}\right)}{Om}} \]

   10. Applied rewrites 39.1%

       \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right)}}{Om}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \left(\color{blue}{\left(n \cdot \ell\right)} \cdot \left(U \cdot \ell\right)\right)}{Om}} \]

       2. associate-*r* N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \ell\right)}}{Om}} \]

       3. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \ell\right)}}{Om}} \]

       4. *-commutative N/A

          \[\leadsto \sqrt{\frac{-4 \cdot \left(\color{blue}{\left(U \cdot \left(n \cdot \ell\right)\right)} \cdot \ell\right)}{Om}} \]

       5. lower-*.f64 39.1

          \[\leadsto \sqrt{\frac{-4 \cdot \left(\color{blue}{\left(U \cdot \left(n \cdot \ell\right)\right)} \cdot \ell\right)}{Om}} \]

   12. Applied rewrites 39.1%

       \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(U \cdot \left(n \cdot \ell\right)\right) \cdot \ell\right)}}{Om}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.65 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right) \cdot -4}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 46.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 9 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot \left(U \cdot l\_m\right)\right) \cdot \left(l\_m \cdot \frac{-4}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 9e+70)
   (sqrt (* t (* U (* 2.0 n))))
   (sqrt (* (* n (* U l_m)) (* l_m (/ -4.0 Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 9e+70) {
		tmp = sqrt((t * (U * (2.0 * n))));
	} else {
		tmp = sqrt(((n * (U * l_m)) * (l_m * (-4.0 / 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 <= 9d+70) then
        tmp = sqrt((t * (u * (2.0d0 * n))))
    else
        tmp = sqrt(((n * (u * l_m)) * (l_m * ((-4.0d0) / 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 <= 9e+70) {
		tmp = Math.sqrt((t * (U * (2.0 * n))));
	} else {
		tmp = Math.sqrt(((n * (U * l_m)) * (l_m * (-4.0 / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 9e+70:
		tmp = math.sqrt((t * (U * (2.0 * n))))
	else:
		tmp = math.sqrt(((n * (U * l_m)) * (l_m * (-4.0 / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 9e+70)
		tmp = sqrt(Float64(t * Float64(U * Float64(2.0 * n))));
	else
		tmp = sqrt(Float64(Float64(n * Float64(U * l_m)) * Float64(l_m * Float64(-4.0 / 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 <= 9e+70)
		tmp = sqrt((t * (U * (2.0 * n))));
	else
		tmp = sqrt(((n * (U * l_m)) * (l_m * (-4.0 / 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, 9e+70], N[Sqrt[N[(t * N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * N[(U * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * N[(-4.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.9999999999999999e70

   1. Initial program 63.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. lower-*.f64 43.2

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Applied rewrites 43.2%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot U\right)} \cdot n\right) \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(U \cdot 2\right)} \cdot n\right) \cdot t} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(2 \cdot n\right)}\right) \cdot t} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      9. lower-*.f64 43.5

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      12. lower-*.f64 43.5

          \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(2 \cdot n\right)}\right) \cdot t} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(n \cdot 2\right)}\right) \cdot t} \]

      15. lower-*.f64 43.5

          \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(n \cdot 2\right)}\right) \cdot t} \]

   7. Applied rewrites 43.5%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot 2\right)\right) \cdot t}} \]

   if 8.9999999999999999e70 < l

   1. Initial program 29.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. lower-*.f64 34.8

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Applied rewrites 34.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   7. 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. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}}{Om}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right)}{Om}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right)}{Om}} \]

      7. lower-*.f64 26.9

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right)}{Om}} \]

   8. Applied rewrites 26.9%

      \[\leadsto \sqrt{\color{blue}{\frac{-4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}}} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)}\right)}{Om}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\left(n \cdot \ell\right)}\right)\right)}{Om}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(U \cdot \left(\ell \cdot \color{blue}{\left(n \cdot \ell\right)}\right)\right)}{Om}} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(n \cdot \ell\right)\right)}}{Om}} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(\color{blue}{\left(\ell \cdot U\right)} \cdot \left(n \cdot \ell\right)\right)}{Om}} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)\right)}}{Om}} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(\ell \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right)}\right)}{Om}} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \left(\ell \cdot \color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right)}\right)}{Om}} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{-4 \cdot \color{blue}{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right)}}{Om}} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{\color{blue}{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right) \cdot -4}}{Om}} \]

      11. associate-/l* N/A

          \[\leadsto \sqrt{\color{blue}{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right) \cdot \frac{-4}{Om}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right)} \cdot \frac{-4}{Om}} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \ell\right)} \cdot \frac{-4}{Om}} \]

      14. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(\ell \cdot \frac{-4}{Om}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(\ell \cdot \frac{-4}{Om}\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot \ell\right) \cdot U\right)} \cdot \left(\ell \cdot \frac{-4}{Om}\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot \ell\right)} \cdot U\right) \cdot \left(\ell \cdot \frac{-4}{Om}\right)} \]

      18. associate-*l* N/A

          \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(\ell \cdot U\right)\right)} \cdot \left(\ell \cdot \frac{-4}{Om}\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(\ell \cdot U\right)\right)} \cdot \left(\ell \cdot \frac{-4}{Om}\right)} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(n \cdot \color{blue}{\left(U \cdot \ell\right)}\right) \cdot \left(\ell \cdot \frac{-4}{Om}\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot \color{blue}{\left(U \cdot \ell\right)}\right) \cdot \left(\ell \cdot \frac{-4}{Om}\right)} \]

      22. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \color{blue}{\left(\ell \cdot \frac{-4}{Om}\right)}} \]

      23. lower-/.f64 39.9

          \[\leadsto \sqrt{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\ell \cdot \color{blue}{\frac{-4}{Om}}\right)} \]

   10. Applied rewrites 39.9%

       \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\ell \cdot \frac{-4}{Om}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot \left(U \cdot \ell\right)\right) \cdot \left(\ell \cdot \frac{-4}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 36.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* n t) (* 2.0 U))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt(((n * t) * (2.0 * U)));
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt(((n * t) * (2.0d0 * u)))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt(((n * t) * (2.0 * U)));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt(((n * t) * (2.0 * U)))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(n * t) * Float64(2.0 * U)))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt(((n * t) * (2.0 * U)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 56.9%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 37.7

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

5. Applied rewrites 37.7%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

6. Final simplification 37.7%

   \[\leadsto \sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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*))))))