Toniolo and Linder, Equation (13)

Percentage Accurate: 48.8% → 64.3%

Time: 20.2s

Alternatives: 23

Speedup: 2.5×

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: 48.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(l\_m, \frac{l\_m}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(\frac{l\_m}{Om} \cdot \left(U* - U\right), n \cdot \frac{l\_m}{Om}, \mathsf{fma}\left(l\_m \cdot -2, \frac{l\_m}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 0.0)
     (* (sqrt (* U (fma l_m (/ l_m (* Om -0.5)) t))) (sqrt (* 2.0 n)))
     (if (<= t_2 1e+307)
       (sqrt
        (*
         t_1
         (fma
          (* (/ l_m Om) (- U* U))
          (* n (/ l_m Om))
          (fma (* l_m -2.0) (/ l_m Om) t))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* (* n U) (fma U* (/ n (* Om 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_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((U * fma(l_m, (l_m / (Om * -0.5)), t))) * sqrt((2.0 * n));
	} else if (t_2 <= 1e+307) {
		tmp = sqrt((t_1 * fma(((l_m / Om) * (U_42_ - U)), (n * (l_m / Om)), fma((l_m * -2.0), (l_m / Om), t))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * fma(U_42_, (n / (Om * 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(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(U * fma(l_m, Float64(l_m / Float64(Om * -0.5)), t))) * sqrt(Float64(2.0 * n)));
	elseif (t_2 <= 1e+307)
		tmp = sqrt(Float64(t_1 * fma(Float64(Float64(l_m / Om) * Float64(U_42_ - U)), Float64(n * Float64(l_m / Om)), fma(Float64(l_m * -2.0), Float64(l_m / Om), t))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * fma(U_42_, Float64(n / Float64(Om * Om)), Float64(-2.0 / Om)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(U * N[(l$95$m * N[(l$95$m / N[(Om * -0.5), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[Sqrt[N[(t$95$1 * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + N[(N[(l$95$m * -2.0), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(U$42$ * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / 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 12.2%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 18.1

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

   5. Simplified 18.1%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot U} \cdot \sqrt{n \cdot 2}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 98.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   4. Applied egg-rr 99.2%

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

   if 9.99999999999999986e306 < (*.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 20.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   4. Applied egg-rr 26.5%

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

   5. Taylor expanded in U around 0

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

   7. Simplified 26.7%

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

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. associate-/l* N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. /-lowering-/.f64 20.0

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

   10. Simplified 20.0%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om} \cdot \left(U* - U\right), n \cdot \frac{\ell}{Om}, \mathsf{fma}\left(\ell \cdot -2, \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(l\_m, \frac{l\_m}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(\frac{l\_m}{Om} \cdot U*, n \cdot \frac{l\_m}{Om}, \mathsf{fma}\left(l\_m \cdot -2, \frac{l\_m}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 0.0)
     (* (sqrt (* U (fma l_m (/ l_m (* Om -0.5)) t))) (sqrt (* 2.0 n)))
     (if (<= t_2 1e+307)
       (sqrt
        (*
         t_1
         (fma
          (* (/ l_m Om) U*)
          (* n (/ l_m Om))
          (fma (* l_m -2.0) (/ l_m Om) t))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* (* n U) (fma U* (/ n (* Om 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_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((U * fma(l_m, (l_m / (Om * -0.5)), t))) * sqrt((2.0 * n));
	} else if (t_2 <= 1e+307) {
		tmp = sqrt((t_1 * fma(((l_m / Om) * U_42_), (n * (l_m / Om)), fma((l_m * -2.0), (l_m / Om), t))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * fma(U_42_, (n / (Om * 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(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(U * fma(l_m, Float64(l_m / Float64(Om * -0.5)), t))) * sqrt(Float64(2.0 * n)));
	elseif (t_2 <= 1e+307)
		tmp = sqrt(Float64(t_1 * fma(Float64(Float64(l_m / Om) * U_42_), Float64(n * Float64(l_m / Om)), fma(Float64(l_m * -2.0), Float64(l_m / Om), t))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * fma(U_42_, Float64(n / Float64(Om * Om)), Float64(-2.0 / Om)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(U * N[(l$95$m * N[(l$95$m / N[(Om * -0.5), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[Sqrt[N[(t$95$1 * N[(N[(N[(l$95$m / Om), $MachinePrecision] * U$42$), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + N[(N[(l$95$m * -2.0), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(U$42$ * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / 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 12.2%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 18.1

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

   5. Simplified 18.1%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot U} \cdot \sqrt{n \cdot 2}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 98.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in U around 0

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

   7. Simplified 98.8%

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

   if 9.99999999999999986e306 < (*.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 20.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   4. Applied egg-rr 26.5%

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

   5. Taylor expanded in U around 0

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

   7. Simplified 26.7%

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

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. associate-/l* N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. /-lowering-/.f64 20.0

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

   10. Simplified 20.0%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om} \cdot U*, n \cdot \frac{\ell}{Om}, \mathsf{fma}\left(\ell \cdot -2, \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(l\_m, \frac{l\_m}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 0.0)
     (* (sqrt (* U (fma l_m (/ l_m (* Om -0.5)) t))) (sqrt (* 2.0 n)))
     (if (<= t_2 1e+307)
       (sqrt (* t_1 (- t (/ (* (* l_m l_m) (fma n (/ (- U U*) Om) 2.0)) Om))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* (* n U) (fma U* (/ n (* Om 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_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((U * fma(l_m, (l_m / (Om * -0.5)), t))) * sqrt((2.0 * n));
	} else if (t_2 <= 1e+307) {
		tmp = sqrt((t_1 * (t - (((l_m * l_m) * fma(n, ((U - U_42_) / Om), 2.0)) / Om))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * fma(U_42_, (n / (Om * 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(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(U * fma(l_m, Float64(l_m / Float64(Om * -0.5)), t))) * sqrt(Float64(2.0 * n)));
	elseif (t_2 <= 1e+307)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Om), 2.0)) / Om))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * fma(U_42_, Float64(n / Float64(Om * Om)), Float64(-2.0 / Om)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(U * N[(l$95$m * N[(l$95$m / N[(Om * -0.5), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(U$42$ * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / 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 12.2%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 18.1

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

   5. Simplified 18.1%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot U} \cdot \sqrt{n \cdot 2}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999986e306

   1. Initial program 98.2%

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

   3. Taylor expanded in t around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. /-lowering-/.f64 N/A

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

   5. Simplified 88.7%

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

   if 9.99999999999999986e306 < (*.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 20.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   4. Applied egg-rr 26.5%

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

   5. Taylor expanded in U around 0

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

   7. Simplified 26.7%

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

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. associate-/l* N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. /-lowering-/.f64 20.0

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

   10. Simplified 20.0%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+307}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot U\right)} \cdot \sqrt{n}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left(l\_m, \frac{l\_m}{Om \cdot -0.5}, t\right)\right) \cdot \left(2 \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (sqrt
          (*
           t_1
           (+
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_2 0.0)
     (* (sqrt (* t (* 2.0 U))) (sqrt n))
     (if (<= t_2 2e+147)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (sqrt (* (* U (fma l_m (/ l_m (* Om -0.5)) t)) (* 2.0 n)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((t * (2.0 * U))) * sqrt(n);
	} else if (t_2 <= 2e+147) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = sqrt(((U * fma(l_m, (l_m / (Om * -0.5)), t)) * (2.0 * n)));
	}
	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(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(t * Float64(2.0 * U))) * sqrt(n));
	elseif (t_2 <= 2e+147)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = sqrt(Float64(Float64(U * fma(l_m, Float64(l_m / Float64(Om * -0.5)), t)) * Float64(2.0 * n)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+147], 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[(U * N[(l$95$m * N[(l$95$m / N[(Om * -0.5), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] * N[(2.0 * n), $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 14.3%

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

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 34.0

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

   5. Simplified 34.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 46.9

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

   7. Applied egg-rr 46.9%

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

   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*))))) < 2e147

   1. Initial program 98.1%

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

   3. Taylor expanded in 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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 87.5

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

   5. Simplified 87.5%

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

   if 2e147 < (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 21.0%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 18.4

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

   5. Simplified 18.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 24.5%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot U\right) \cdot \left(n \cdot 2\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot U\right)} \cdot \sqrt{n}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right)\right) \cdot \left(2 \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot U\right)} \cdot \sqrt{n}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{t\_1 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (sqrt
          (*
           t_1
           (+
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_2 0.0)
     (* (sqrt (* t (* 2.0 U))) (sqrt n))
     (if (<= t_2 5e+153)
       (sqrt (* t_1 t))
       (sqrt (/ (* -4.0 (* n (* U (* l_m l_m)))) Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((t * (2.0 * U))) * sqrt(n);
	} else if (t_2 <= 5e+153) {
		tmp = sqrt((t_1 * t));
	} else {
		tmp = sqrt(((-4.0 * (n * (U * (l_m * l_m)))) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = u * (2.0d0 * n)
    t_2 = sqrt((t_1 * ((t - (2.0d0 * ((l_m * l_m) / om))) + ((n * ((l_m / om) ** 2.0d0)) * (u_42 - u)))))
    if (t_2 <= 0.0d0) then
        tmp = sqrt((t * (2.0d0 * u))) * sqrt(n)
    else if (t_2 <= 5d+153) then
        tmp = sqrt((t_1 * t))
    else
        tmp = sqrt((((-4.0d0) * (n * (u * (l_m * l_m)))) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = Math.sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt((t * (2.0 * U))) * Math.sqrt(n);
	} else if (t_2 <= 5e+153) {
		tmp = Math.sqrt((t_1 * t));
	} else {
		tmp = Math.sqrt(((-4.0 * (n * (U * (l_m * l_m)))) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (2.0 * n)
	t_2 = math.sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)))))
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt((t * (2.0 * U))) * math.sqrt(n)
	elif t_2 <= 5e+153:
		tmp = math.sqrt((t_1 * t))
	else:
		tmp = math.sqrt(((-4.0 * (n * (U * (l_m * l_m)))) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(t * Float64(2.0 * U))) * sqrt(n));
	elseif (t_2 <= 5e+153)
		tmp = sqrt(Float64(t_1 * t));
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(n * Float64(U * Float64(l_m * l_m)))) / Om));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U * (2.0 * n);
	t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U)))));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt((t * (2.0 * U))) * sqrt(n);
	elseif (t_2 <= 5e+153)
		tmp = sqrt((t_1 * t));
	else
		tmp = sqrt(((-4.0 * (n * (U * (l_m * l_m)))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+153], N[Sqrt[N[(t$95$1 * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-4.0 * N[(n * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 14.3%

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

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 34.0

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

   5. Simplified 34.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 46.9

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

   7. Applied egg-rr 46.9%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000018e153

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 73.2%

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

   if 5.00000000000000018e153 < (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 19.1%

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

   3. 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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 16.5

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

   5. Simplified 16.5%

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

   6. Taylor expanded in l around inf

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

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

      3. *-lowering-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 17.3

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

   8. Simplified 17.3%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot U\right)} \cdot \sqrt{n}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(n \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot U\right)} \cdot \sqrt{n}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{t\_1 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(U \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (sqrt
          (*
           t_1
           (+
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_2 0.0)
     (* (sqrt (* t (* 2.0 U))) (sqrt n))
     (if (<= t_2 5e+153)
       (sqrt (* t_1 t))
       (sqrt (/ (* -4.0 (* U (* n (* l_m l_m)))) Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((t * (2.0 * U))) * sqrt(n);
	} else if (t_2 <= 5e+153) {
		tmp = sqrt((t_1 * t));
	} else {
		tmp = sqrt(((-4.0 * (U * (n * (l_m * l_m)))) / Om));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = u * (2.0d0 * n)
    t_2 = sqrt((t_1 * ((t - (2.0d0 * ((l_m * l_m) / om))) + ((n * ((l_m / om) ** 2.0d0)) * (u_42 - u)))))
    if (t_2 <= 0.0d0) then
        tmp = sqrt((t * (2.0d0 * u))) * sqrt(n)
    else if (t_2 <= 5d+153) then
        tmp = sqrt((t_1 * t))
    else
        tmp = sqrt((((-4.0d0) * (u * (n * (l_m * l_m)))) / om))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = Math.sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt((t * (2.0 * U))) * Math.sqrt(n);
	} else if (t_2 <= 5e+153) {
		tmp = Math.sqrt((t_1 * t));
	} else {
		tmp = Math.sqrt(((-4.0 * (U * (n * (l_m * l_m)))) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (2.0 * n)
	t_2 = math.sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)))))
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt((t * (2.0 * U))) * math.sqrt(n)
	elif t_2 <= 5e+153:
		tmp = math.sqrt((t_1 * t))
	else:
		tmp = math.sqrt(((-4.0 * (U * (n * (l_m * l_m)))) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(t * Float64(2.0 * U))) * sqrt(n));
	elseif (t_2 <= 5e+153)
		tmp = sqrt(Float64(t_1 * t));
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(U * Float64(n * Float64(l_m * l_m)))) / Om));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U * (2.0 * n);
	t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U)))));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt((t * (2.0 * U))) * sqrt(n);
	elseif (t_2 <= 5e+153)
		tmp = sqrt((t_1 * t));
	else
		tmp = sqrt(((-4.0 * (U * (n * (l_m * l_m)))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+153], N[Sqrt[N[(t$95$1 * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-4.0 * N[(U * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 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 14.3%

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

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 34.0

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

   5. Simplified 34.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 46.9

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

   7. Applied egg-rr 46.9%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000018e153

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 73.2%

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

   if 5.00000000000000018e153 < (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 19.1%

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

   3. 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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 17.0

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

   5. Simplified 17.0%

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

   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. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.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. *-lowering-*.f64 17.0

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

   8. Simplified 17.0%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot U\right)} \cdot \sqrt{n}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m, \frac{l\_m}{Om \cdot -0.5}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \left(\frac{n \cdot \sqrt{2}}{Om} \cdot \left(-l\_m\right)\right)\\ \end{array} \end{array} \]

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
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U t)))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma l_m (/ l_m (* Om -0.5)) t)))
       (* (sqrt (* U (- U* U))) (* (/ (* n (sqrt 2.0)) Om) (- l_m)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(l_m, (l_m / (Om * -0.5)), t)));
	} else {
		tmp = sqrt((U * (U_42_ - U))) * (((n * sqrt(2.0)) / Om) * -l_m);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(l_m, Float64(l_m / Float64(Om * -0.5)), t)));
	else
		tmp = Float64(sqrt(Float64(U * Float64(U_42_ - U))) * Float64(Float64(Float64(n * sqrt(2.0)) / Om) * Float64(-l_m)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(l$95$m * N[(l$95$m / N[(Om * -0.5), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * (-l$95$m)), $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 12.2%

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

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 29.4

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

   5. Simplified 29.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. rem-square-sqrt N/A

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

      4. *-lowering-*.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 29.4

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

   7. Applied egg-rr 29.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-lowering-sqrt.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 42.7

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

   9. Applied egg-rr 42.7%

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

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

   1. Initial program 66.3%

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

   3. Taylor expanded in 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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 57.2

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

   5. Simplified 57.2%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. associate-*r/ N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 60.5%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}} \]

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

   1. Initial program 0.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   4. Applied egg-rr 0.6%

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

   5. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   7. Simplified 5.2%

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

   8. Taylor expanded in n around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 15.4

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

   10. Simplified 15.4%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(U* - U\right)} \cdot \left(\frac{n \cdot \sqrt{2}}{Om} \cdot \left(-\ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m, \frac{l\_m}{Om \cdot -0.5}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \sqrt{U \cdot U*}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U t)))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma l_m (/ l_m (* Om -0.5)) t)))
       (* (* l_m (/ (* n (sqrt 2.0)) Om)) (sqrt (* U U*)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(l_m, (l_m / (Om * -0.5)), t)));
	} else {
		tmp = (l_m * ((n * sqrt(2.0)) / Om)) * sqrt((U * U_42_));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(l_m, Float64(l_m / Float64(Om * -0.5)), t)));
	else
		tmp = Float64(Float64(l_m * Float64(Float64(n * sqrt(2.0)) / Om)) * sqrt(Float64(U * U_42_)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(l$95$m * N[(l$95$m / N[(Om * -0.5), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[(N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $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 12.2%

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

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 29.4

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

   5. Simplified 29.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. rem-square-sqrt N/A

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

      4. *-lowering-*.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 29.4

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

   7. Applied egg-rr 29.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. sqrt-lowering-sqrt.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 42.7

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

   9. Applied egg-rr 42.7%

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

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

   1. Initial program 66.3%

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

   3. Taylor expanded in 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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 57.2

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

   5. Simplified 57.2%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. associate-*r/ N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 60.5%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}} \]

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

   1. Initial program 0.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   4. Applied egg-rr 0.6%

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

   5. Taylor expanded in U* around inf

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

      1. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. *-lowering-*.f64 31.3

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

   7. Simplified 31.3%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \sqrt{U \cdot U*}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.5% accurate, 0.5× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))
        (t_3 (sqrt (fabs (* U (* 2.0 (* n t)))))))
   (if (<= t_2 5e-307) t_3 (if (<= t_2 1e+307) (sqrt (* t_1 t)) t_3))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double t_3 = sqrt(fabs((U * (2.0 * (n * t)))));
	double tmp;
	if (t_2 <= 5e-307) {
		tmp = t_3;
	} else if (t_2 <= 1e+307) {
		tmp = sqrt((t_1 * t));
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = u * (2.0d0 * n)
    t_2 = t_1 * ((t - (2.0d0 * ((l_m * l_m) / om))) + ((n * ((l_m / om) ** 2.0d0)) * (u_42 - u)))
    t_3 = sqrt(abs((u * (2.0d0 * (n * t)))))
    if (t_2 <= 5d-307) then
        tmp = t_3
    else if (t_2 <= 1d+307) then
        tmp = sqrt((t_1 * t))
    else
        tmp = t_3
    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_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double t_3 = Math.sqrt(Math.abs((U * (2.0 * (n * t)))));
	double tmp;
	if (t_2 <= 5e-307) {
		tmp = t_3;
	} else if (t_2 <= 1e+307) {
		tmp = Math.sqrt((t_1 * t));
	} else {
		tmp = t_3;
	}
	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_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)))
	t_3 = math.sqrt(math.fabs((U * (2.0 * (n * t)))))
	tmp = 0
	if t_2 <= 5e-307:
		tmp = t_3
	elif t_2 <= 1e+307:
		tmp = math.sqrt((t_1 * t))
	else:
		tmp = t_3
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	t_3 = sqrt(abs(Float64(U * Float64(2.0 * Float64(n * t)))))
	tmp = 0.0
	if (t_2 <= 5e-307)
		tmp = t_3;
	elseif (t_2 <= 1e+307)
		tmp = sqrt(Float64(t_1 * t));
	else
		tmp = t_3;
	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_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U)));
	t_3 = sqrt(abs((U * (2.0 * (n * t)))));
	tmp = 0.0;
	if (t_2 <= 5e-307)
		tmp = t_3;
	elseif (t_2 <= 1e+307)
		tmp = sqrt((t_1 * t));
	else
		tmp = t_3;
	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[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Abs[N[(U * N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 5e-307], t$95$3, If[LessEqual[t$95$2, 1e+307], N[Sqrt[N[(t$95$1 * t), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]

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*)))) < 5.00000000000000014e-307 or 9.99999999999999986e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 18.6%

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

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

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 12.2

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

   5. Simplified 12.2%

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

      1. associate-*r* N/A

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

      2. rem-square-sqrt N/A

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

      3. sqrt-unprod N/A

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

      4. rem-sqrt-square N/A

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

      5. fabs-lowering-fabs.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 17.0

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

   7. Applied egg-rr 17.0%

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

   if 5.00000000000000014e-307 < (*.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*)))) < 9.99999999999999986e306

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 73.9%

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

4. Final simplification 37.7%

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

Alternative 10: 48.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;\sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot U\right)} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m, \frac{l\_m}{Om \cdot -0.5}, t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n))))
   (if (<=
        (sqrt
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))
        0.0)
     (* (sqrt (* t (* 2.0 U))) (sqrt n))
     (sqrt (* t_1 (fma l_m (/ l_m (* Om -0.5)) t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double tmp;
	if (sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))))) <= 0.0) {
		tmp = sqrt((t * (2.0 * U))) * sqrt(n);
	} else {
		tmp = sqrt((t_1 * fma(l_m, (l_m / (Om * -0.5)), t)));
	}
	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 (sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))) <= 0.0)
		tmp = Float64(sqrt(Float64(t * Float64(2.0 * U))) * sqrt(n));
	else
		tmp = sqrt(Float64(t_1 * fma(l_m, Float64(l_m / Float64(Om * -0.5)), t)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[Sqrt[N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(l$95$m * N[(l$95$m / N[(Om * -0.5), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 14.3%

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

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 34.0

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

   5. Simplified 34.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 46.9

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

   7. Applied egg-rr 46.9%

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

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

   1. Initial program 52.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 46.7

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

   5. Simplified 46.7%

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. associate-*r/ N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 49.4%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot U\right)} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.3% accurate, 0.8× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n))))
   (if (<=
        (sqrt
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))
        0.0)
     (* (sqrt (* t (* 2.0 U))) (sqrt n))
     (sqrt (* t_1 (fma (* l_m 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 t_1 = U * (2.0 * n);
	double tmp;
	if (sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))))) <= 0.0) {
		tmp = sqrt((t * (2.0 * U))) * sqrt(n);
	} else {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	}
	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 (sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))) <= 0.0)
		tmp = Float64(sqrt(Float64(t * Float64(2.0 * U))) * sqrt(n));
	else
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 14.3%

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

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 34.0

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

   5. Simplified 34.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 46.9

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

   7. Applied egg-rr 46.9%

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

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

   1. Initial program 52.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 46.7

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

   5. Simplified 46.7%

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

4. Final simplification 46.7%

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

Alternative 12: 38.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double tmp;
	if (Math.sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U))))) <= 0.0) {
		tmp = Math.sqrt(((n * t) * (2.0 * U)));
	} else {
		tmp = Math.sqrt((t_1 * t));
	}
	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 math.sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U))))) <= 0.0:
		tmp = math.sqrt(((n * t) * (2.0 * U)))
	else:
		tmp = math.sqrt((t_1 * t))
	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 (sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))) <= 0.0)
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * U)));
	else
		tmp = sqrt(Float64(t_1 * t));
	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 (sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U))))) <= 0.0)
		tmp = sqrt(((n * t) * (2.0 * U)));
	else
		tmp = sqrt((t_1 * t));
	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[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 14.3%

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

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 34.0

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

   5. Simplified 34.0%

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

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

   1. Initial program 52.6%

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

   3. Taylor expanded in t around inf

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

   5. Simplified 34.3%

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

4. Final simplification 34.3%

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

Alternative 13: 61.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.3 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{\mathsf{fma}\left(n \cdot \left(\frac{l\_m}{Om} \cdot \left(U - U*\right)\right), l\_m, 2 \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.3e+152)
   (sqrt
    (*
     (* 2.0 n)
     (*
      U
      (-
       t
       (/ (fma (* n (* (/ l_m Om) (- U U*))) l_m (* 2.0 (* l_m l_m))) Om)))))
   (*
    (* l_m (sqrt 2.0))
    (sqrt (* (* n U) (fma U* (/ n (* Om 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 tmp;
	if (l_m <= 2.3e+152) {
		tmp = sqrt(((2.0 * n) * (U * (t - (fma((n * ((l_m / Om) * (U - U_42_))), l_m, (2.0 * (l_m * l_m))) / Om)))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * fma(U_42_, (n / (Om * Om)), (-2.0 / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.3e+152)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(fma(Float64(n * Float64(Float64(l_m / Om) * Float64(U - U_42_))), l_m, Float64(2.0 * Float64(l_m * l_m))) / Om)))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * fma(U_42_, Float64(n / Float64(Om * Om)), Float64(-2.0 / Om)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.3e+152], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(N[(N[(n * N[(N[(l$95$m / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m + N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(U$42$ * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.29999999999999985e152

   1. Initial program 51.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   4. Applied egg-rr 56.2%

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

      1. associate-*l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. associate-/l* N/A

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

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

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

      12. associate-+l- N/A

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

      13. --lowering--.f64 N/A

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

   6. Applied egg-rr 53.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. distribute-frac-neg N/A

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

      5. remove-double-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 56.0

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

   8. Applied egg-rr 56.0%

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

   if 2.29999999999999985e152 < l

   1. Initial program 12.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   4. Applied egg-rr 19.9%

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

   5. Taylor expanded in U around 0

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

   7. Simplified 20.4%

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

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. associate-/l* N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. /-lowering-/.f64 59.2

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

   10. Simplified 59.2%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{\mathsf{fma}\left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right), \ell, 2 \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 4 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\frac{l\_m \cdot U*}{Om} \cdot \left(n \cdot l\_m\right) - 2 \cdot \left(l\_m \cdot l\_m\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 4e+144)
   (sqrt
    (*
     (* 2.0 n)
     (*
      U
      (+ t (/ (- (* (/ (* l_m U*) Om) (* n l_m)) (* 2.0 (* l_m l_m))) Om)))))
   (*
    (* l_m (sqrt 2.0))
    (sqrt (* (* n U) (fma U* (/ n (* Om 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 tmp;
	if (l_m <= 4e+144) {
		tmp = sqrt(((2.0 * n) * (U * (t + (((((l_m * U_42_) / Om) * (n * l_m)) - (2.0 * (l_m * l_m))) / Om)))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * fma(U_42_, (n / (Om * Om)), (-2.0 / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 4e+144)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(Float64(Float64(l_m * U_42_) / Om) * Float64(n * l_m)) - Float64(2.0 * Float64(l_m * l_m))) / Om)))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * fma(U_42_, Float64(n / Float64(Om * Om)), Float64(-2.0 / Om)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 4e+144], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(N[(N[(l$95$m * U$42$), $MachinePrecision] / Om), $MachinePrecision] * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(U$42$ * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.00000000000000009e144

   1. Initial program 51.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   4. Applied egg-rr 56.2%

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

      1. associate-*l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. associate-/l* N/A

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

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

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

      12. associate-+l- N/A

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

      13. --lowering--.f64 N/A

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

   6. Applied egg-rr 53.8%

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

   7. Taylor expanded in U around 0

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

      1. *-lowering-*.f64 54.3

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

   9. Simplified 54.3%

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

   if 4.00000000000000009e144 < l

   1. Initial program 12.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   4. Applied egg-rr 19.9%

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

   5. Taylor expanded in U around 0

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

   7. Simplified 20.4%

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

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. associate-/l* N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. /-lowering-/.f64 59.2

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

   10. Simplified 59.2%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\frac{\ell \cdot U*}{Om} \cdot \left(n \cdot \ell\right) - 2 \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(n \cdot l\_m\right) \cdot \left(U \cdot l\_m\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(l\_m, \frac{l\_m}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -3.8e+22)
   (sqrt
    (*
     (* U (* 2.0 n))
     (- t (/ (* (* l_m l_m) (fma n (/ (- U U*) Om) 2.0)) Om))))
   (if (<= n 5.5e-277)
     (sqrt (fma 2.0 (* U (* n t)) (/ (* (* (* n l_m) (* U l_m)) -4.0) Om)))
     (* (sqrt (* U (fma l_m (/ l_m (* Om -0.5)) t))) (sqrt (* 2.0 n))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -3.8e+22) {
		tmp = sqrt(((U * (2.0 * n)) * (t - (((l_m * l_m) * fma(n, ((U - U_42_) / Om), 2.0)) / Om))));
	} else if (n <= 5.5e-277) {
		tmp = sqrt(fma(2.0, (U * (n * t)), ((((n * l_m) * (U * l_m)) * -4.0) / Om)));
	} else {
		tmp = sqrt((U * fma(l_m, (l_m / (Om * -0.5)), t))) * sqrt((2.0 * n));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -3.8e+22)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(t - Float64(Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Om), 2.0)) / Om))));
	elseif (n <= 5.5e-277)
		tmp = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(Float64(Float64(n * l_m) * Float64(U * l_m)) * -4.0) / Om)));
	else
		tmp = Float64(sqrt(Float64(U * fma(l_m, Float64(l_m / Float64(Om * -0.5)), t))) * sqrt(Float64(2.0 * n)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -3.8e+22], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 5.5e-277], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(n * l$95$m), $MachinePrecision] * N[(U * l$95$m), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(l$95$m * N[(l$95$m / N[(Om * -0.5), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.8000000000000004e22

   1. Initial program 52.9%

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

   3. Taylor expanded in t around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. /-lowering-/.f64 N/A

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

   5. Simplified 60.8%

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

   if -3.8000000000000004e22 < n < 5.49999999999999952e-277

   1. Initial program 44.2%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 44.4

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

   5. Simplified 44.4%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 56.1

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

   7. Applied egg-rr 56.1%

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

   if 5.49999999999999952e-277 < n

   1. Initial program 46.8%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 42.2

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

   5. Simplified 42.2%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot U} \cdot \sqrt{n \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 53.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\frac{U* \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(n \cdot l\_m\right) \cdot \left(U \cdot l\_m\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(l\_m, \frac{l\_m}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -8.2e+72)
   (sqrt (* (* 2.0 n) (* U (+ t (/ (/ (* U* (* n (* l_m l_m))) Om) Om)))))
   (if (<= n 3.6e-277)
     (sqrt (fma 2.0 (* U (* n t)) (/ (* (* (* n l_m) (* U l_m)) -4.0) Om)))
     (* (sqrt (* U (fma l_m (/ l_m (* Om -0.5)) t))) (sqrt (* 2.0 n))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -8.2e+72) {
		tmp = sqrt(((2.0 * n) * (U * (t + (((U_42_ * (n * (l_m * l_m))) / Om) / Om)))));
	} else if (n <= 3.6e-277) {
		tmp = sqrt(fma(2.0, (U * (n * t)), ((((n * l_m) * (U * l_m)) * -4.0) / Om)));
	} else {
		tmp = sqrt((U * fma(l_m, (l_m / (Om * -0.5)), t))) * sqrt((2.0 * n));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -8.2e+72)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(U_42_ * Float64(n * Float64(l_m * l_m))) / Om) / Om)))));
	elseif (n <= 3.6e-277)
		tmp = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(Float64(Float64(n * l_m) * Float64(U * l_m)) * -4.0) / Om)));
	else
		tmp = Float64(sqrt(Float64(U * fma(l_m, Float64(l_m / Float64(Om * -0.5)), t))) * sqrt(Float64(2.0 * n)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -8.2e+72], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(U$42$ * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.6e-277], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(n * l$95$m), $MachinePrecision] * N[(U * l$95$m), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(l$95$m * N[(l$95$m / N[(Om * -0.5), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.19999999999999926e72

   1. Initial program 51.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

   4. Applied egg-rr 57.4%

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

      1. associate-*l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. associate-/l* N/A

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

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

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

      12. associate-+l- N/A

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

      13. --lowering--.f64 N/A

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

   6. Applied egg-rr 60.1%

      \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \left(\ell \cdot \ell\right) - \frac{-\left(U - U*\right) \cdot \ell}{Om} \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}} \]

   7. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{\color{blue}{-1 \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{Om}}}{Om}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{\color{blue}{\mathsf{neg}\left(\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}}{Om}\right)\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{\color{blue}{\mathsf{neg}\left(\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}}{Om}\right)\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{\mathsf{neg}\left(\color{blue}{\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{Om}}\right)}{Om}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{\mathsf{neg}\left(\frac{\color{blue}{U* \cdot \left({\ell}^{2} \cdot n\right)}}{Om}\right)}{Om}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{\mathsf{neg}\left(\frac{U* \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}}{Om}\right)}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{\mathsf{neg}\left(\frac{U* \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)}{Om}\right)}{Om}\right)\right)} \]

      7. *-lowering-*.f64 61.9

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{-\frac{U* \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)}{Om}}{Om}\right)\right)} \]

   9. Simplified 61.9%

      \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{\color{blue}{-\frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}}}{Om}\right)\right)} \]

   if -8.19999999999999926e72 < n < 3.59999999999999984e-277

   1. Initial program 46.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 43.5

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Simplified 43.5%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)}\right) \cdot -4}{Om}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(U \cdot \ell\right)} \cdot \left(\ell \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      5. *-lowering-*.f64 54.4

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(U \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

   7. Applied egg-rr 54.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

   if 3.59999999999999984e-277 < n

   1. Initial program 46.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 42.2

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \]

   5. Simplified 42.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}} \]
   6. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)}^{\frac{1}{2}}} \]

      2. associate-*l* N/A

         \[\leadsto {\color{blue}{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)\right)}}^{\frac{1}{2}} \]

      3. *-commutative N/A

         \[\leadsto {\color{blue}{\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right)}}^{\frac{1}{2}} \]

      4. unpow-prod-down N/A

         \[\leadsto \color{blue}{{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)}^{\frac{1}{2}} \cdot {\left(2 \cdot n\right)}^{\frac{1}{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)}^{\frac{1}{2}} \cdot {\left(2 \cdot n\right)}^{\frac{1}{2}}} \]

   7. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot U} \cdot \sqrt{n \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 54.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \left(U* - U\right) \cdot \left(n \cdot \left(l\_m \cdot \frac{l\_m}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(n \cdot l\_m\right) \cdot \left(U \cdot l\_m\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(l\_m, \frac{l\_m}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -3.8e+22)
   (sqrt
    (* (* U (* 2.0 n)) (+ t (* (- U* U) (* n (* l_m (/ l_m (* Om Om))))))))
   (if (<= n 3.1e-276)
     (sqrt (fma 2.0 (* U (* n t)) (/ (* (* (* n l_m) (* U l_m)) -4.0) Om)))
     (* (sqrt (* U (fma l_m (/ l_m (* Om -0.5)) t))) (sqrt (* 2.0 n))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -3.8e+22) {
		tmp = sqrt(((U * (2.0 * n)) * (t + ((U_42_ - U) * (n * (l_m * (l_m / (Om * Om))))))));
	} else if (n <= 3.1e-276) {
		tmp = sqrt(fma(2.0, (U * (n * t)), ((((n * l_m) * (U * l_m)) * -4.0) / Om)));
	} else {
		tmp = sqrt((U * fma(l_m, (l_m / (Om * -0.5)), t))) * sqrt((2.0 * n));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -3.8e+22)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(t + Float64(Float64(U_42_ - U) * Float64(n * Float64(l_m * Float64(l_m / Float64(Om * Om))))))));
	elseif (n <= 3.1e-276)
		tmp = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(Float64(Float64(n * l_m) * Float64(U * l_m)) * -4.0) / Om)));
	else
		tmp = Float64(sqrt(Float64(U * fma(l_m, Float64(l_m / Float64(Om * -0.5)), t))) * sqrt(Float64(2.0 * n)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -3.8e+22], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[(l$95$m * N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.1e-276], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(n * l$95$m), $MachinePrecision] * N[(U * l$95$m), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(l$95$m * N[(l$95$m / N[(Om * -0.5), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.8000000000000004e22

   1. Initial program 52.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 51.2

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 51.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 56.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]

   if -3.8000000000000004e22 < n < 3.09999999999999989e-276

   1. Initial program 44.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 44.4

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Simplified 44.4%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)}\right) \cdot -4}{Om}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(U \cdot \ell\right)} \cdot \left(\ell \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      5. *-lowering-*.f64 56.1

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(U \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

   7. Applied egg-rr 56.1%

      \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

   if 3.09999999999999989e-276 < n

   1. Initial program 46.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 42.2

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \]

   5. Simplified 42.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}} \]
   6. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)}^{\frac{1}{2}}} \]

      2. associate-*l* N/A

         \[\leadsto {\color{blue}{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)\right)}}^{\frac{1}{2}} \]

      3. *-commutative N/A

         \[\leadsto {\color{blue}{\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right)}}^{\frac{1}{2}} \]

      4. unpow-prod-down N/A

         \[\leadsto \color{blue}{{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)}^{\frac{1}{2}} \cdot {\left(2 \cdot n\right)}^{\frac{1}{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)}^{\frac{1}{2}} \cdot {\left(2 \cdot n\right)}^{\frac{1}{2}}} \]

   7. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot U} \cdot \sqrt{n \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \left(U* - U\right) \cdot \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 53.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + U* \cdot \frac{n \cdot \left(l\_m \cdot l\_m\right)}{Om \cdot Om}\right)\right)}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(n \cdot l\_m\right) \cdot \left(U \cdot l\_m\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(l\_m, \frac{l\_m}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -1e+73)
   (sqrt (* (* 2.0 n) (* U (+ t (* U* (/ (* n (* l_m l_m)) (* Om Om)))))))
   (if (<= n 5.2e-277)
     (sqrt (fma 2.0 (* U (* n t)) (/ (* (* (* n l_m) (* U l_m)) -4.0) Om)))
     (* (sqrt (* U (fma l_m (/ l_m (* Om -0.5)) t))) (sqrt (* 2.0 n))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -1e+73) {
		tmp = sqrt(((2.0 * n) * (U * (t + (U_42_ * ((n * (l_m * l_m)) / (Om * Om)))))));
	} else if (n <= 5.2e-277) {
		tmp = sqrt(fma(2.0, (U * (n * t)), ((((n * l_m) * (U * l_m)) * -4.0) / Om)));
	} else {
		tmp = sqrt((U * fma(l_m, (l_m / (Om * -0.5)), t))) * sqrt((2.0 * n));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -1e+73)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(U_42_ * Float64(Float64(n * Float64(l_m * l_m)) / Float64(Om * Om)))))));
	elseif (n <= 5.2e-277)
		tmp = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(Float64(Float64(n * l_m) * Float64(U * l_m)) * -4.0) / Om)));
	else
		tmp = Float64(sqrt(Float64(U * fma(l_m, Float64(l_m / Float64(Om * -0.5)), t))) * sqrt(Float64(2.0 * n)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -1e+73], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(U$42$ * N[(N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 5.2e-277], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(n * l$95$m), $MachinePrecision] * N[(U * l$95$m), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(l$95$m * N[(l$95$m / N[(Om * -0.5), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.99999999999999983e72

   1. Initial program 51.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 57.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right)\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right)} \cdot \left(U \cdot \left(\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right)} \cdot \left(U \cdot \left(\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \color{blue}{\left(U \cdot \left(\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right)\right)}} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \color{blue}{\left(\left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right) + \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}\right)} \]

      7. +-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)} + \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t + \color{blue}{-2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)}\right) + \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)} \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right) + \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \]

      11. cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} + \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \]

      12. associate-+l- N/A

          \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)}\right)} \]

      13. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)}\right)} \]

   6. Applied egg-rr 60.1%

      \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot \left(\ell \cdot \ell\right) - \frac{-\left(U - U*\right) \cdot \ell}{Om} \cdot \left(\ell \cdot n\right)}{Om}\right)\right)}} \]

   7. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \color{blue}{-1 \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(\mathsf{neg}\left(\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)\right)}\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(\mathsf{neg}\left(\color{blue}{U* \cdot \frac{{\ell}^{2} \cdot n}{{Om}^{2}}}\right)\right)\right)\right)} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(\mathsf{neg}\left(U*\right)\right) \cdot \frac{{\ell}^{2} \cdot n}{{Om}^{2}}}\right)\right)} \]

      4. neg-mul-1 N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(-1 \cdot U*\right)} \cdot \frac{{\ell}^{2} \cdot n}{{Om}^{2}}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(-1 \cdot U*\right) \cdot \frac{{\ell}^{2} \cdot n}{{Om}^{2}}}\right)\right)} \]

      6. neg-mul-1 N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(\mathsf{neg}\left(U*\right)\right)} \cdot \frac{{\ell}^{2} \cdot n}{{Om}^{2}}\right)\right)} \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(\mathsf{neg}\left(U*\right)\right)} \cdot \frac{{\ell}^{2} \cdot n}{{Om}^{2}}\right)\right)} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(\mathsf{neg}\left(U*\right)\right) \cdot \color{blue}{\frac{{\ell}^{2} \cdot n}{{Om}^{2}}}\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(\mathsf{neg}\left(U*\right)\right) \cdot \frac{\color{blue}{{\ell}^{2} \cdot n}}{{Om}^{2}}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(\mathsf{neg}\left(U*\right)\right) \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot n}{{Om}^{2}}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(\mathsf{neg}\left(U*\right)\right) \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot n}{{Om}^{2}}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(\mathsf{neg}\left(U*\right)\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot n}{\color{blue}{Om \cdot Om}}\right)\right)} \]

      13. *-lowering-*.f64 57.5

          \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(-U*\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot n}{\color{blue}{Om \cdot Om}}\right)\right)} \]

   9. Simplified 57.5%

      \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \color{blue}{\left(-U*\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot n}{Om \cdot Om}}\right)\right)} \]

   if -9.99999999999999983e72 < n < 5.2e-277

   1. Initial program 46.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 43.5

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Simplified 43.5%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)}\right) \cdot -4}{Om}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(U \cdot \ell\right)} \cdot \left(\ell \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      5. *-lowering-*.f64 54.4

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(U \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

   7. Applied egg-rr 54.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

   if 5.2e-277 < n

   1. Initial program 46.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 42.2

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \]

   5. Simplified 42.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}} \]
   6. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)}^{\frac{1}{2}}} \]

      2. associate-*l* N/A

         \[\leadsto {\color{blue}{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)\right)}}^{\frac{1}{2}} \]

      3. *-commutative N/A

         \[\leadsto {\color{blue}{\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right)}}^{\frac{1}{2}} \]

      4. unpow-prod-down N/A

         \[\leadsto \color{blue}{{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)}^{\frac{1}{2}} \cdot {\left(2 \cdot n\right)}^{\frac{1}{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)}^{\frac{1}{2}} \cdot {\left(2 \cdot n\right)}^{\frac{1}{2}}} \]

   7. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot U} \cdot \sqrt{n \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + U* \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}\right)\right)}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \mathsf{fma}\left(l\_m, \frac{l\_m}{Om \cdot -0.5}, t\right)\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(n \cdot l\_m\right) \cdot \left(U \cdot l\_m\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \sqrt{2 \cdot n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (fma l_m (/ l_m (* Om -0.5)) t))))
   (if (<= n -9.5e+72)
     (sqrt (* t_1 (* 2.0 n)))
     (if (<= n 3.1e-276)
       (sqrt (fma 2.0 (* U (* n t)) (/ (* (* (* n l_m) (* U l_m)) -4.0) Om)))
       (* (sqrt t_1) (sqrt (* 2.0 n)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * fma(l_m, (l_m / (Om * -0.5)), t);
	double tmp;
	if (n <= -9.5e+72) {
		tmp = sqrt((t_1 * (2.0 * n)));
	} else if (n <= 3.1e-276) {
		tmp = sqrt(fma(2.0, (U * (n * t)), ((((n * l_m) * (U * l_m)) * -4.0) / Om)));
	} else {
		tmp = sqrt(t_1) * sqrt((2.0 * n));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * fma(l_m, Float64(l_m / Float64(Om * -0.5)), t))
	tmp = 0.0
	if (n <= -9.5e+72)
		tmp = sqrt(Float64(t_1 * Float64(2.0 * n)));
	elseif (n <= 3.1e-276)
		tmp = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(Float64(Float64(n * l_m) * Float64(U * l_m)) * -4.0) / Om)));
	else
		tmp = Float64(sqrt(t_1) * sqrt(Float64(2.0 * n)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(l$95$m * N[(l$95$m / N[(Om * -0.5), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9.5e+72], N[Sqrt[N[(t$95$1 * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3.1e-276], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(n * l$95$m), $MachinePrecision] * N[(U * l$95$m), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.50000000000000054e72

   1. Initial program 51.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 42.7

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \]

   5. Simplified 42.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)}} \]

   7. Applied egg-rr 46.8%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot U\right) \cdot \left(n \cdot 2\right)}} \]

   if -9.50000000000000054e72 < n < 3.09999999999999989e-276

   1. Initial program 46.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 43.5

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Simplified 43.5%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)}\right) \cdot -4}{Om}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\color{blue}{\left(U \cdot \ell\right)} \cdot \left(\ell \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      5. *-lowering-*.f64 54.4

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(U \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

   7. Applied egg-rr 54.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(\left(U \cdot \ell\right) \cdot \left(\ell \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

   if 3.09999999999999989e-276 < n

   1. Initial program 46.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.f64 42.2

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \]

   5. Simplified 42.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}} \]
   6. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)}^{\frac{1}{2}}} \]

      2. associate-*l* N/A

         \[\leadsto {\color{blue}{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)\right)}}^{\frac{1}{2}} \]

      3. *-commutative N/A

         \[\leadsto {\color{blue}{\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right) \cdot \left(2 \cdot n\right)\right)}}^{\frac{1}{2}} \]

      4. unpow-prod-down N/A

         \[\leadsto \color{blue}{{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)}^{\frac{1}{2}} \cdot {\left(2 \cdot n\right)}^{\frac{1}{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-2}{Om} + t\right)\right)}^{\frac{1}{2}} \cdot {\left(2 \cdot n\right)}^{\frac{1}{2}}} \]

   7. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot U} \cdot \sqrt{n \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(\left(n \cdot \ell\right) \cdot \left(U \cdot \ell\right)\right) \cdot -4}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right)} \cdot \sqrt{2 \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot U} \cdot \sqrt{2 \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 1.15e+130)
   (sqrt (* (* 2.0 U) (* n (fma -2.0 (/ (* l_m l_m) Om) t))))
   (* (sqrt (* n U)) (sqrt (* 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 (t <= 1.15e+130) {
		tmp = sqrt(((2.0 * U) * (n * fma(-2.0, ((l_m * l_m) / Om), t))));
	} else {
		tmp = sqrt((n * U)) * sqrt((2.0 * t));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 1.15e+130)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * fma(-2.0, Float64(Float64(l_m * l_m) / Om), t))));
	else
		tmp = Float64(sqrt(Float64(n * U)) * sqrt(Float64(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[t, 1.15e+130], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.15000000000000011e130

   1. Initial program 45.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 50.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   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. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}\right)} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)}\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)} \]

      13. *-lowering-*.f64 40.9

          \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)} \]

   7. Simplified 40.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

   if 1.15000000000000011e130 < t

   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 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. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 43.3

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 43.3%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(U \cdot n\right) \cdot t\right)} \cdot 2} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot U\right)} \cdot t\right) \cdot 2} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(t \cdot 2\right)}} \]

      6. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{n \cdot U} \cdot \sqrt{t \cdot 2}} \]

      7. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(n \cdot U\right)}^{\frac{1}{2}}} \cdot \sqrt{t \cdot 2} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\left(n \cdot U\right)}^{\frac{1}{2}} \cdot \sqrt{t \cdot 2}} \]

      9. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{n \cdot U}} \cdot \sqrt{t \cdot 2} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{n \cdot U}} \cdot \sqrt{t \cdot 2} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{U \cdot n}} \cdot \sqrt{t \cdot 2} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{U \cdot n}} \cdot \sqrt{t \cdot 2} \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{U \cdot n} \cdot \color{blue}{\sqrt{t \cdot 2}} \]

      14. *-lowering-*.f64 66.1

          \[\leadsto \sqrt{U \cdot n} \cdot \sqrt{\color{blue}{t \cdot 2}} \]

   7. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{\sqrt{U \cdot n} \cdot \sqrt{t \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot U} \cdot \sqrt{2 \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 39.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 2.2 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{\left|U \cdot \left(2 \cdot \left(n \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 2.2e-304)
   (sqrt (fabs (* U (* 2.0 (* n t)))))
   (* (sqrt (* 2.0 n)) (sqrt (* U t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 2.2e-304) {
		tmp = sqrt(fabs((U * (2.0 * (n * t)))));
	} else {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 2.2d-304) then
        tmp = sqrt(abs((u * (2.0d0 * (n * t)))))
    else
        tmp = sqrt((2.0d0 * n)) * sqrt((u * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 2.2e-304) {
		tmp = Math.sqrt(Math.abs((U * (2.0 * (n * t)))));
	} else {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 2.2e-304:
		tmp = math.sqrt(math.fabs((U * (2.0 * (n * t)))))
	else:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 2.2e-304)
		tmp = sqrt(abs(Float64(U * Float64(2.0 * Float64(n * t)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= 2.2e-304)
		tmp = sqrt(abs((U * (2.0 * (n * t)))));
	else
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 2.2e-304], N[Sqrt[N[Abs[N[(U * N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.2e-304

   1. Initial program 49.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 28.5

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 28.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      3. sqrt-unprod N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right) \cdot \left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}} \]

      4. rem-sqrt-square N/A

         \[\leadsto \sqrt{\color{blue}{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}} \]

      5. fabs-lowering-fabs.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left|\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right|} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left|\color{blue}{\left(U \cdot 2\right)} \cdot \left(n \cdot t\right)\right|} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{\left|\color{blue}{U \cdot \left(2 \cdot \left(n \cdot t\right)\right)}\right|} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left|\color{blue}{U \cdot \left(2 \cdot \left(n \cdot t\right)\right)}\right|} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left|U \cdot \color{blue}{\left(2 \cdot \left(n \cdot t\right)\right)}\right|} \]

      11. *-lowering-*.f64 32.8

          \[\leadsto \sqrt{\left|U \cdot \left(2 \cdot \color{blue}{\left(n \cdot t\right)}\right)\right|} \]

   7. Applied egg-rr 32.8%

      \[\leadsto \sqrt{\color{blue}{\left|U \cdot \left(2 \cdot \left(n \cdot t\right)\right)\right|}} \]

   if 2.2e-304 < n

   1. Initial program 45.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 32.5

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 32.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \sqrt{U \cdot \left(n \cdot t\right)}\right)} \cdot 2} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \sqrt{U \cdot \left(n \cdot t\right)}\right) \cdot 2}} \]

      5. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right)} \cdot 2} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right)} \cdot 2} \]

      7. *-lowering-*.f64 32.5

         \[\leadsto \sqrt{\left(U \cdot \color{blue}{\left(n \cdot t\right)}\right) \cdot 2} \]

   7. Applied egg-rr 32.5%

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t}} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot t} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{t \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{t \cdot \color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)}} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(t \cdot U\right) \cdot \left(2 \cdot n\right)}} \]

      9. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{t \cdot U} \cdot \sqrt{2 \cdot n}} \]

      10. pow1/2 N/A

          \[\leadsto \sqrt{t \cdot U} \cdot \color{blue}{{\left(2 \cdot n\right)}^{\frac{1}{2}}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{t \cdot U} \cdot {\left(2 \cdot n\right)}^{\frac{1}{2}}} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{t \cdot U}} \cdot {\left(2 \cdot n\right)}^{\frac{1}{2}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{t \cdot U}} \cdot {\left(2 \cdot n\right)}^{\frac{1}{2}} \]

      14. pow1/2 N/A

          \[\leadsto \sqrt{t \cdot U} \cdot \color{blue}{\sqrt{2 \cdot n}} \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{t \cdot U} \cdot \color{blue}{\sqrt{2 \cdot n}} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{t \cdot U} \cdot \sqrt{\color{blue}{n \cdot 2}} \]

      17. *-lowering-*.f64 43.3

          \[\leadsto \sqrt{t \cdot U} \cdot \sqrt{\color{blue}{n \cdot 2}} \]

   9. Applied egg-rr 43.3%

      \[\leadsto \color{blue}{\sqrt{t \cdot U} \cdot \sqrt{n \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.2 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{\left|U \cdot \left(2 \cdot \left(n \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 40.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left|U \cdot \left(2 \cdot \left(n \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot U} \cdot \sqrt{2 \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 2.6e-308)
   (sqrt (fabs (* U (* 2.0 (* n t)))))
   (* (sqrt (* n U)) (sqrt (* 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 (t <= 2.6e-308) {
		tmp = sqrt(fabs((U * (2.0 * (n * t)))));
	} else {
		tmp = sqrt((n * U)) * sqrt((2.0 * t));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 2.6d-308) then
        tmp = sqrt(abs((u * (2.0d0 * (n * t)))))
    else
        tmp = sqrt((n * u)) * sqrt((2.0d0 * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= 2.6e-308) {
		tmp = Math.sqrt(Math.abs((U * (2.0 * (n * t)))));
	} else {
		tmp = Math.sqrt((n * U)) * Math.sqrt((2.0 * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= 2.6e-308:
		tmp = math.sqrt(math.fabs((U * (2.0 * (n * t)))))
	else:
		tmp = math.sqrt((n * U)) * math.sqrt((2.0 * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 2.6e-308)
		tmp = sqrt(abs(Float64(U * Float64(2.0 * Float64(n * t)))));
	else
		tmp = Float64(sqrt(Float64(n * U)) * sqrt(Float64(2.0 * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= 2.6e-308)
		tmp = sqrt(abs((U * (2.0 * (n * t)))));
	else
		tmp = sqrt((n * U)) * sqrt((2.0 * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, 2.6e-308], N[Sqrt[N[Abs[N[(U * N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.6e-308

   1. Initial program 47.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 31.3

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 31.3%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      3. sqrt-unprod N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right) \cdot \left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}} \]

      4. rem-sqrt-square N/A

         \[\leadsto \sqrt{\color{blue}{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}} \]

      5. fabs-lowering-fabs.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left|\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right|} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left|\color{blue}{\left(U \cdot 2\right)} \cdot \left(n \cdot t\right)\right|} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{\left|\color{blue}{U \cdot \left(2 \cdot \left(n \cdot t\right)\right)}\right|} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left|\color{blue}{U \cdot \left(2 \cdot \left(n \cdot t\right)\right)}\right|} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left|U \cdot \color{blue}{\left(2 \cdot \left(n \cdot t\right)\right)}\right|} \]

      11. *-lowering-*.f64 35.2

          \[\leadsto \sqrt{\left|U \cdot \left(2 \cdot \color{blue}{\left(n \cdot t\right)}\right)\right|} \]

   7. Applied egg-rr 35.2%

      \[\leadsto \sqrt{\color{blue}{\left|U \cdot \left(2 \cdot \left(n \cdot t\right)\right)\right|}} \]

   if 2.6e-308 < t

   1. Initial program 47.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 29.7

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 29.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(U \cdot n\right) \cdot t\right)} \cdot 2} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot U\right)} \cdot t\right) \cdot 2} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot U\right) \cdot \left(t \cdot 2\right)}} \]

      6. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{n \cdot U} \cdot \sqrt{t \cdot 2}} \]

      7. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(n \cdot U\right)}^{\frac{1}{2}}} \cdot \sqrt{t \cdot 2} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{\left(n \cdot U\right)}^{\frac{1}{2}} \cdot \sqrt{t \cdot 2}} \]

      9. pow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{n \cdot U}} \cdot \sqrt{t \cdot 2} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{n \cdot U}} \cdot \sqrt{t \cdot 2} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{U \cdot n}} \cdot \sqrt{t \cdot 2} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{U \cdot n}} \cdot \sqrt{t \cdot 2} \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \sqrt{U \cdot n} \cdot \color{blue}{\sqrt{t \cdot 2}} \]

      14. *-lowering-*.f64 40.8

          \[\leadsto \sqrt{U \cdot n} \cdot \sqrt{\color{blue}{t \cdot 2}} \]

   7. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{\sqrt{U \cdot n} \cdot \sqrt{t \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left|U \cdot \left(2 \cdot \left(n \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot U} \cdot \sqrt{2 \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 35.6% 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 47.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

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. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

   4. *-lowering-*.f64 30.5

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

5. Simplified 30.5%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

6. Final simplification 30.5%

   \[\leadsto \sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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*))))))