Toniolo and Linder, Equation (13)

Percentage Accurate: 49.3% → 66.3%

Time: 15.5s

Alternatives: 24

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 66.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\\ t_3 := U \cdot \left(n \cdot 2\right)\\ t_4 := \sqrt{\left(t\_2 - \left(U - U*\right) \cdot \left(t\_1 \cdot n\right)\right) \cdot t\_3}\\ \mathbf{if}\;t\_4 \leq 10^{-156}:\\ \;\;\;\;\sqrt{U \cdot 2} \cdot \sqrt{\mathsf{fma}\left(-t\_1, \left(U - U*\right) \cdot n, \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\right) \cdot n}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(t\_2 - \left(U* - U\right) \cdot \left(\left(\left(\frac{l\_m}{Om} \cdot n\right) \cdot l\_m\right) \cdot \frac{-1}{Om}\right)\right) \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l_m Om) 2.0))
        (t_2 (- t (* (/ (* l_m l_m) Om) 2.0)))
        (t_3 (* U (* n 2.0)))
        (t_4 (sqrt (* (- t_2 (* (- U U*) (* t_1 n))) t_3))))
   (if (<= t_4 1e-156)
     (*
      (sqrt (* U 2.0))
      (sqrt
       (* (fma (- t_1) (* (- U U*) n) (fma (* (/ l_m Om) l_m) -2.0 t)) n)))
     (if (<= t_4 2e+152)
       (sqrt
        (* (- t_2 (* (- U* U) (* (* (* (/ l_m Om) n) l_m) (/ -1.0 Om)))) t_3))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (* U n) (- (* (/ (- U* U) Om) (/ n 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 = pow((l_m / Om), 2.0);
	double t_2 = t - (((l_m * l_m) / Om) * 2.0);
	double t_3 = U * (n * 2.0);
	double t_4 = sqrt(((t_2 - ((U - U_42_) * (t_1 * n))) * t_3));
	double tmp;
	if (t_4 <= 1e-156) {
		tmp = sqrt((U * 2.0)) * sqrt((fma(-t_1, ((U - U_42_) * n), fma(((l_m / Om) * l_m), -2.0, t)) * n));
	} else if (t_4 <= 2e+152) {
		tmp = sqrt(((t_2 - ((U_42_ - U) * ((((l_m / Om) * n) * l_m) * (-1.0 / Om)))) * t_3));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt(((U * n) * ((((U_42_ - U) / Om) * (n / Om)) - (2.0 / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(l_m / Om) ^ 2.0
	t_2 = Float64(t - Float64(Float64(Float64(l_m * l_m) / Om) * 2.0))
	t_3 = Float64(U * Float64(n * 2.0))
	t_4 = sqrt(Float64(Float64(t_2 - Float64(Float64(U - U_42_) * Float64(t_1 * n))) * t_3))
	tmp = 0.0
	if (t_4 <= 1e-156)
		tmp = Float64(sqrt(Float64(U * 2.0)) * sqrt(Float64(fma(Float64(-t_1), Float64(Float64(U - U_42_) * n), fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)) * n)));
	elseif (t_4 <= 2e+152)
		tmp = sqrt(Float64(Float64(t_2 - Float64(Float64(U_42_ - U) * Float64(Float64(Float64(Float64(l_m / Om) * n) * l_m) * Float64(-1.0 / Om)))) * t_3));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(U * n) * Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / 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[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(t$95$2 - N[(N[(U - U$42$), $MachinePrecision] * N[(t$95$1 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 1e-156], N[(N[Sqrt[N[(U * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[((-t$95$1) * N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] + N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+152], N[Sqrt[N[(N[(t$95$2 - N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] * l$95$m), $MachinePrecision] * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(U * n), $MachinePrecision] * N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.00000000000000004e-156

   1. Initial program 8.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 8.4

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 8.4%

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

   6. Applied rewrites 49.7%

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

   if 1.00000000000000004e-156 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.0000000000000001e152

   1. Initial program 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. frac-2neg N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. lower-/.f64 97.2

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

   4. Applied rewrites 97.2%

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

   if 2.0000000000000001e152 < (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 25.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 31.1

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 31.1%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 26.0%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 10^{-156}:\\ \;\;\;\;\sqrt{U \cdot 2} \cdot \sqrt{\mathsf{fma}\left(-{\left(\frac{\ell}{Om}\right)}^{2}, \left(U - U*\right) \cdot n, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot n}\\ \mathbf{elif}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U* - U\right) \cdot \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\right) \cdot \frac{-1}{Om}\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m}{Om} \cdot n\\ t_2 := t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\\ t_3 := U \cdot \left(n \cdot 2\right)\\ t_4 := \left(t\_2 - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_3\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-313}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(t\_1, \frac{U*}{Om} \cdot l\_m, \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\left(t\_2 - \left(U* - U\right) \cdot \left(\left(t\_1 \cdot l\_m\right) \cdot \frac{-1}{Om}\right)\right) \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (/ l_m Om) n))
        (t_2 (- t (* (/ (* l_m l_m) Om) 2.0)))
        (t_3 (* U (* n 2.0)))
        (t_4 (* (- t_2 (* (- U U*) (* (pow (/ l_m Om) 2.0) n))) t_3)))
   (if (<= t_4 5e-313)
     (*
      (sqrt U)
      (sqrt
       (*
        (fma t_1 (* (/ U* Om) l_m) (fma (* (/ l_m Om) l_m) -2.0 t))
        (* n 2.0))))
     (if (<= t_4 5e+304)
       (sqrt (* (- t_2 (* (- U* U) (* (* t_1 l_m) (/ -1.0 Om)))) t_3))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (* U n) (- (* (/ (- U* U) Om) (/ n 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 = (l_m / Om) * n;
	double t_2 = t - (((l_m * l_m) / Om) * 2.0);
	double t_3 = U * (n * 2.0);
	double t_4 = (t_2 - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_3;
	double tmp;
	if (t_4 <= 5e-313) {
		tmp = sqrt(U) * sqrt((fma(t_1, ((U_42_ / Om) * l_m), fma(((l_m / Om) * l_m), -2.0, t)) * (n * 2.0)));
	} else if (t_4 <= 5e+304) {
		tmp = sqrt(((t_2 - ((U_42_ - U) * ((t_1 * l_m) * (-1.0 / Om)))) * t_3));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt(((U * n) * ((((U_42_ - U) / Om) * (n / Om)) - (2.0 / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m / Om) * n)
	t_2 = Float64(t - Float64(Float64(Float64(l_m * l_m) / Om) * 2.0))
	t_3 = Float64(U * Float64(n * 2.0))
	t_4 = Float64(Float64(t_2 - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_3)
	tmp = 0.0
	if (t_4 <= 5e-313)
		tmp = Float64(sqrt(U) * sqrt(Float64(fma(t_1, Float64(Float64(U_42_ / Om) * l_m), fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)) * Float64(n * 2.0))));
	elseif (t_4 <= 5e+304)
		tmp = sqrt(Float64(Float64(t_2 - Float64(Float64(U_42_ - U) * Float64(Float64(t_1 * l_m) * Float64(-1.0 / Om)))) * t_3));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(U * n) * Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / 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[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, 5e-313], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(N[(t$95$1 * N[(N[(U$42$ / Om), $MachinePrecision] * l$95$m), $MachinePrecision] + N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+304], N[Sqrt[N[(N[(t$95$2 - N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(t$95$1 * l$95$m), $MachinePrecision] * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(U * n), $MachinePrecision] * N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / 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*)))) < 5.00000000002e-313

   1. Initial program 7.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 16.8

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 16.8%

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

   5. Taylor expanded in U around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 16.8

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

   7. Applied rewrites 16.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 43.8%

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

   if 5.00000000002e-313 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999999999999997e304

   1. Initial program 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. frac-2neg N/A

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

      14. metadata-eval N/A

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

      15. remove-double-neg N/A

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

      16. lower-/.f64 97.2

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

   4. Applied rewrites 97.2%

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

   if 4.9999999999999997e304 < (*.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 27.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 29.5

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 29.5%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 27.6%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{-313}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot n, \frac{U*}{Om} \cdot \ell, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U* - U\right) \cdot \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\right) \cdot \frac{-1}{Om}\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m}{Om} \cdot n\\ t_2 := U \cdot \left(n \cdot 2\right)\\ t_3 := \frac{l\_m \cdot l\_m}{Om}\\ t_4 := \left(\left(t - t\_3 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_2\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-313}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(t\_1, \frac{U*}{Om} \cdot l\_m, \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* \cdot \frac{l\_m}{Om}, t\_1, \mathsf{fma}\left(-2, t\_3, t\right)\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (/ l_m Om) n))
        (t_2 (* U (* n 2.0)))
        (t_3 (/ (* l_m l_m) Om))
        (t_4
         (*
          (- (- t (* t_3 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_2)))
   (if (<= t_4 5e-313)
     (*
      (sqrt U)
      (sqrt
       (*
        (fma t_1 (* (/ U* Om) l_m) (fma (* (/ l_m Om) l_m) -2.0 t))
        (* n 2.0))))
     (if (<= t_4 5e+304)
       (sqrt (* (fma (* U* (/ l_m Om)) t_1 (fma -2.0 t_3 t)) t_2))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (* U n) (- (* (/ (- U* U) Om) (/ n 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 = (l_m / Om) * n;
	double t_2 = U * (n * 2.0);
	double t_3 = (l_m * l_m) / Om;
	double t_4 = ((t - (t_3 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_2;
	double tmp;
	if (t_4 <= 5e-313) {
		tmp = sqrt(U) * sqrt((fma(t_1, ((U_42_ / Om) * l_m), fma(((l_m / Om) * l_m), -2.0, t)) * (n * 2.0)));
	} else if (t_4 <= 5e+304) {
		tmp = sqrt((fma((U_42_ * (l_m / Om)), t_1, fma(-2.0, t_3, t)) * t_2));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt(((U * n) * ((((U_42_ - U) / Om) * (n / Om)) - (2.0 / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m / Om) * n)
	t_2 = Float64(U * Float64(n * 2.0))
	t_3 = Float64(Float64(l_m * l_m) / Om)
	t_4 = Float64(Float64(Float64(t - Float64(t_3 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_2)
	tmp = 0.0
	if (t_4 <= 5e-313)
		tmp = Float64(sqrt(U) * sqrt(Float64(fma(t_1, Float64(Float64(U_42_ / Om) * l_m), fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)) * Float64(n * 2.0))));
	elseif (t_4 <= 5e+304)
		tmp = sqrt(Float64(fma(Float64(U_42_ * Float64(l_m / Om)), t_1, fma(-2.0, t_3, t)) * t_2));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(U * n) * Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / 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[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t - N[(t$95$3 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, 5e-313], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(N[(t$95$1 * N[(N[(U$42$ / Om), $MachinePrecision] * l$95$m), $MachinePrecision] + N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+304], N[Sqrt[N[(N[(N[(U$42$ * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(-2.0 * t$95$3 + t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(U * n), $MachinePrecision] * N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / 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*)))) < 5.00000000002e-313

   1. Initial program 7.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 16.8

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 16.8%

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

   5. Taylor expanded in U around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 16.8

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

   7. Applied rewrites 16.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 43.8%

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

   if 5.00000000002e-313 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999999999999997e304

   1. Initial program 96.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 96.2

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in U around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 96.8

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

   7. Applied rewrites 96.8%

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

   if 4.9999999999999997e304 < (*.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 27.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 29.5

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 29.5%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 27.6%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{-313}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot n, \frac{U*}{Om} \cdot \ell, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-313}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-l\_m, \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right) \cdot \frac{l\_m}{Om}, t\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* \cdot \frac{l\_m}{Om}, \frac{l\_m}{Om} \cdot n, \mathsf{fma}\left(-2, t\_2, t\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3
         (*
          (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_3 5e-313)
     (*
      (sqrt
       (*
        (fma (- l_m) (* (fma (- U U*) (/ n Om) 2.0) (/ l_m Om)) t)
        (* n 2.0)))
      (sqrt U))
     (if (<= t_3 5e+304)
       (sqrt (* (fma (* U* (/ l_m Om)) (* (/ l_m Om) n) (fma -2.0 t_2 t)) t_1))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (* U n) (- (* (/ (- U* U) Om) (/ n 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 * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_3 <= 5e-313) {
		tmp = sqrt((fma(-l_m, (fma((U - U_42_), (n / Om), 2.0) * (l_m / Om)), t) * (n * 2.0))) * sqrt(U);
	} else if (t_3 <= 5e+304) {
		tmp = sqrt((fma((U_42_ * (l_m / Om)), ((l_m / Om) * n), fma(-2.0, t_2, t)) * t_1));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt(((U * n) * ((((U_42_ - U) / Om) * (n / 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(n * 2.0))
	t_2 = Float64(Float64(l_m * l_m) / Om)
	t_3 = Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	tmp = 0.0
	if (t_3 <= 5e-313)
		tmp = Float64(sqrt(Float64(fma(Float64(-l_m), Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) * Float64(l_m / Om)), t) * Float64(n * 2.0))) * sqrt(U));
	elseif (t_3 <= 5e+304)
		tmp = sqrt(Float64(fma(Float64(U_42_ * Float64(l_m / Om)), Float64(Float64(l_m / Om) * n), fma(-2.0, t_2, t)) * t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(U * n) * Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / 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[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-313], N[(N[Sqrt[N[(N[((-l$95$m) * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+304], N[Sqrt[N[(N[(N[(U$42$ * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * t$95$2 + t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(U * n), $MachinePrecision] * N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / 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*)))) < 5.00000000002e-313

   1. Initial program 7.0%

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

   3. Taylor expanded in n around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

   5. Applied rewrites 11.9%

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

   7. Applied rewrites 14.4%

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

   9. Applied rewrites 41.5%

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

   if 5.00000000002e-313 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999999999999997e304

   1. Initial program 96.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 96.2

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in U around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 96.8

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

   7. Applied rewrites 96.8%

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

   if 4.9999999999999997e304 < (*.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 27.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 29.5

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 29.5%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 27.6%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{-313}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-\ell, \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right) \cdot \frac{\ell}{Om}, t\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-313}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-l\_m, \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right) \cdot \frac{l\_m}{Om}, t\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right) \cdot \left(\frac{l\_m}{Om} \cdot l\_m\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2
         (*
          (-
           (- t (* (/ (* l_m l_m) Om) 2.0))
           (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_2 5e-313)
     (*
      (sqrt
       (*
        (fma (- l_m) (* (fma (- U U*) (/ n Om) 2.0) (/ l_m Om)) t)
        (* n 2.0)))
      (sqrt U))
     (if (<= t_2 5e+304)
       (sqrt (* (- t (* (fma (/ n Om) (- U U*) 2.0) (* (/ l_m Om) l_m))) t_1))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (* U n) (- (* (/ (- U* U) Om) (/ n 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 * (n * 2.0);
	double t_2 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_2 <= 5e-313) {
		tmp = sqrt((fma(-l_m, (fma((U - U_42_), (n / Om), 2.0) * (l_m / Om)), t) * (n * 2.0))) * sqrt(U);
	} else if (t_2 <= 5e+304) {
		tmp = sqrt(((t - (fma((n / Om), (U - U_42_), 2.0) * ((l_m / Om) * l_m))) * t_1));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt(((U * n) * ((((U_42_ - U) / Om) * (n / 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(n * 2.0))
	t_2 = Float64(Float64(Float64(t - Float64(Float64(Float64(l_m * l_m) / Om) * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	tmp = 0.0
	if (t_2 <= 5e-313)
		tmp = Float64(sqrt(Float64(fma(Float64(-l_m), Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) * Float64(l_m / Om)), t) * Float64(n * 2.0))) * sqrt(U));
	elseif (t_2 <= 5e+304)
		tmp = sqrt(Float64(Float64(t - Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) * Float64(Float64(l_m / Om) * l_m))) * t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(U * n) * Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / 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[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-313], N[(N[Sqrt[N[(N[((-l$95$m) * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+304], N[Sqrt[N[(N[(t - N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(U * n), $MachinePrecision] * N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / 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*)))) < 5.00000000002e-313

   1. Initial program 7.0%

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

   3. Taylor expanded in n around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

   5. Applied rewrites 11.9%

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

   7. Applied rewrites 14.4%

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

   9. Applied rewrites 41.5%

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

   if 5.00000000002e-313 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999999999999997e304

   1. Initial program 96.4%

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

   3. Taylor expanded in n around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

   5. Applied rewrites 95.3%

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

   7. Applied rewrites 96.6%

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

   if 4.9999999999999997e304 < (*.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 27.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 29.5

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 29.5%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 27.6%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{-313}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-\ell, \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right) \cdot \frac{\ell}{Om}, t\right) \cdot \left(n \cdot 2\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right) \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-313}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(l\_m \cdot n\right) \cdot l\_m\right) \cdot U}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right) \cdot \left(\frac{l\_m}{Om} \cdot l\_m\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right) \cdot n\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2
         (*
          (-
           (- t (* (/ (* l_m l_m) Om) 2.0))
           (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_2 5e-313)
     (sqrt (fma (/ (* (* (* l_m n) l_m) U) Om) -4.0 (* (* (* t n) U) 2.0)))
     (if (<= t_2 5e+304)
       (sqrt (* (- t (* (fma (/ n Om) (- U U*) 2.0) (* (/ l_m Om) l_m))) t_1))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (* (fma n (/ (- U* U) (* Om Om)) (/ -2.0 Om)) n) U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = ((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_2 <= 5e-313) {
		tmp = sqrt(fma(((((l_m * n) * l_m) * U) / Om), -4.0, (((t * n) * U) * 2.0)));
	} else if (t_2 <= 5e+304) {
		tmp = sqrt(((t - (fma((n / Om), (U - U_42_), 2.0) * ((l_m / Om) * l_m))) * t_1));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt(((fma(n, ((U_42_ - U) / (Om * Om)), (-2.0 / Om)) * n) * U));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(Float64(t - Float64(Float64(Float64(l_m * l_m) / Om) * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)
	tmp = 0.0
	if (t_2 <= 5e-313)
		tmp = sqrt(fma(Float64(Float64(Float64(Float64(l_m * n) * l_m) * U) / Om), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	elseif (t_2 <= 5e+304)
		tmp = sqrt(Float64(Float64(t - Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) * Float64(Float64(l_m / Om) * l_m))) * t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(fma(n, Float64(Float64(U_42_ - U) / Float64(Om * Om)), Float64(-2.0 / Om)) * n) * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-313], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * n), $MachinePrecision] * l$95$m), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+304], N[Sqrt[N[(N[(t - N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(n * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.00000000002e-313

   1. Initial program 7.0%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 38.5

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

   5. Applied rewrites 38.5%

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

   7. Applied rewrites 29.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.0%

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

   if 5.00000000002e-313 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999999999999997e304

   1. Initial program 96.4%

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

   3. Taylor expanded in n around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

   5. Applied rewrites 95.3%

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

   7. Applied rewrites 96.6%

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

   if 4.9999999999999997e304 < (*.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 27.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 29.5

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 29.5%

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

   5. Taylor expanded in U around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 29.4

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

   7. Applied rewrites 29.4%

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

   8. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

   10. Applied rewrites 20.8%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{-313}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot n\right) \cdot \ell\right) \cdot U}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right) \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right) \cdot n\right) \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.3% accurate, 0.4× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3
         (sqrt
          (*
           (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
           t_1))))
   (if (<= t_3 0.0)
     (sqrt (* (fma -4.0 (* t_2 U) (* (* t U) 2.0)) n))
     (if (<= t_3 INFINITY)
       (sqrt (* (- t (* (* (/ 2.0 Om) l_m) l_m)) t_1))
       (sqrt (* (/ (* (* (* l_m n) (* l_m n)) (* U* U)) (* Om Om)) 2.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = sqrt((((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((fma(-4.0, (t_2 * U), ((t * U) * 2.0)) * n));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((t - (((2.0 / Om) * l_m) * l_m)) * t_1));
	} else {
		tmp = sqrt((((((l_m * n) * (l_m * n)) * (U_42_ * U)) / (Om * Om)) * 2.0));
	}
	return tmp;
}

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(-4.0 * N[(t$95$2 * U), $MachinePrecision] + N[(N[(t * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(t - N[(N[(N[(2.0 / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * n), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * N[(U$42$ * U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * 2.0), $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 5.6%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 40.7%

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

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

   1. Initial program 72.8%

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

   3. Taylor expanded in n around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 66.8%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 10.4

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 10.4%

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

   5. Taylor expanded in U around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 10.4

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

   7. Applied rewrites 10.4%

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

   8. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 29.0

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

   10. Applied rewrites 29.0%

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

4. Final simplification 57.0%

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

Alternative 8: 53.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.6%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 37.7

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

   5. Applied rewrites 37.7%

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

   7. Applied rewrites 28.8%

      \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.3%

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

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

   1. Initial program 72.8%

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

   3. Taylor expanded in n around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 66.8%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 1.4

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in U around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 1.4

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

   7. Applied rewrites 1.4%

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

   8. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 28.7

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

   10. Applied rewrites 28.7%

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

4. Final simplification 57.2%

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

Alternative 9: 52.7% accurate, 0.4× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3
         (*
          (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_3 0.0)
     (sqrt (* (fma -4.0 (* t_2 U) (* (* t U) 2.0)) n))
     (if (<= t_3 INFINITY)
       (sqrt (* (- t (* (* (/ 2.0 Om) l_m) l_m)) t_1))
       (sqrt (* (/ (* (* (* n n) (* l_m l_m)) (* U* U)) (* Om Om)) 2.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((fma(-4.0, (t_2 * U), ((t * U) * 2.0)) * n));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((t - (((2.0 / Om) * l_m) * l_m)) * t_1));
	} else {
		tmp = sqrt((((((n * n) * (l_m * l_m)) * (U_42_ * U)) / (Om * Om)) * 2.0));
	}
	return tmp;
}

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(-4.0 * N[(t$95$2 * U), $MachinePrecision] + N[(N[(t * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(t - N[(N[(N[(2.0 / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(n * n), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(U$42$ * U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.6%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 37.7

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

   5. Applied rewrites 37.7%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 38.7%

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

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

   1. Initial program 72.8%

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

   3. Taylor expanded in n around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 66.8%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 1.4

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 25.6

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

   7. Applied rewrites 25.6%

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

4. Final simplification 56.5%

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

Alternative 10: 52.2% accurate, 0.4× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3
         (*
          (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1)))
   (if (<= t_3 0.0)
     (sqrt (* (fma -4.0 (* t_2 U) (* (* t U) 2.0)) n))
     (if (<= t_3 INFINITY)
       (sqrt (* (- t (* (* (/ 2.0 Om) l_m) l_m)) t_1))
       (* (/ (* (* (sqrt 2.0) n) l_m) Om) (sqrt (* U* U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = ((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((fma(-4.0, (t_2 * U), ((t * U) * 2.0)) * n));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((t - (((2.0 / Om) * l_m) * l_m)) * t_1));
	} else {
		tmp = (((sqrt(2.0) * n) * l_m) / Om) * sqrt((U_42_ * U));
	}
	return tmp;
}

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(-4.0 * N[(t$95$2 * U), $MachinePrecision] + N[(N[(t * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(t - N[(N[(N[(2.0 / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.6%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 37.7

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

   5. Applied rewrites 37.7%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 38.7%

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

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

   1. Initial program 72.8%

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

   3. Taylor expanded in n around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 66.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 17.5

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

   5. Applied rewrites 17.5%

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

4. Final simplification 55.4%

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

Alternative 11: 52.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 38.5

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

   5. Applied rewrites 38.5%

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

   if 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 72.8%

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

   3. Taylor expanded in n around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in n around 0

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

   10. Applied rewrites 66.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 17.5

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

   5. Applied rewrites 17.5%

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

4. Final simplification 55.3%

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

Alternative 12: 49.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 38.5

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

   5. Applied rewrites 38.5%

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

   if 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 72.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      2. unsub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)} - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      3. associate--r+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)}\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} + \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      10. cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)\right)} \]

   5. Applied rewrites 68.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \ell \cdot \color{blue}{\left(\ell \cdot \frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om}\right)}\right)} \]

   8. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \ell \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.8%

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \ell \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 0.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right)} \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      11. times-frac N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U - U*}{Om}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, 2 \cdot \frac{1}{Om}\right)}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{n}{Om}}, \frac{U - U*}{Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \color{blue}{\frac{U - U*}{Om}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{\color{blue}{U - U*}}{Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      16. associate-*r/ N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{\color{blue}{2}}{Om}\right)\right)} \]

      18. lower-/.f64 46.1

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \color{blue}{\frac{2}{Om}}\right)\right)} \]

   5. Applied rewrites 46.1%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot n}{Om}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 18.3%

      \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(2 \cdot \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot n}{Om}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(t - \left(\frac{2}{Om} \cdot \ell\right) \cdot \ell\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om} \cdot 2\right) \cdot \left(-2 \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ \mathbf{if}\;\sqrt{\left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1} \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, t\_2 \cdot U, \left(t \cdot U\right) \cdot 2\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right) \cdot \left(\frac{l\_m}{Om} \cdot l\_m\right)\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0))) (t_2 (/ (* l_m l_m) Om)))
   (if (<=
        (sqrt
         (* (- (- t (* t_2 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n))) t_1))
        0.0)
     (sqrt (* (fma -4.0 (* t_2 U) (* (* t U) 2.0)) n))
     (sqrt (* (- t (* (fma (/ n Om) (- U U*) 2.0) (* (/ l_m Om) l_m))) t_1)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double tmp;
	if (sqrt((((t - (t_2 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1)) <= 0.0) {
		tmp = sqrt((fma(-4.0, (t_2 * U), ((t * U) * 2.0)) * n));
	} else {
		tmp = sqrt(((t - (fma((n / Om), (U - U_42_), 2.0) * ((l_m / Om) * l_m))) * t_1));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(l_m * l_m) / Om)
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * t_1)) <= 0.0)
		tmp = sqrt(Float64(fma(-4.0, Float64(t_2 * U), Float64(Float64(t * U) * 2.0)) * n));
	else
		tmp = sqrt(Float64(Float64(t - Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) * Float64(Float64(l_m / Om) * l_m))) * t_1));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], 0.0], N[Sqrt[N[(N[(-4.0 * N[(t$95$2 * U), $MachinePrecision] + N[(N[(t * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(t - N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $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 5.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} \cdot -4} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      13. lower-*.f64 39.8

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2\right)} \]

   5. Applied rewrites 39.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{n \cdot \color{blue}{\left(-4 \cdot \frac{U \cdot {\ell}^{2}}{Om} + 2 \cdot \left(U \cdot t\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.7%

      \[\leadsto \sqrt{n \cdot \color{blue}{\mathsf{fma}\left(-4, U \cdot \frac{\ell \cdot \ell}{Om}, 2 \cdot \left(U \cdot t\right)\right)}} \]

   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 58.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      2. unsub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)} - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      3. associate--r+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)}\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} + \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      10. cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)\right)} \]

   5. Applied rewrites 60.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 64.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{\ell \cdot \ell}{Om} \cdot U, \left(t \cdot U\right) \cdot 2\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right) \cdot \left(\frac{\ell}{Om} \cdot \ell\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (*
       (-
        (- t (* (/ (* l_m l_m) Om) 2.0))
        (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
       (* U (* n 2.0)))
      0.0)
   (sqrt (* (* (* t n) U) 2.0))
   (sqrt (* (* (* U n) t) 2.0))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * (U * (n * 2.0))) <= 0.0) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((((U * n) * t) * 2.0));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((((t - (((l_m * l_m) / om) * 2.0d0)) - ((u - u_42) * (((l_m / om) ** 2.0d0) * n))) * (u * (n * 2.0d0))) <= 0.0d0) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((((u * n) * t) * 2.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (Math.pow((l_m / Om), 2.0) * n))) * (U * (n * 2.0))) <= 0.0) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((((U * n) * t) * 2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (math.pow((l_m / Om), 2.0) * n))) * (U * (n * 2.0))) <= 0.0:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((((U * n) * t) * 2.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Float64(Float64(Float64(t - Float64(Float64(Float64(l_m * l_m) / Om) * 2.0)) - Float64(Float64(U - U_42_) * Float64((Float64(l_m / Om) ^ 2.0) * n))) * Float64(U * Float64(n * 2.0))) <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(U * n) * t) * 2.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (((l_m / Om) ^ 2.0) * n))) * (U * (n * 2.0))) <= 0.0)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((((U * n) * t) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[N[(N[(N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.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 4.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 26.5

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 26.5%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 60.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 37.6

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 37.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.2%

      \[\leadsto \sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.12 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot 2 - \left(l\_m \cdot n\right) \cdot \left(\frac{U*}{Om} \cdot l\_m\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;l\_m \leq 1.35 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right) \cdot \left(\left(l\_m \cdot n\right) \cdot l\_m\right)\right) \cdot \left(-2 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right) \cdot n\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.12e+74)
   (sqrt
    (*
     (- t (/ (- (* (* l_m l_m) 2.0) (* (* l_m n) (* (/ U* Om) l_m))) Om))
     (* U (* n 2.0))))
   (if (<= l_m 1.35e+198)
     (sqrt
      (*
       (* (fma (/ n Om) (/ (- U U*) Om) (/ 2.0 Om)) (* (* l_m n) l_m))
       (* -2.0 U)))
     (*
      (* (sqrt 2.0) l_m)
      (sqrt (* (* (fma n (/ (- U* U) (* Om Om)) (/ -2.0 Om)) n) U))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.12e+74) {
		tmp = sqrt(((t - ((((l_m * l_m) * 2.0) - ((l_m * n) * ((U_42_ / Om) * l_m))) / Om)) * (U * (n * 2.0))));
	} else if (l_m <= 1.35e+198) {
		tmp = sqrt(((fma((n / Om), ((U - U_42_) / Om), (2.0 / Om)) * ((l_m * n) * l_m)) * (-2.0 * U)));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt(((fma(n, ((U_42_ - U) / (Om * Om)), (-2.0 / Om)) * n) * U));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.12e+74)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(Float64(Float64(l_m * l_m) * 2.0) - Float64(Float64(l_m * n) * Float64(Float64(U_42_ / Om) * l_m))) / Om)) * Float64(U * Float64(n * 2.0))));
	elseif (l_m <= 1.35e+198)
		tmp = sqrt(Float64(Float64(fma(Float64(n / Om), Float64(Float64(U - U_42_) / Om), Float64(2.0 / Om)) * Float64(Float64(l_m * n) * l_m)) * Float64(-2.0 * U)));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(fma(n, Float64(Float64(U_42_ - U) / Float64(Om * Om)), Float64(-2.0 / Om)) * n) * U)));
	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, 2.12e+74], N[Sqrt[N[(N[(t - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[(l$95$m * n), $MachinePrecision] * N[(N[(U$42$ / Om), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.35e+198], N[Sqrt[N[(N[(N[(N[(n / Om), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m * n), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(n * N[(N[(U$42$ - U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.11999999999999998e74

   1. Initial program 55.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(-\left(U - U*\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. lower-*.f64 58.3

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. lift--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t - 2 \cdot \frac{\ell \cdot \ell}{Om}}\right)} \]

   4. Applied rewrites 58.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{U* \cdot \ell}{Om}}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U* \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U* \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

      3. lower-/.f64 58.6

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

   7. Applied rewrites 58.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U* \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(-2 \cdot \frac{\ell \cdot \ell}{Om} + t\right)} + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right)} + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)} \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}}\right) + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      9. associate-+l- N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)}} \]

      10. lower--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)}} \]

   9. Applied rewrites 58.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right) - \left(n \cdot \ell\right) \cdot \left(\frac{U*}{Om} \cdot \ell\right)}{Om}\right)}} \]

   if 2.11999999999999998e74 < l < 1.35e198

   1. Initial program 37.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right)} \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      11. times-frac N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U - U*}{Om}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, 2 \cdot \frac{1}{Om}\right)}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{n}{Om}}, \frac{U - U*}{Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \color{blue}{\frac{U - U*}{Om}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{\color{blue}{U - U*}}{Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      16. associate-*r/ N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{\color{blue}{2}}{Om}\right)\right)} \]

      18. lower-/.f64 46.9

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \color{blue}{\frac{2}{Om}}\right)\right)} \]

   5. Applied rewrites 46.9%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.5%

      \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(n \cdot \ell\right) \cdot \ell\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{n}{Om}}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)} \]

   if 1.35e198 < l

   1. Initial program 24.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(-\left(U - U*\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. lower-*.f64 25.7

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      17. lift--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t - 2 \cdot \frac{\ell \cdot \ell}{Om}}\right)} \]

   4. Applied rewrites 25.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{U* \cdot \ell}{Om}}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U* \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U* \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

      3. lower-/.f64 25.7

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

   7. Applied rewrites 25.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U* \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \]

   8. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]

   10. Applied rewrites 92.2%

       \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right) \cdot n\right) \cdot U} \cdot \left(\sqrt{2} \cdot \ell\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.12 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot 2 - \left(\ell \cdot n\right) \cdot \left(\frac{U*}{Om} \cdot \ell\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right) \cdot \left(\left(\ell \cdot n\right) \cdot \ell\right)\right) \cdot \left(-2 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(\mathsf{fma}\left(n, \frac{U* - U}{Om \cdot Om}, \frac{-2}{Om}\right) \cdot n\right) \cdot U}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(t - \frac{\left(2 - \frac{U* \cdot n}{Om}\right) \cdot \left(l\_m \cdot l\_m\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{if}\;n \leq -3.6 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U}{Om} \cdot \left(\left(l\_m \cdot n\right) \cdot l\_m\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{+162}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t \cdot U} \cdot \sqrt{2}\right) \cdot \sqrt{n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (- t (/ (* (- 2.0 (/ (* U* n) Om)) (* l_m l_m)) Om))
           (* U (* n 2.0))))))
   (if (<= n -3.6e-41)
     t_1
     (if (<= n 7.2e-158)
       (sqrt (fma (* (/ U Om) (* (* l_m n) l_m)) -4.0 (* (* (* t n) U) 2.0)))
       (if (<= n 8e+162) t_1 (* (* (sqrt (* t U)) (sqrt 2.0)) (sqrt 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 = sqrt(((t - (((2.0 - ((U_42_ * n) / Om)) * (l_m * l_m)) / Om)) * (U * (n * 2.0))));
	double tmp;
	if (n <= -3.6e-41) {
		tmp = t_1;
	} else if (n <= 7.2e-158) {
		tmp = sqrt(fma(((U / Om) * ((l_m * n) * l_m)), -4.0, (((t * n) * U) * 2.0)));
	} else if (n <= 8e+162) {
		tmp = t_1;
	} else {
		tmp = (sqrt((t * U)) * sqrt(2.0)) * sqrt(n);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(t - Float64(Float64(Float64(2.0 - Float64(Float64(U_42_ * n) / Om)) * Float64(l_m * l_m)) / Om)) * Float64(U * Float64(n * 2.0))))
	tmp = 0.0
	if (n <= -3.6e-41)
		tmp = t_1;
	elseif (n <= 7.2e-158)
		tmp = sqrt(fma(Float64(Float64(U / Om) * Float64(Float64(l_m * n) * l_m)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	elseif (n <= 8e+162)
		tmp = t_1;
	else
		tmp = Float64(Float64(sqrt(Float64(t * U)) * sqrt(2.0)) * sqrt(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[Sqrt[N[(N[(t - N[(N[(N[(2.0 - N[(N[(U$42$ * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -3.6e-41], t$95$1, If[LessEqual[n, 7.2e-158], N[Sqrt[N[(N[(N[(U / Om), $MachinePrecision] * N[(N[(l$95$m * n), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 8e+162], t$95$1, N[(N[(N[Sqrt[N[(t * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.6e-41 or 7.19999999999999982e-158 < n < 7.9999999999999995e162

   1. Initial program 62.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      2. unsub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)} - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      3. associate--r+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)}\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} + \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      10. cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)\right)} \]

   5. Applied rewrites 64.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)}} \]

   6. Taylor expanded in U around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{{\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)}{Om}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{\left(2 - \frac{n \cdot U*}{Om}\right) \cdot \left(\ell \cdot \ell\right)}{Om}\right)} \]

   if -3.6e-41 < n < 7.19999999999999982e-158

   1. Initial program 41.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} \cdot -4} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      13. lower-*.f64 53.2

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2\right)} \]

   5. Applied rewrites 53.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 57.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 60.3%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]

   if 7.9999999999999995e162 < n

   1. Initial program 42.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      8. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{n}} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}} \]

   5. Taylor expanded in t around inf

      \[\leadsto \sqrt{n} \cdot \color{blue}{\left(\sqrt{U \cdot t} \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \color{blue}{\left(\sqrt{U \cdot t} \cdot \sqrt{2}\right)} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{n} \cdot \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{2}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \left(\sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{2}\right) \]

      4. lower-sqrt.f64 60.3

         \[\leadsto \sqrt{n} \cdot \left(\sqrt{U \cdot t} \cdot \color{blue}{\sqrt{2}}\right) \]

   7. Applied rewrites 60.3%

      \[\leadsto \sqrt{n} \cdot \color{blue}{\left(\sqrt{U \cdot t} \cdot \sqrt{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(2 - \frac{U* \cdot n}{Om}\right) \cdot \left(\ell \cdot \ell\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U}{Om} \cdot \left(\left(\ell \cdot n\right) \cdot \ell\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(2 - \frac{U* \cdot n}{Om}\right) \cdot \left(\ell \cdot \ell\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t \cdot U} \cdot \sqrt{2}\right) \cdot \sqrt{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;n \leq -3.6 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(2 - \frac{U* \cdot n}{Om}\right) \cdot \left(l\_m \cdot l\_m\right)}{Om}\right) \cdot t\_1}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U}{Om} \cdot \left(\left(l\_m \cdot n\right) \cdot l\_m\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \frac{\mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right) \cdot l\_m}{Om} \cdot l\_m\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0))))
   (if (<= n -3.6e-41)
     (sqrt (* (- t (/ (* (- 2.0 (/ (* U* n) Om)) (* l_m l_m)) Om)) t_1))
     (if (<= n 4.5e-164)
       (sqrt (fma (* (/ U Om) (* (* l_m n) l_m)) -4.0 (* (* (* t n) U) 2.0)))
       (sqrt
        (* (- t (* (/ (* (fma n (/ (- U U*) Om) 2.0) l_m) Om) l_m)) t_1))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double tmp;
	if (n <= -3.6e-41) {
		tmp = sqrt(((t - (((2.0 - ((U_42_ * n) / Om)) * (l_m * l_m)) / Om)) * t_1));
	} else if (n <= 4.5e-164) {
		tmp = sqrt(fma(((U / Om) * ((l_m * n) * l_m)), -4.0, (((t * n) * U) * 2.0)));
	} else {
		tmp = sqrt(((t - (((fma(n, ((U - U_42_) / Om), 2.0) * l_m) / Om) * l_m)) * t_1));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	tmp = 0.0
	if (n <= -3.6e-41)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(Float64(2.0 - Float64(Float64(U_42_ * n) / Om)) * Float64(l_m * l_m)) / Om)) * t_1));
	elseif (n <= 4.5e-164)
		tmp = sqrt(fma(Float64(Float64(U / Om) * Float64(Float64(l_m * n) * l_m)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	else
		tmp = sqrt(Float64(Float64(t - Float64(Float64(Float64(fma(n, Float64(Float64(U - U_42_) / Om), 2.0) * l_m) / Om) * l_m)) * t_1));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.6e-41], N[Sqrt[N[(N[(t - N[(N[(N[(2.0 - N[(N[(U$42$ * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 4.5e-164], N[Sqrt[N[(N[(N[(U / Om), $MachinePrecision] * N[(N[(l$95$m * n), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(t - N[(N[(N[(N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision] * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.6e-41

   1. Initial program 54.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      2. unsub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)} - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      3. associate--r+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)}\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} + \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      10. cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)\right)} \]

   5. Applied rewrites 60.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)}} \]

   6. Taylor expanded in U around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{{\ell}^{2} \cdot \left(2 + -1 \cdot \frac{U* \cdot n}{Om}\right)}{Om}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{\left(2 - \frac{n \cdot U*}{Om}\right) \cdot \left(\ell \cdot \ell\right)}{Om}\right)} \]

   if -3.6e-41 < n < 4.4999999999999997e-164

   1. Initial program 42.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} \cdot -4} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      13. lower-*.f64 53.3

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2\right)} \]

   5. Applied rewrites 53.3%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 57.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 60.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]

   if 4.4999999999999997e-164 < n

   1. Initial program 60.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      2. unsub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)} - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      3. associate--r+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)}\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} + \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      10. cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)\right)} \]

   5. Applied rewrites 59.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \ell \cdot \color{blue}{\left(\ell \cdot \frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om}\right)}\right)} \]

   8. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \ell \cdot \frac{\ell \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)}{\color{blue}{Om}}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 63.8%

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \ell \cdot \frac{\ell \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{\color{blue}{Om}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(2 - \frac{U* \cdot n}{Om}\right) \cdot \left(\ell \cdot \ell\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U}{Om} \cdot \left(\left(\ell \cdot n\right) \cdot \ell\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \frac{\mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right) \cdot \ell}{Om} \cdot \ell\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 51.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;n \leq -9.2 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(\left(l\_m \cdot l\_m\right) \cdot n\right) \cdot \left(U - U*\right)}{Om \cdot Om}\right) \cdot t\_1}\\ \mathbf{elif}\;n \leq 9.8 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U}{Om} \cdot \left(\left(l\_m \cdot n\right) \cdot l\_m\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \left(\frac{n}{Om \cdot Om} \cdot \left(l\_m \cdot l\_m\right)\right) \cdot \left(-U*\right)\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0))))
   (if (<= n -9.2e-39)
     (sqrt (* (- t (/ (* (* (* l_m l_m) n) (- U U*)) (* Om Om))) t_1))
     (if (<= n 9.8e+44)
       (sqrt (fma (* (/ U Om) (* (* l_m n) l_m)) -4.0 (* (* (* t n) U) 2.0)))
       (sqrt (* (- t (* (* (/ n (* Om Om)) (* l_m l_m)) (- U*))) t_1))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double tmp;
	if (n <= -9.2e-39) {
		tmp = sqrt(((t - ((((l_m * l_m) * n) * (U - U_42_)) / (Om * Om))) * t_1));
	} else if (n <= 9.8e+44) {
		tmp = sqrt(fma(((U / Om) * ((l_m * n) * l_m)), -4.0, (((t * n) * U) * 2.0)));
	} else {
		tmp = sqrt(((t - (((n / (Om * Om)) * (l_m * l_m)) * -U_42_)) * t_1));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	tmp = 0.0
	if (n <= -9.2e-39)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(Float64(Float64(l_m * l_m) * n) * Float64(U - U_42_)) / Float64(Om * Om))) * t_1));
	elseif (n <= 9.8e+44)
		tmp = sqrt(fma(Float64(Float64(U / Om) * Float64(Float64(l_m * n) * l_m)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	else
		tmp = sqrt(Float64(Float64(t - Float64(Float64(Float64(n / Float64(Om * Om)) * Float64(l_m * l_m)) * Float64(-U_42_))) * t_1));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9.2e-39], N[Sqrt[N[(N[(t - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 9.8e+44], N[Sqrt[N[(N[(N[(U / Om), $MachinePrecision] * N[(N[(l$95$m * n), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(t - N[(N[(N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * (-U$42$)), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.20000000000000033e-39

   1. Initial program 54.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      2. unsub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)} - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      3. associate--r+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)}\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} + \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      10. cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)\right)} \]

   5. Applied rewrites 60.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)}} \]

   6. Taylor expanded in n around -inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{{Om}^{2}}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}\right)} \]

   if -9.20000000000000033e-39 < n < 9.80000000000000071e44

   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 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

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} \cdot -4} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      13. lower-*.f64 55.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2\right)} \]

   5. Applied rewrites 55.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 60.2%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]

   if 9.80000000000000071e44 < n

   1. Initial program 55.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      2. unsub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)} - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      3. associate--r+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)}\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} + \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      10. cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)\right)} \]

   5. Applied rewrites 50.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)}} \]

   6. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - -1 \cdot \color{blue}{\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(-U*\right) \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om \cdot Om}\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.2 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U - U*\right)}{Om \cdot Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;n \leq 9.8 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U}{Om} \cdot \left(\left(\ell \cdot n\right) \cdot \ell\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \left(\frac{n}{Om \cdot Om} \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(-U*\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 51.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(t - \left(\frac{n}{Om \cdot Om} \cdot \left(l\_m \cdot l\_m\right)\right) \cdot \left(-U*\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{if}\;n \leq -3.6 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 9.8 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U}{Om} \cdot \left(\left(l\_m \cdot n\right) \cdot l\_m\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (- t (* (* (/ n (* Om Om)) (* l_m l_m)) (- U*)))
           (* U (* n 2.0))))))
   (if (<= n -3.6e-41)
     t_1
     (if (<= n 9.8e+44)
       (sqrt (fma (* (/ U Om) (* (* l_m n) l_m)) -4.0 (* (* (* t n) U) 2.0)))
       t_1))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt(((t - (((n / (Om * Om)) * (l_m * l_m)) * -U_42_)) * (U * (n * 2.0))));
	double tmp;
	if (n <= -3.6e-41) {
		tmp = t_1;
	} else if (n <= 9.8e+44) {
		tmp = sqrt(fma(((U / Om) * ((l_m * n) * l_m)), -4.0, (((t * n) * U) * 2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(t - Float64(Float64(Float64(n / Float64(Om * Om)) * Float64(l_m * l_m)) * Float64(-U_42_))) * Float64(U * Float64(n * 2.0))))
	tmp = 0.0
	if (n <= -3.6e-41)
		tmp = t_1;
	elseif (n <= 9.8e+44)
		tmp = sqrt(fma(Float64(Float64(U / Om) * Float64(Float64(l_m * n) * l_m)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(t - N[(N[(N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * (-U$42$)), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -3.6e-41], t$95$1, If[LessEqual[n, 9.8e+44], N[Sqrt[N[(N[(N[(U / Om), $MachinePrecision] * N[(N[(l$95$m * n), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.6e-41 or 9.80000000000000071e44 < n

   1. Initial program 54.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t + -1 \cdot \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}\right) - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      2. unsub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)} - 2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      3. associate--r+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)}\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} + \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      10. cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      11. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)\right)} \]

   5. Applied rewrites 56.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)}} \]

   6. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - -1 \cdot \color{blue}{\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(-U*\right) \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om \cdot Om}\right)}\right)} \]

   if -3.6e-41 < n < 9.80000000000000071e44

   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 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

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} \cdot -4} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      13. lower-*.f64 55.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2\right)} \]

   5. Applied rewrites 55.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 60.2%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{\left(t - \left(\frac{n}{Om \cdot Om} \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(-U*\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;n \leq 9.8 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U}{Om} \cdot \left(\left(\ell \cdot n\right) \cdot \ell\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \left(\frac{n}{Om \cdot Om} \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(-U*\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 43.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{l\_m \cdot l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t \cdot U} \cdot \sqrt{2}\right) \cdot \sqrt{n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 2e+163)
   (sqrt (* (* (* (fma -2.0 (/ (* l_m l_m) Om) t) n) U) 2.0))
   (* (* (sqrt (* t U)) (sqrt 2.0)) (sqrt 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 <= 2e+163) {
		tmp = sqrt((((fma(-2.0, ((l_m * l_m) / Om), t) * n) * U) * 2.0));
	} else {
		tmp = (sqrt((t * U)) * sqrt(2.0)) * sqrt(n);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 2e+163)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(Float64(l_m * l_m) / Om), t) * n) * U) * 2.0));
	else
		tmp = Float64(Float64(sqrt(Float64(t * U)) * sqrt(2.0)) * sqrt(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, 2e+163], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(t * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.9999999999999999e163

   1. Initial program 53.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right)} \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U\right) \cdot 2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U\right) \cdot 2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot n\right) \cdot U\right) \cdot 2} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      13. lower-*.f64 51.9

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

   5. Applied rewrites 51.9%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}} \]

   if 1.9999999999999999e163 < n

   1. Initial program 42.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot 2\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{n \cdot \left(2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      8. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{n}} \cdot \sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}} \]

   5. Taylor expanded in t around inf

      \[\leadsto \sqrt{n} \cdot \color{blue}{\left(\sqrt{U \cdot t} \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \color{blue}{\left(\sqrt{U \cdot t} \cdot \sqrt{2}\right)} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{n} \cdot \left(\color{blue}{\sqrt{U \cdot t}} \cdot \sqrt{2}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \left(\sqrt{\color{blue}{U \cdot t}} \cdot \sqrt{2}\right) \]

      4. lower-sqrt.f64 60.3

         \[\leadsto \sqrt{n} \cdot \left(\sqrt{U \cdot t} \cdot \color{blue}{\sqrt{2}}\right) \]

   7. Applied rewrites 60.3%

      \[\leadsto \sqrt{n} \cdot \color{blue}{\left(\sqrt{U \cdot t} \cdot \sqrt{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t \cdot U} \cdot \sqrt{2}\right) \cdot \sqrt{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 39.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.06 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left(l\_m \cdot l\_m\right) \cdot n}{Om} \cdot 2\right) \cdot \left(-2 \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.06e+40)
   (sqrt (* (* (* t U) n) 2.0))
   (sqrt (* (* (/ (* (* l_m l_m) n) Om) 2.0) (* -2.0 U)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.06e+40) {
		tmp = sqrt((((t * U) * n) * 2.0));
	} else {
		tmp = sqrt((((((l_m * l_m) * n) / Om) * 2.0) * (-2.0 * U)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.06d+40) then
        tmp = sqrt((((t * u) * n) * 2.0d0))
    else
        tmp = sqrt((((((l_m * l_m) * n) / om) * 2.0d0) * ((-2.0d0) * u)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.06e+40) {
		tmp = Math.sqrt((((t * U) * n) * 2.0));
	} else {
		tmp = Math.sqrt((((((l_m * l_m) * n) / Om) * 2.0) * (-2.0 * U)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.06e+40:
		tmp = math.sqrt((((t * U) * n) * 2.0))
	else:
		tmp = math.sqrt((((((l_m * l_m) * n) / Om) * 2.0) * (-2.0 * U)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.06e+40)
		tmp = sqrt(Float64(Float64(Float64(t * U) * n) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l_m * l_m) * n) / Om) * 2.0) * Float64(-2.0 * U)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.06e+40)
		tmp = sqrt((((t * U) * n) * 2.0));
	else
		tmp = sqrt((((((l_m * l_m) * n) / Om) * 2.0) * (-2.0 * U)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.06e+40], N[Sqrt[N[(N[(N[(t * U), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision] * N[(-2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.05999999999999996e40

   1. Initial program 55.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 40.6

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 40.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.7%

      \[\leadsto \sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2} \]

   if 1.05999999999999996e40 < l

   1. Initial program 37.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right)} \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      11. times-frac N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U - U*}{Om}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, 2 \cdot \frac{1}{Om}\right)}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{n}{Om}}, \frac{U - U*}{Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \color{blue}{\frac{U - U*}{Om}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{\color{blue}{U - U*}}{Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      16. associate-*r/ N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{\color{blue}{2}}{Om}\right)\right)} \]

      18. lower-/.f64 44.5

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \color{blue}{\frac{2}{Om}}\right)\right)} \]

   5. Applied rewrites 44.5%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot n}{Om}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.6%

      \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(2 \cdot \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot n}{Om}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.06 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left(\ell \cdot \ell\right) \cdot n}{Om} \cdot 2\right) \cdot \left(-2 \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 39.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.06 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(l\_m \cdot l\_m\right) \cdot n\right) \cdot U}{Om} \cdot -4}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.06e+40)
   (sqrt (* (* (* t U) n) 2.0))
   (sqrt (* (/ (* (* (* l_m l_m) n) U) Om) -4.0))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.06e+40) {
		tmp = sqrt((((t * U) * n) * 2.0));
	} else {
		tmp = sqrt((((((l_m * l_m) * n) * U) / Om) * -4.0));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.06d+40) then
        tmp = sqrt((((t * u) * n) * 2.0d0))
    else
        tmp = sqrt((((((l_m * l_m) * n) * u) / om) * (-4.0d0)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.06e+40) {
		tmp = Math.sqrt((((t * U) * n) * 2.0));
	} else {
		tmp = Math.sqrt((((((l_m * l_m) * n) * U) / Om) * -4.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.06e+40:
		tmp = math.sqrt((((t * U) * n) * 2.0))
	else:
		tmp = math.sqrt((((((l_m * l_m) * n) * U) / Om) * -4.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.06e+40)
		tmp = sqrt(Float64(Float64(Float64(t * U) * n) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l_m * l_m) * n) * U) / Om) * -4.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.06e+40)
		tmp = sqrt((((t * U) * n) * 2.0));
	else
		tmp = sqrt((((((l_m * l_m) * n) * U) / Om) * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.06e+40], N[Sqrt[N[(N[(N[(t * U), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.05999999999999996e40

   1. Initial program 55.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 40.6

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 40.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.7%

      \[\leadsto \sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2} \]

   if 1.05999999999999996e40 < l

   1. Initial program 37.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} \cdot -4} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right) \cdot U}}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot U}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2\right)} \]

      13. lower-*.f64 41.7

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2\right)} \]

   5. Applied rewrites 41.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}, -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 38.7%

      \[\leadsto \sqrt{\mathsf{fma}\left(\ell \cdot \left(\left(n \cdot \ell\right) \cdot \frac{U}{Om}\right), -4, \left(\left(n \cdot t\right) \cdot U\right) \cdot 2\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto \sqrt{-4 \cdot \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 38.1%

       \[\leadsto \sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om} \cdot \color{blue}{-4}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 38.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.3 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(l\_m \cdot l\_m\right) \cdot U\right) \cdot n}{Om} \cdot -4}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.3e+40)
   (sqrt (* (* (* t U) n) 2.0))
   (sqrt (* (/ (* (* (* l_m l_m) U) n) Om) -4.0))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.3e+40) {
		tmp = sqrt((((t * U) * n) * 2.0));
	} else {
		tmp = sqrt((((((l_m * l_m) * U) * n) / Om) * -4.0));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.3d+40) then
        tmp = sqrt((((t * u) * n) * 2.0d0))
    else
        tmp = sqrt((((((l_m * l_m) * u) * n) / om) * (-4.0d0)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.3e+40) {
		tmp = Math.sqrt((((t * U) * n) * 2.0));
	} else {
		tmp = Math.sqrt((((((l_m * l_m) * U) * n) / Om) * -4.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.3e+40:
		tmp = math.sqrt((((t * U) * n) * 2.0))
	else:
		tmp = math.sqrt((((((l_m * l_m) * U) * n) / Om) * -4.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.3e+40)
		tmp = sqrt(Float64(Float64(Float64(t * U) * n) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l_m * l_m) * U) * n) / Om) * -4.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.3e+40)
		tmp = sqrt((((t * U) * n) * 2.0));
	else
		tmp = sqrt((((((l_m * l_m) * U) * n) / Om) * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.3e+40], N[Sqrt[N[(N[(N[(t * U), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * U), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e40

   1. Initial program 55.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

      5. lower-*.f64 40.6

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 40.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.7%

      \[\leadsto \sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2} \]

   if 1.3e40 < l

   1. Initial program 37.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right)} \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left({\ell}^{2} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\color{blue}{\left({\ell}^{2} \cdot n\right)} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      11. times-frac N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(\color{blue}{\frac{n}{Om} \cdot \frac{U - U*}{Om}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, 2 \cdot \frac{1}{Om}\right)}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{n}{Om}}, \frac{U - U*}{Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \color{blue}{\frac{U - U*}{Om}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{\color{blue}{U - U*}}{Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      16. associate-*r/ N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{\color{blue}{2}}{Om}\right)\right)} \]

      18. lower-/.f64 44.5

          \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \color{blue}{\frac{2}{Om}}\right)\right)} \]

   5. Applied rewrites 44.5%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(\frac{n}{Om}, \frac{U - U*}{Om}, \frac{2}{Om}\right)\right)}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{-4 \cdot \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.1%

      \[\leadsto \sqrt{-4 \cdot \color{blue}{\frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot U\right) \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om} \cdot -4}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 35.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* (* U n) t) 2.0)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt((((U * n) * t) * 2.0));
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((u * n) * t) * 2.0d0))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt((((U * n) * t) * 2.0));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((((U * n) * t) * 2.0))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(Float64(U * n) * t) * 2.0))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((((U * n) * t) * 2.0));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
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. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

   3. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

   5. lower-*.f64 35.9

      \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

5. Applied rewrites 35.9%

   \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
6. Step-by-step derivation

7. Applied rewrites 36.5%

   \[\leadsto \sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot 2} \]

8. Final simplification 36.5%

   \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot t\right) \cdot 2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(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*))))))