Toniolo and Linder, Equation (13)

Percentage Accurate: 50.0% → 67.8%

Time: 18.0s

Alternatives: 21

Speedup: 2.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.0% 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: 67.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \mathsf{fma}\left(-2, t\_2, t\right)\\ t_4 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_5 := \sqrt{\left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left(t\_4 \cdot n\right)\right) \cdot t\_1}\\ \mathbf{if}\;t\_5 \leq 0:\\ \;\;\;\;\sqrt{\left(t\_3 - \left(\left(U - U*\right) \cdot n\right) \cdot t\_4\right) \cdot \left(U \cdot 2\right)} \cdot \sqrt{n}\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{l\_m}{Om} \cdot n\right) \cdot \left(U* - U\right), \frac{l\_m}{Om}, t\_3\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l_m l_m) Om))
        (t_3 (fma -2.0 t_2 t))
        (t_4 (pow (/ l_m Om) 2.0))
        (t_5 (sqrt (* (- (- t (* t_2 2.0)) (* (- U U*) (* t_4 n))) t_1))))
   (if (<= t_5 0.0)
     (* (sqrt (* (- t_3 (* (* (- U U*) n) t_4)) (* U 2.0))) (sqrt n))
     (if (<= t_5 5e+153)
       (sqrt (* (fma (* (* (/ l_m Om) n) (- U* U)) (/ l_m Om) t_3) t_1))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (- (* (/ (- U* U) Om) (/ n Om)) (/ 2.0 Om)) (* U n))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (l_m * l_m) / Om;
	double t_3 = fma(-2.0, t_2, t);
	double t_4 = pow((l_m / Om), 2.0);
	double t_5 = sqrt((((t - (t_2 * 2.0)) - ((U - U_42_) * (t_4 * n))) * t_1));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = sqrt(((t_3 - (((U - U_42_) * n) * t_4)) * (U * 2.0))) * sqrt(n);
	} else if (t_5 <= 5e+153) {
		tmp = sqrt((fma((((l_m / Om) * n) * (U_42_ - U)), (l_m / Om), t_3) * t_1));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt((((((U_42_ - U) / Om) * (n / Om)) - (2.0 / Om)) * (U * n)));
	}
	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 = fma(-2.0, t_2, t)
	t_4 = Float64(l_m / Om) ^ 2.0
	t_5 = sqrt(Float64(Float64(Float64(t - Float64(t_2 * 2.0)) - Float64(Float64(U - U_42_) * Float64(t_4 * n))) * t_1))
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(t_3 - Float64(Float64(Float64(U - U_42_) * n) * t_4)) * Float64(U * 2.0))) * sqrt(n));
	elseif (t_5 <= 5e+153)
		tmp = sqrt(Float64(fma(Float64(Float64(Float64(l_m / Om) * n) * Float64(U_42_ - U)), Float64(l_m / Om), t_3) * t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / Om)) - Float64(2.0 / Om)) * Float64(U * n))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(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[(-2.0 * t$95$2 + t), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(N[(N[(t - N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(t$95$4 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 0.0], N[(N[Sqrt[N[(N[(t$95$3 - N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 5e+153], N[Sqrt[N[(N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.7%

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

      1. lift-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 50.3%

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

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

   1. Initial program 96.5%

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

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 97.6

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in U* around 0

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

      1. lower--.f64 97.6

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

   7. Applied rewrites 97.6%

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

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

   1. Initial program 22.4%

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

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 25.3

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 25.3%

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

   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 29.7%

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

4. Final simplification 61.3%

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

Alternative 2: 67.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\\ t_2 := U \cdot \left(n \cdot 2\right)\\ t_3 := \frac{l\_m \cdot l\_m}{Om}\\ t_4 := \sqrt{\left(\left(t - t\_3 \cdot 2\right) - \left(U - U*\right) \cdot t\_1\right) \cdot t\_2}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-U, t\_1, \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\right) \cdot U\right) \cdot 2} \cdot \sqrt{n}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{l\_m}{Om} \cdot n\right) \cdot \left(U* - U\right), \frac{l\_m}{Om}, \mathsf{fma}\left(-2, t\_3, t\right)\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (pow (/ l_m Om) 2.0) n))
        (t_2 (* U (* n 2.0)))
        (t_3 (/ (* l_m l_m) Om))
        (t_4 (sqrt (* (- (- t (* t_3 2.0)) (* (- U U*) t_1)) t_2))))
   (if (<= t_4 0.0)
     (*
      (sqrt (* (* (fma (- U) t_1 (fma (* (/ l_m Om) l_m) -2.0 t)) U) 2.0))
      (sqrt n))
     (if (<= t_4 5e+153)
       (sqrt
        (*
         (fma (* (* (/ l_m Om) n) (- U* U)) (/ l_m Om) (fma -2.0 t_3 t))
         t_2))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (- (* (/ (- U* U) Om) (/ n Om)) (/ 2.0 Om)) (* U 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 = pow((l_m / Om), 2.0) * n;
	double t_2 = U * (n * 2.0);
	double t_3 = (l_m * l_m) / Om;
	double t_4 = sqrt((((t - (t_3 * 2.0)) - ((U - U_42_) * t_1)) * t_2));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((fma(-U, t_1, fma(((l_m / Om) * l_m), -2.0, t)) * U) * 2.0)) * sqrt(n);
	} else if (t_4 <= 5e+153) {
		tmp = sqrt((fma((((l_m / Om) * n) * (U_42_ - U)), (l_m / Om), fma(-2.0, t_3, t)) * t_2));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt((((((U_42_ - U) / Om) * (n / Om)) - (2.0 / Om)) * (U * n)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64((Float64(l_m / Om) ^ 2.0) * n)
	t_2 = Float64(U * Float64(n * 2.0))
	t_3 = Float64(Float64(l_m * l_m) / Om)
	t_4 = sqrt(Float64(Float64(Float64(t - Float64(t_3 * 2.0)) - Float64(Float64(U - U_42_) * t_1)) * t_2))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(fma(Float64(-U), t_1, fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)) * U) * 2.0)) * sqrt(n));
	elseif (t_4 <= 5e+153)
		tmp = sqrt(Float64(fma(Float64(Float64(Float64(l_m / Om) * n) * Float64(U_42_ - U)), Float64(l_m / Om), fma(-2.0, t_3, t)) * t_2));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / Om)) - Float64(2.0 / Om)) * Float64(U * 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[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $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[Sqrt[N[(N[(N[(t - N[(t$95$3 * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(N[(N[((-U) * t$95$1 + N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+153], N[Sqrt[N[(N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + 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[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.7%

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

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 5.7

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 5.7%

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

   5. Taylor expanded in U* around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 5.7

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

   7. Applied rewrites 5.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

      9. lower-*.f64 N/A

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

   9. Applied rewrites 48.7%

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

   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*))))) < 5.00000000000000018e153

   1. Initial program 96.5%

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

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 97.6

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in U* around 0

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

      1. lower--.f64 97.6

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

   7. Applied rewrites 97.6%

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

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

   1. Initial program 22.4%

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

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 25.3

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 25.3%

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

   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 29.7%

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

4. Final simplification 61.1%

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

Alternative 3: 67.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \sqrt{\left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{l\_m}{Om} \cdot n\right) \cdot \left(U* - U\right), \frac{l\_m}{Om}, \mathsf{fma}\left(-2, t\_2, t\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 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 (* n 2.0)) (sqrt (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))
     (if (<= t_3 5e+153)
       (sqrt
        (*
         (fma (* (* (/ l_m Om) n) (- U* U)) (/ l_m Om) (fma -2.0 t_2 t))
         t_1))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (- (* (/ (- U* U) Om) (/ n Om)) (/ 2.0 Om)) (* U n))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (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((n * 2.0)) * sqrt((fma(((l_m / Om) * l_m), -2.0, t) * U));
	} else if (t_3 <= 5e+153) {
		tmp = sqrt((fma((((l_m / Om) * n) * (U_42_ - U)), (l_m / Om), fma(-2.0, t_2, t)) * t_1));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt((((((U_42_ - U) / Om) * (n / Om)) - (2.0 / Om)) * (U * n)));
	}
	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 = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)));
	elseif (t_3 <= 5e+153)
		tmp = sqrt(Float64(fma(Float64(Float64(Float64(l_m / Om) * n) * Float64(U_42_ - U)), Float64(l_m / Om), fma(-2.0, t_2, t)) * t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / Om)) - Float64(2.0 / Om)) * Float64(U * n))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(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[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+153], N[Sqrt[N[(N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $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[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.7%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 29.2

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

   5. Applied rewrites 29.2%

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

   7. Applied rewrites 48.5%

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

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

   1. Initial program 96.5%

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

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 97.6

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in U* around 0

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

      1. lower--.f64 97.6

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

   7. Applied rewrites 97.6%

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

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

   1. Initial program 22.4%

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

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 25.3

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 25.3%

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

   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 29.7%

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

4. Final simplification 61.1%

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

Alternative 4: 64.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m}{Om} \cdot l\_m\\ t_2 := U \cdot \left(n \cdot 2\right)\\ t_3 := \sqrt{\left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_2}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{\mathsf{fma}\left(t\_1, -2, t\right) \cdot U}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot 2 - \left(\left(U* - U\right) \cdot n\right) \cdot t\_1}{Om}\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot l\_m\right) \cdot \sqrt{\left(\frac{U* - U}{Om} \cdot \frac{n}{Om} - \frac{2}{Om}\right) \cdot \left(U \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (/ l_m Om) l_m))
        (t_2 (* U (* n 2.0)))
        (t_3
         (sqrt
          (*
           (-
            (- t (* (/ (* l_m l_m) Om) 2.0))
            (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
           t_2))))
   (if (<= t_3 0.0)
     (* (sqrt (* n 2.0)) (sqrt (* (fma t_1 -2.0 t) U)))
     (if (<= t_3 5e+153)
       (sqrt
        (* (- t (/ (- (* (* l_m l_m) 2.0) (* (* (- U* U) n) t_1)) Om)) t_2))
       (*
        (* (sqrt 2.0) l_m)
        (sqrt (* (- (* (/ (- U* U) Om) (/ n Om)) (/ 2.0 Om)) (* U 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 = (l_m / Om) * l_m;
	double t_2 = U * (n * 2.0);
	double t_3 = sqrt((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_2));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((n * 2.0)) * sqrt((fma(t_1, -2.0, t) * U));
	} else if (t_3 <= 5e+153) {
		tmp = sqrt(((t - ((((l_m * l_m) * 2.0) - (((U_42_ - U) * n) * t_1)) / Om)) * t_2));
	} else {
		tmp = (sqrt(2.0) * l_m) * sqrt((((((U_42_ - U) / Om) * (n / Om)) - (2.0 / Om)) * (U * n)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m / Om) * l_m)
	t_2 = Float64(U * Float64(n * 2.0))
	t_3 = sqrt(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_2))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(fma(t_1, -2.0, t) * U)));
	elseif (t_3 <= 5e+153)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(Float64(Float64(l_m * l_m) * 2.0) - Float64(Float64(Float64(U_42_ - U) * n) * t_1)) / Om)) * t_2));
	else
		tmp = Float64(Float64(sqrt(2.0) * l_m) * sqrt(Float64(Float64(Float64(Float64(Float64(U_42_ - U) / Om) * Float64(n / Om)) - Float64(2.0 / Om)) * Float64(U * 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[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[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$2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(t$95$1 * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+153], N[Sqrt[N[(N[(t - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[(N[(U$42$ - U), $MachinePrecision] * n), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.7%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 29.2

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

   5. Applied rewrites 29.2%

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

   7. Applied rewrites 48.5%

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

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

   1. Initial program 96.5%

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 96.5

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 94.7%

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

   6. Applied rewrites 93.3%

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

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

   1. Initial program 22.4%

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

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 25.3

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 25.3%

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

   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 29.7%

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

4. Final simplification 59.2%

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

Alternative 5: 58.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m}{Om} \cdot l\_m\\ t_2 := U \cdot \left(n \cdot 2\right)\\ t_3 := \sqrt{\left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_2}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{\mathsf{fma}\left(t\_1, -2, t\right) \cdot U}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot 2 - \left(\left(U* - U\right) \cdot n\right) \cdot t\_1}{Om}\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right) \cdot t\_1\right) \cdot n\right) \cdot \left(-2 \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (/ l_m Om) l_m))
        (t_2 (* U (* n 2.0)))
        (t_3
         (sqrt
          (*
           (-
            (- t (* (/ (* l_m l_m) Om) 2.0))
            (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
           t_2))))
   (if (<= t_3 0.0)
     (* (sqrt (* n 2.0)) (sqrt (* (fma t_1 -2.0 t) U)))
     (if (<= t_3 5e+153)
       (sqrt
        (* (- t (/ (- (* (* l_m l_m) 2.0) (* (* (- U* U) n) t_1)) Om)) t_2))
       (sqrt (* (* (* (fma (/ n Om) (- U U*) 2.0) t_1) n) (* -2.0 U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m / Om) * l_m;
	double t_2 = U * (n * 2.0);
	double t_3 = sqrt((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_2));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((n * 2.0)) * sqrt((fma(t_1, -2.0, t) * U));
	} else if (t_3 <= 5e+153) {
		tmp = sqrt(((t - ((((l_m * l_m) * 2.0) - (((U_42_ - U) * n) * t_1)) / Om)) * t_2));
	} else {
		tmp = sqrt((((fma((n / Om), (U - U_42_), 2.0) * t_1) * n) * (-2.0 * U)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m / Om) * l_m)
	t_2 = Float64(U * Float64(n * 2.0))
	t_3 = sqrt(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_2))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(fma(t_1, -2.0, t) * U)));
	elseif (t_3 <= 5e+153)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(Float64(Float64(l_m * l_m) * 2.0) - Float64(Float64(Float64(U_42_ - U) * n) * t_1)) / Om)) * t_2));
	else
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) * t_1) * n) * Float64(-2.0 * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[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$2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(t$95$1 * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+153], N[Sqrt[N[(N[(t - N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[(N[(U$42$ - U), $MachinePrecision] * n), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * n), $MachinePrecision] * N[(-2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.7%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 29.2

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

   5. Applied rewrites 29.2%

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

   7. Applied rewrites 48.5%

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

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

   1. Initial program 96.5%

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 96.5

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 94.7%

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

   6. Applied rewrites 93.3%

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

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

   1. Initial program 22.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 30.2%

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

   7. Applied rewrites 40.3%

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

4. Final simplification 64.0%

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

Alternative 6: 56.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m}{Om} \cdot l\_m\\ t_2 := U \cdot \left(n \cdot 2\right)\\ t_3 := \frac{l\_m \cdot l\_m}{Om}\\ t_4 := \sqrt{\left(\left(t - t\_3 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_2}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{\mathsf{fma}\left(t\_1, -2, t\right) \cdot U}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_3, t\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right) \cdot t\_1\right) \cdot n\right) \cdot \left(-2 \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (/ l_m Om) l_m))
        (t_2 (* U (* n 2.0)))
        (t_3 (/ (* l_m l_m) Om))
        (t_4
         (sqrt
          (*
           (- (- t (* t_3 2.0)) (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
           t_2))))
   (if (<= t_4 0.0)
     (* (sqrt (* n 2.0)) (sqrt (* (fma t_1 -2.0 t) U)))
     (if (<= t_4 5e+153)
       (sqrt (* (fma -2.0 t_3 t) t_2))
       (sqrt (* (* (* (fma (/ n Om) (- U U*) 2.0) t_1) n) (* -2.0 U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m / Om) * l_m;
	double t_2 = U * (n * 2.0);
	double t_3 = (l_m * l_m) / Om;
	double t_4 = sqrt((((t - (t_3 * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_2));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt((n * 2.0)) * sqrt((fma(t_1, -2.0, t) * U));
	} else if (t_4 <= 5e+153) {
		tmp = sqrt((fma(-2.0, t_3, t) * t_2));
	} else {
		tmp = sqrt((((fma((n / Om), (U - U_42_), 2.0) * t_1) * n) * (-2.0 * U)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m / Om) * l_m)
	t_2 = Float64(U * Float64(n * 2.0))
	t_3 = Float64(Float64(l_m * l_m) / Om)
	t_4 = sqrt(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 <= 0.0)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(fma(t_1, -2.0, t) * U)));
	elseif (t_4 <= 5e+153)
		tmp = sqrt(Float64(fma(-2.0, t_3, t) * t_2));
	else
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) * t_1) * n) * Float64(-2.0 * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $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[Sqrt[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]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(t$95$1 * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+153], N[Sqrt[N[(N[(-2.0 * t$95$3 + t), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * n), $MachinePrecision] * N[(-2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.7%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 29.2

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

   5. Applied rewrites 29.2%

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

   7. Applied rewrites 48.5%

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

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

   1. Initial program 96.5%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

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

   1. Initial program 22.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 30.2%

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

   7. Applied rewrites 40.3%

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

4. Final simplification 61.2%

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

Alternative 7: 54.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[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]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$1 * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(t$95$1 * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(U / Om), $MachinePrecision] * N[(N[(n * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision] * N[(U$42$ * l$95$m), $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 5.7%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 29.2

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

   5. Applied rewrites 29.2%

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

   7. Applied rewrites 48.5%

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

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

   1. Initial program 70.3%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 55.3

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

   5. Applied rewrites 55.3%

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

   7. Applied rewrites 66.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around inf

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

      1. *-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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 19.5

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

   5. Applied rewrites 19.5%

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

   7. Applied rewrites 29.4%

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

4. Final simplification 58.9%

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

Alternative 8: 53.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.7%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 29.2

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

   5. Applied rewrites 29.2%

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

   7. Applied rewrites 48.5%

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

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

   1. Initial program 70.3%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 55.3

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

   5. Applied rewrites 55.3%

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

   7. Applied rewrites 66.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in 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 7.5

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

   5. Applied rewrites 7.5%

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

   6. Taylor expanded in U around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 11.4%

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

   9. Taylor expanded in U* around inf

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

   11. Applied rewrites 23.0%

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

4. Final simplification 58.0%

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

Alternative 9: 51.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{l\_m \cdot l\_m}{Om}\\ t_3 := \sqrt{\left(\left(t - t\_2 \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1}\\ t_4 := \sqrt{\left(\left(\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{if}\;t\_3 \leq 4 \cdot 10^{-129}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

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)))
        (t_4 (sqrt (* (* (* (fma (* (/ l_m Om) l_m) -2.0 t) n) U) 2.0))))
   (if (<= t_3 4e-129)
     t_4
     (if (<= t_3 5e+135) (sqrt (* (fma -2.0 t_2 t) t_1)) t_4))))

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 t_4 = sqrt((((fma(((l_m / Om) * l_m), -2.0, t) * n) * U) * 2.0));
	double tmp;
	if (t_3 <= 4e-129) {
		tmp = t_4;
	} else if (t_3 <= 5e+135) {
		tmp = sqrt((fma(-2.0, t_2, t) * t_1));
	} else {
		tmp = t_4;
	}
	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))
	t_4 = sqrt(Float64(Float64(Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * n) * U) * 2.0))
	tmp = 0.0
	if (t_3 <= 4e-129)
		tmp = t_4;
	elseif (t_3 <= 5e+135)
		tmp = sqrt(Float64(fma(-2.0, t_2, t) * t_1));
	else
		tmp = t_4;
	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]}, Block[{t$95$4 = N[Sqrt[N[(N[(N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 4e-129], t$95$4, If[LessEqual[t$95$3, 5e+135], N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], t$95$4]]]]]]

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*))))) < 3.9999999999999997e-129 or 5.00000000000000029e135 < (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 23.9%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 24.3

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

   5. Applied rewrites 24.3%

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

   7. Applied rewrites 33.1%

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

   if 3.9999999999999997e-129 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000029e135

   1. Initial program 97.3%

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

   3. Taylor expanded in Om around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 4 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 5 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1\\ t_3 := \sqrt{\left(\left(\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{if}\;t\_2 \leq 10^{+71}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2.5 \cdot 10^{+271}:\\ \;\;\;\;\sqrt{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2
         (*
          (-
           (- t (* (/ (* l_m l_m) Om) 2.0))
           (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1))
        (t_3 (sqrt (* (* (* (fma (* (/ l_m Om) l_m) -2.0 t) n) U) 2.0))))
   (if (<= t_2 1e+71) t_3 (if (<= t_2 2.5e+271) (sqrt (* t t_1)) t_3))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (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 t_3 = sqrt((((fma(((l_m / Om) * l_m), -2.0, t) * n) * U) * 2.0));
	double tmp;
	if (t_2 <= 1e+71) {
		tmp = t_3;
	} else if (t_2 <= 2.5e+271) {
		tmp = sqrt((t * t_1));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(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)
	t_3 = sqrt(Float64(Float64(Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * n) * U) * 2.0))
	tmp = 0.0
	if (t_2 <= 1e+71)
		tmp = t_3;
	elseif (t_2 <= 2.5e+271)
		tmp = sqrt(Float64(t * t_1));
	else
		tmp = t_3;
	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]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1e+71], t$95$3, If[LessEqual[t$95$2, 2.5e+271], N[Sqrt[N[(t * t$95$1), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e71 or 2.5000000000000002e271 < (*.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 46.0%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 42.5

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

   5. Applied rewrites 42.5%

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

   7. Applied rewrites 48.7%

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

   if 1e71 < (*.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*)))) < 2.5000000000000002e271

   1. Initial program 99.6%

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

   3. Taylor expanded in t around inf

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

      1. *-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 61.3

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

   5. Applied rewrites 61.3%

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

   7. Applied rewrites 89.9%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 10^{+71}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 2.5 \cdot 10^{+271}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;\sqrt{\left(\left(t - \frac{l\_m \cdot l\_m}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot n\right)\right) \cdot t\_1} \leq 4 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0))))
   (if (<=
        (sqrt
         (*
          (-
           (- t (* (/ (* l_m l_m) Om) 2.0))
           (* (- U U*) (* (pow (/ l_m Om) 2.0) n)))
          t_1))
        4e-129)
     (sqrt (* (* (* t n) U) 2.0))
     (sqrt (* t t_1)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double tmp;
	if (sqrt((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (pow((l_m / Om), 2.0) * n))) * t_1)) <= 4e-129) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((t * t_1));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = u * (n * 2.0d0)
    if (sqrt((((t - (((l_m * l_m) / om) * 2.0d0)) - ((u - u_42) * (((l_m / om) ** 2.0d0) * n))) * t_1)) <= 4d-129) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((t * t_1))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double tmp;
	if (Math.sqrt((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (Math.pow((l_m / Om), 2.0) * n))) * t_1)) <= 4e-129) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((t * t_1));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (n * 2.0)
	tmp = 0
	if math.sqrt((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (math.pow((l_m / Om), 2.0) * n))) * t_1)) <= 4e-129:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((t * t_1))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	tmp = 0.0
	if (sqrt(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)) <= 4e-129)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(t * t_1));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U * (n * 2.0);
	tmp = 0.0;
	if (sqrt((((t - (((l_m * l_m) / Om) * 2.0)) - ((U - U_42_) * (((l_m / Om) ^ 2.0) * n))) * t_1)) <= 4e-129)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((t * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[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]], $MachinePrecision], 4e-129], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t * 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*))))) < 3.9999999999999997e-129

   1. Initial program 18.2%

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

   3. Taylor expanded in t around inf

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

      1. *-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 30.4

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

   5. Applied rewrites 30.4%

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

   if 3.9999999999999997e-129 < (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 57.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 37.7

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

   5. Applied rewrites 37.7%

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

   7. Applied rewrites 42.2%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 4 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[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]], $MachinePrecision], 0.0], N[Sqrt[N[(N[(N[(t * U), $MachinePrecision] * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t * 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 5.7%

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

   3. Taylor expanded in t around inf

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

      1. *-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 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 20.0%

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

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

   1. 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 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 38.9

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

   5. Applied rewrites 38.9%

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

   7. Applied rewrites 43.2%

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

4. Final simplification 40.5%

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

Alternative 13: 54.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(\left(t - \left(\frac{n}{Om \cdot Om} \cdot \left(l\_m \cdot l\_m\right)\right) \cdot \left(-U*\right)\right) \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{elif}\;n \leq 30000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(l\_m \cdot \left(n \cdot 2\right), \left(-2 \cdot \frac{l\_m}{Om}\right) \cdot U, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\frac{\left(l\_m \cdot l\_m\right) \cdot n}{Om} \cdot \left(-U*\right)}{Om} \cdot n\right) \cdot \left(-2 \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -2.2e+68)
   (sqrt (* (* (- t (* (* (/ n (* Om Om)) (* l_m l_m)) (- U*))) n) (* U 2.0)))
   (if (<= n 30000000000.0)
     (sqrt
      (fma (* l_m (* n 2.0)) (* (* -2.0 (/ l_m Om)) U) (* (* (* t n) U) 2.0)))
     (if (<= n 2.5e+228)
       (* (sqrt (* n 2.0)) (sqrt (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))
       (sqrt
        (* (* (/ (* (/ (* (* l_m l_m) n) Om) (- U*)) Om) n) (* -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 (n <= -2.2e+68) {
		tmp = sqrt((((t - (((n / (Om * Om)) * (l_m * l_m)) * -U_42_)) * n) * (U * 2.0)));
	} else if (n <= 30000000000.0) {
		tmp = sqrt(fma((l_m * (n * 2.0)), ((-2.0 * (l_m / Om)) * U), (((t * n) * U) * 2.0)));
	} else if (n <= 2.5e+228) {
		tmp = sqrt((n * 2.0)) * sqrt((fma(((l_m / Om) * l_m), -2.0, t) * U));
	} else {
		tmp = sqrt((((((((l_m * l_m) * n) / Om) * -U_42_) / Om) * n) * (-2.0 * U)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -2.2e+68)
		tmp = sqrt(Float64(Float64(Float64(t - Float64(Float64(Float64(n / Float64(Om * Om)) * Float64(l_m * l_m)) * Float64(-U_42_))) * n) * Float64(U * 2.0)));
	elseif (n <= 30000000000.0)
		tmp = sqrt(fma(Float64(l_m * Float64(n * 2.0)), Float64(Float64(-2.0 * Float64(l_m / Om)) * U), Float64(Float64(Float64(t * n) * U) * 2.0)));
	elseif (n <= 2.5e+228)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(l_m * l_m) * n) / Om) * Float64(-U_42_)) / Om) * n) * Float64(-2.0 * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -2.2e+68], N[Sqrt[N[(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), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 30000000000.0], N[Sqrt[N[(N[(l$95$m * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2.5e+228], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * (-U$42$)), $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision] * N[(-2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.19999999999999987e68

   1. Initial program 55.2%

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

   3. Taylor expanded in t around inf

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

      1. *-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%

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

   6. Taylor expanded in U around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 48.1%

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

   9. Taylor expanded in U* around inf

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

   11. Applied rewrites 50.5%

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

   if -2.19999999999999987e68 < n < 3e10

   1. Initial program 45.3%

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 51.9

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 51.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 53.1

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

   6. Applied rewrites 53.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 56.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 56.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

   8. Applied rewrites 56.9%

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

   9. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 62.8

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

   11. Applied rewrites 62.8%

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

   if 3e10 < n < 2.5e228

   1. Initial program 71.4%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 50.6

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

   5. Applied rewrites 50.6%

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

   7. Applied rewrites 63.9%

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

   if 2.5e228 < n

   1. Initial program 48.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 63.2%

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

   6. Taylor expanded in U* around inf

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

   8. Applied rewrites 63.4%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(\left(t - \left(\frac{n}{Om \cdot Om} \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(-U*\right)\right) \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{elif}\;n \leq 30000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \left(n \cdot 2\right), \left(-2 \cdot \frac{\ell}{Om}\right) \cdot U, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\frac{\left(\ell \cdot \ell\right) \cdot n}{Om} \cdot \left(-U*\right)}{Om} \cdot n\right) \cdot \left(-2 \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(t - \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right) \cdot \left(l\_m \cdot l\_m\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{if}\;n \leq -2.05 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1050000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(l\_m \cdot \left(n \cdot 2\right), \left(-2 \cdot \frac{l\_m}{Om}\right) \cdot U, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (- t (/ (* (fma (- U U*) (/ n Om) 2.0) (* l_m l_m)) Om))
           (* U (* n 2.0))))))
   (if (<= n -2.05e+68)
     t_1
     (if (<= n 1050000.0)
       (sqrt
        (fma
         (* l_m (* n 2.0))
         (* (* -2.0 (/ l_m Om)) U)
         (* (* (* 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 - ((fma((U - U_42_), (n / Om), 2.0) * (l_m * l_m)) / Om)) * (U * (n * 2.0))));
	double tmp;
	if (n <= -2.05e+68) {
		tmp = t_1;
	} else if (n <= 1050000.0) {
		tmp = sqrt(fma((l_m * (n * 2.0)), ((-2.0 * (l_m / Om)) * U), (((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(fma(Float64(U - U_42_), Float64(n / Om), 2.0) * Float64(l_m * l_m)) / Om)) * Float64(U * Float64(n * 2.0))))
	tmp = 0.0
	if (n <= -2.05e+68)
		tmp = t_1;
	elseif (n <= 1050000.0)
		tmp = sqrt(fma(Float64(l_m * Float64(n * 2.0)), Float64(Float64(-2.0 * Float64(l_m / Om)) * U), 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[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $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, -2.05e+68], t$95$1, If[LessEqual[n, 1050000.0], N[Sqrt[N[(N[(l$95$m * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] + 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 < -2.05e68 or 1.05e6 < n

   1. Initial program 62.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 64.4%

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

   if -2.05e68 < n < 1.05e6

   1. Initial program 45.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 51.5

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 51.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.8

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

   6. Applied rewrites 52.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 56.6

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 56.6

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

   8. Applied rewrites 56.6%

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

   9. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 62.6

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

   11. Applied rewrites 62.6%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.05 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(t - \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right) \cdot \left(\ell \cdot \ell\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;n \leq 1050000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \left(n \cdot 2\right), \left(-2 \cdot \frac{\ell}{Om}\right) \cdot U, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right) \cdot \left(\ell \cdot \ell\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m}{Om} \cdot l\_m\\ \mathbf{if}\;n \leq -2.2 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(\left(t - \left(\frac{n}{Om \cdot Om} \cdot \left(l\_m \cdot l\_m\right)\right) \cdot \left(-U*\right)\right) \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{elif}\;n \leq 30000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(l\_m \cdot \left(n \cdot 2\right), \left(-2 \cdot \frac{l\_m}{Om}\right) \cdot U, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{\mathsf{fma}\left(t\_1, -2, t\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U \cdot n}{Om} \cdot \left(\left(t\_1 \cdot U*\right) \cdot n\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (/ l_m Om) l_m)))
   (if (<= n -2.2e+68)
     (sqrt
      (* (* (- t (* (* (/ n (* Om Om)) (* l_m l_m)) (- U*))) n) (* U 2.0)))
     (if (<= n 30000000000.0)
       (sqrt
        (fma
         (* l_m (* n 2.0))
         (* (* -2.0 (/ l_m Om)) U)
         (* (* (* t n) U) 2.0)))
       (if (<= n 2.5e+228)
         (* (sqrt (* n 2.0)) (sqrt (* (fma t_1 -2.0 t) U)))
         (sqrt (* (* (/ (* U n) Om) (* (* t_1 U*) n)) 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m / Om) * l_m;
	double tmp;
	if (n <= -2.2e+68) {
		tmp = sqrt((((t - (((n / (Om * Om)) * (l_m * l_m)) * -U_42_)) * n) * (U * 2.0)));
	} else if (n <= 30000000000.0) {
		tmp = sqrt(fma((l_m * (n * 2.0)), ((-2.0 * (l_m / Om)) * U), (((t * n) * U) * 2.0)));
	} else if (n <= 2.5e+228) {
		tmp = sqrt((n * 2.0)) * sqrt((fma(t_1, -2.0, t) * U));
	} else {
		tmp = sqrt(((((U * n) / Om) * ((t_1 * U_42_) * n)) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m / Om) * l_m)
	tmp = 0.0
	if (n <= -2.2e+68)
		tmp = sqrt(Float64(Float64(Float64(t - Float64(Float64(Float64(n / Float64(Om * Om)) * Float64(l_m * l_m)) * Float64(-U_42_))) * n) * Float64(U * 2.0)));
	elseif (n <= 30000000000.0)
		tmp = sqrt(fma(Float64(l_m * Float64(n * 2.0)), Float64(Float64(-2.0 * Float64(l_m / Om)) * U), Float64(Float64(Float64(t * n) * U) * 2.0)));
	elseif (n <= 2.5e+228)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(fma(t_1, -2.0, t) * U)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U * n) / Om) * Float64(Float64(t_1 * U_42_) * n)) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision]}, If[LessEqual[n, -2.2e+68], N[Sqrt[N[(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), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 30000000000.0], N[Sqrt[N[(N[(l$95$m * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2.5e+228], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(t$95$1 * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(N[(U * n), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(t$95$1 * U$42$), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.19999999999999987e68

   1. Initial program 55.2%

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

   3. Taylor expanded in t around inf

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

      1. *-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%

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

   6. Taylor expanded in U around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 48.1%

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

   9. Taylor expanded in U* around inf

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

   11. Applied rewrites 50.5%

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

   if -2.19999999999999987e68 < n < 3e10

   1. Initial program 45.3%

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

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 51.9

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

      18. lift-*.f64 N/A

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

   4. Applied rewrites 51.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 53.1

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

   6. Applied rewrites 53.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 56.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 56.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

   8. Applied rewrites 56.9%

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

   9. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 62.8

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

   11. Applied rewrites 62.8%

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

   if 3e10 < n < 2.5e228

   1. Initial program 71.4%

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

   3. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 50.6

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

   5. Applied rewrites 50.6%

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

   7. Applied rewrites 63.9%

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

   if 2.5e228 < n

   1. Initial program 48.5%

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

   3. Taylor expanded in U* around inf

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

      1. *-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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)\right) \cdot \color{blue}{\frac{U}{{Om}^{2}}}\right) \cdot 2} \]

      14. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)\right) \cdot \frac{U}{\color{blue}{Om \cdot Om}}\right) \cdot 2} \]

      15. lower-*.f64 62.0

          \[\leadsto \sqrt{\left(\left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)\right) \cdot \frac{U}{\color{blue}{Om \cdot Om}}\right) \cdot 2} \]

   5. Applied rewrites 62.0%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\left(U* \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot n\right)\right) \cdot \frac{U}{Om \cdot Om}\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 63.2%

      \[\leadsto \sqrt{\left(\left(\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot U*\right) \cdot n\right) \cdot \frac{U \cdot n}{Om}\right) \cdot 2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(\left(t - \left(\frac{n}{Om \cdot Om} \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(-U*\right)\right) \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{elif}\;n \leq 30000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \left(n \cdot 2\right), \left(-2 \cdot \frac{\ell}{Om}\right) \cdot U, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U \cdot n}{Om} \cdot \left(\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot U*\right) \cdot n\right)\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\\ \mathbf{if}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(t\_1 \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{t\_1 \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left(\left(l\_m \cdot l\_m\right) \cdot n\right) \cdot U*}{Om \cdot Om} \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* (/ l_m Om) l_m) -2.0 t)))
   (if (<= n -2e-310)
     (sqrt (* (* (* t_1 n) U) 2.0))
     (if (<= n 2.7e+228)
       (* (sqrt (* n 2.0)) (sqrt (* t_1 U)))
       (sqrt (* (* (/ (* (* (* l_m l_m) n) U*) (* Om Om)) n) (* U 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma(((l_m / Om) * l_m), -2.0, t);
	double tmp;
	if (n <= -2e-310) {
		tmp = sqrt((((t_1 * n) * U) * 2.0));
	} else if (n <= 2.7e+228) {
		tmp = sqrt((n * 2.0)) * sqrt((t_1 * U));
	} else {
		tmp = sqrt(((((((l_m * l_m) * n) * U_42_) / (Om * Om)) * n) * (U * 2.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)
	tmp = 0.0
	if (n <= -2e-310)
		tmp = sqrt(Float64(Float64(Float64(t_1 * n) * U) * 2.0));
	elseif (n <= 2.7e+228)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(t_1 * U)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(l_m * l_m) * n) * U_42_) / Float64(Om * Om)) * n) * Float64(U * 2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, If[LessEqual[n, -2e-310], N[Sqrt[N[(N[(N[(t$95$1 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2.7e+228], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$1 * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] * U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * N[(U * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.999999999999994e-310

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      14. lower-sqrt.f64 43.5

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \color{blue}{\sqrt{2}} \]

   5. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.9%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}} \]

   if -1.999999999999994e-310 < n < 2.7000000000000002e228

   1. Initial program 54.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      14. lower-sqrt.f64 48.0

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \color{blue}{\sqrt{2}} \]

   5. Applied rewrites 48.0%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.0%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U} \cdot \color{blue}{\sqrt{2 \cdot n}} \]

   if 2.7000000000000002e228 < n

   1. Initial program 48.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-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 16.3

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 16.3%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]

   6. Taylor expanded in U around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - \left(-1 \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(-1 \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(-1 \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot \left(t - \left(-1 \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot \left(t - \left(-1 \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \color{blue}{\left(t - \left(-1 \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + -1 \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)}\right)\right)} \]

      7. mul-1-neg N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{\left(\mathsf{neg}\left(\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)\right)}\right)\right)\right)} \]

      8. unsub-neg N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} - \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}} - \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)\right)\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}} - \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} - \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)\right)\right)} \]

   8. Applied rewrites 48.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(2 \cdot \frac{\ell \cdot \ell}{Om} - \frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{n}{Om}\right)\right)\right)}} \]

   9. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{\color{blue}{{Om}^{2}}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.6%

       \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \frac{U* \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{Om \cdot Om}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U*}{Om \cdot Om} \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 51.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\\ \mathbf{if}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(U \cdot n\right)} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{t\_1 \cdot \left(n \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* (/ l_m Om) l_m) -2.0 t)))
   (if (<= U -5e-310)
     (* (sqrt (* t_1 (* U n))) (sqrt 2.0))
     (* (sqrt U) (sqrt (* t_1 (* n 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma(((l_m / Om) * l_m), -2.0, t);
	double tmp;
	if (U <= -5e-310) {
		tmp = sqrt((t_1 * (U * n))) * sqrt(2.0);
	} else {
		tmp = sqrt(U) * sqrt((t_1 * (n * 2.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)
	tmp = 0.0
	if (U <= -5e-310)
		tmp = Float64(sqrt(Float64(t_1 * Float64(U * n))) * sqrt(2.0));
	else
		tmp = Float64(sqrt(U) * sqrt(Float64(t_1 * Float64(n * 2.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, If[LessEqual[U, -5e-310], N[(N[Sqrt[N[(t$95$1 * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -4.999999999999985e-310

   1. Initial program 52.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      14. lower-sqrt.f64 43.2

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \color{blue}{\sqrt{2}} \]

   5. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(U \cdot n\right)} \cdot \sqrt{2} \]

   if -4.999999999999985e-310 < U

   1. Initial program 52.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      14. lower-sqrt.f64 46.3

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \color{blue}{\sqrt{2}} \]

   5. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(2 \cdot n\right)} \cdot \color{blue}{\sqrt{U}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(U \cdot n\right)} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(n \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right)\\ \mathbf{if}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(t\_1 \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{t\_1 \cdot \left(n \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (fma (* (/ l_m Om) l_m) -2.0 t)))
   (if (<= U -5e-310)
     (sqrt (* (* (* t_1 n) U) 2.0))
     (* (sqrt U) (sqrt (* t_1 (* n 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma(((l_m / Om) * l_m), -2.0, t);
	double tmp;
	if (U <= -5e-310) {
		tmp = sqrt((((t_1 * n) * U) * 2.0));
	} else {
		tmp = sqrt(U) * sqrt((t_1 * (n * 2.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = fma(Float64(Float64(l_m / Om) * l_m), -2.0, t)
	tmp = 0.0
	if (U <= -5e-310)
		tmp = sqrt(Float64(Float64(Float64(t_1 * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(U) * sqrt(Float64(t_1 * Float64(n * 2.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, If[LessEqual[U, -5e-310], N[Sqrt[N[(N[(N[(t$95$1 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -4.999999999999985e-310

   1. Initial program 52.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      14. lower-sqrt.f64 43.2

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \color{blue}{\sqrt{2}} \]

   5. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}} \]

   if -4.999999999999985e-310 < U

   1. Initial program 52.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      14. lower-sqrt.f64 46.3

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \color{blue}{\sqrt{2}} \]

   5. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(2 \cdot n\right)} \cdot \color{blue}{\sqrt{U}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \left(n \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot n\\ \mathbf{if}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(t\_1 \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot 2} \cdot \sqrt{t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (fma (* (/ l_m Om) l_m) -2.0 t) n)))
   (if (<= U -5e-310)
     (sqrt (* (* t_1 U) 2.0))
     (* (sqrt (* U 2.0)) (sqrt 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 = fma(((l_m / Om) * l_m), -2.0, t) * n;
	double tmp;
	if (U <= -5e-310) {
		tmp = sqrt(((t_1 * U) * 2.0));
	} else {
		tmp = sqrt((U * 2.0)) * sqrt(t_1);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * n)
	tmp = 0.0
	if (U <= -5e-310)
		tmp = sqrt(Float64(Float64(t_1 * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(U * 2.0)) * sqrt(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[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[U, -5e-310], N[Sqrt[N[(N[(t$95$1 * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -4.999999999999985e-310

   1. Initial program 52.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      14. lower-sqrt.f64 43.2

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \color{blue}{\sqrt{2}} \]

   5. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}} \]

   if -4.999999999999985e-310 < U

   1. Initial program 52.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \cdot \sqrt{2} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U}} \cdot \sqrt{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right)} \cdot U} \cdot \sqrt{2} \]

      7. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)} \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2} \]

      14. lower-sqrt.f64 46.3

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \color{blue}{\sqrt{2}} \]

   5. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot n} \cdot \color{blue}{\sqrt{2 \cdot U}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot 2} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 38.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.9 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{\left(\left(l\_m \cdot l\_m\right) \cdot n\right) \cdot U}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.9e+140)
   (sqrt (* t (* U (* n 2.0))))
   (sqrt (* -4.0 (/ (* (* (* l_m l_m) n) U) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.9e+140) {
		tmp = sqrt((t * (U * (n * 2.0))));
	} else {
		tmp = sqrt((-4.0 * ((((l_m * l_m) * n) * U) / Om)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.9d+140) then
        tmp = sqrt((t * (u * (n * 2.0d0))))
    else
        tmp = sqrt(((-4.0d0) * ((((l_m * l_m) * n) * u) / om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.9e+140) {
		tmp = Math.sqrt((t * (U * (n * 2.0))));
	} else {
		tmp = Math.sqrt((-4.0 * ((((l_m * l_m) * n) * U) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.9e+140:
		tmp = math.sqrt((t * (U * (n * 2.0))))
	else:
		tmp = math.sqrt((-4.0 * ((((l_m * l_m) * n) * U) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.9e+140)
		tmp = sqrt(Float64(t * Float64(U * Float64(n * 2.0))));
	else
		tmp = sqrt(Float64(-4.0 * Float64(Float64(Float64(Float64(l_m * l_m) * n) * U) / Om)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.9e+140)
		tmp = sqrt((t * (U * (n * 2.0))));
	else
		tmp = sqrt((-4.0 * ((((l_m * l_m) * n) * U) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.9e+140], N[Sqrt[N[(t * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.9e140

   1. Initial program 58.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. *-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 43.2

         \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

   5. Applied rewrites 43.2%

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.7%

      \[\leadsto \sqrt{t \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)}} \]

   if 1.9e140 < l

   1. Initial program 23.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(n \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(n \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right)} \cdot \left(n \cdot \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) \cdot n\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-2 \cdot U\right) \cdot \color{blue}{\left(\left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right) \cdot n\right)}} \]

   5. Applied rewrites 33.9%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot U\right) \cdot \left(\frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om} \cdot n\right)}} \]

   6. Taylor expanded in Om around inf

      \[\leadsto \sqrt{-4 \cdot \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 26.1%

      \[\leadsto \sqrt{\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U}{Om} \cdot \color{blue}{-4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 36.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* t (* U (* n 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((t * (U * (n * 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((t * (u * (n * 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((t * (U * (n * 2.0))));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((t * (U * (n * 2.0))))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(t * Float64(U * Float64(n * 2.0))))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((t * (U * (n * 2.0))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(t * N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 52.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2}} \]

   3. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right)} \cdot 2} \]

   5. lower-*.f64 36.7

      \[\leadsto \sqrt{\left(\color{blue}{\left(n \cdot t\right)} \cdot U\right) \cdot 2} \]

5. Applied rewrites 36.7%

   \[\leadsto \sqrt{\color{blue}{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2}} \]
6. Step-by-step derivation

7. Applied rewrites 38.8%

   \[\leadsto \sqrt{t \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)}} \]

8. Final simplification 38.8%

   \[\leadsto \sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(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*))))))