Toniolo and Linder, Equation (13)

Percentage Accurate: 49.4% → 63.3%

Time: 37.2s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 63.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \sqrt{2}\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_2\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - t\_1 \cdot \frac{t\_1}{Om}\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_2 - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (sqrt 2.0)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) t_2))))
   (if (<= t_3 0.0)
     (sqrt (* (* 2.0 n) (* U (- t (* t_1 (/ t_1 Om))))))
     (if (<= t_3 5e+293)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_2 (* 2.0 (* l (/ l Om)))))))
       (fabs (/ (* l (sqrt (* U* (* 2.0 U)))) (/ Om n)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * sqrt(2.0);
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_2);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * (t - (t_1 * (t_1 / Om))))));
	} else if (t_3 <= 5e+293) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = fabs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = l * sqrt(2.0d0)
    t_2 = (n * ((l / om) ** 2.0d0)) * (u_42 - u)
    t_3 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) + t_2)
    if (t_3 <= 0.0d0) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (t_1 * (t_1 / om))))))
    else if (t_3 <= 5d+293) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (t_2 - (2.0d0 * (l * (l / om)))))))
    else
        tmp = abs(((l * sqrt((u_42 * (2.0d0 * u)))) / (om / n)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * Math.sqrt(2.0);
	double t_2 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_2);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (t_1 * (t_1 / Om))))));
	} else if (t_3 <= 5e+293) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.abs(((l * Math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * math.sqrt(2.0)
	t_2 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_2)
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (t_1 * (t_1 / Om))))))
	elif t_3 <= 5e+293:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.fabs(((l * math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * sqrt(2.0))
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_2))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(t_1 * Float64(t_1 / Om))))));
	elseif (t_3 <= 5e+293)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_2 - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = abs(Float64(Float64(l * sqrt(Float64(U_42_ * Float64(2.0 * U)))) / Float64(Om / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * sqrt(2.0);
	t_2 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_2);
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = sqrt(((2.0 * n) * (U * (t - (t_1 * (t_1 / Om))))));
	elseif (t_3 <= 5e+293)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l * (l / Om)))))));
	else
		tmp = abs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(t$95$1 * N[(t$95$1 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 5e+293], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$2 - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(l * N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.6%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in Om around inf 42.6%

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

      1. associate-*r/ 42.6%

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

   7. Simplified 42.6%

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

      1. add-sqr-sqrt 42.6%

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

      2. *-un-lft-identity 42.6%

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

      3. times-frac 42.6%

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

      4. *-commutative 42.6%

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

      5. sqrt-prod 42.6%

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

      6. sqrt-pow1 42.7%

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

      7. metadata-eval 42.7%

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

      8. pow1 42.7%

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

      9. *-commutative 42.7%

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

      10. sqrt-prod 42.7%

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

      11. sqrt-pow1 46.2%

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

      12. metadata-eval 46.2%

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

      13. pow1 46.2%

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

   9. Applied egg-rr 46.2%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.00000000000000033e293

   1. Initial program 97.8%

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

   3. Simplified 97.8%

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

   if 5.00000000000000033e293 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 18.7%

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

   3. Simplified 27.7%

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

      1. associate-*r* 27.7%

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

      2. fma-define 31.5%

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

      3. associate-*r* 32.1%

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

   6. Applied egg-rr 32.1%

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

   7. Taylor expanded in U around 0 20.5%

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

      1. mul-1-neg 20.5%

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

      2. associate-/l* 20.6%

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

      3. unpow2 20.6%

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

      4. unpow2 20.6%

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

      5. times-frac 32.3%

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

      6. unpow2 32.3%

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

   9. Simplified 32.3%

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

   10. Taylor expanded in n around inf 24.4%

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

       1. associate-*r/ 24.4%

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

       2. associate-*r* 24.2%

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

   12. Simplified 24.2%

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

       1. pow1/2 24.3%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 33.9%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 33.9%

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

       2. unpow2 33.9%

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

       3. rem-sqrt-square 44.7%

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

       4. associate-/l* 44.0%

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

       5. *-commutative 44.0%

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

       6. associate-/l* 45.8%

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

   16. Simplified 45.8%

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

       1. associate-*r* 45.7%

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

       2. clear-num 45.7%

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

       3. un-div-inv 45.7%

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

       4. associate-*r* 45.9%

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

   18. Applied egg-rr 45.9%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(\ell \cdot \sqrt{2}\right) \cdot \frac{\ell \cdot \sqrt{2}}{Om}\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_2 (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) t_1))))
   (if (<= t_2 0.0)
     (sqrt (* 2.0 (* U (* n t))))
     (if (<= t_2 5e+293)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l (/ l Om)))))))
       (fabs (/ (* l (sqrt (* U* (* 2.0 U)))) (/ Om n)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else if (t_2 <= 5e+293) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = fabs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (n * ((l / om) ** 2.0d0)) * (u_42 - u)
    t_2 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) + t_1)
    if (t_2 <= 0.0d0) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else if (t_2 <= 5d+293) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (t_1 - (2.0d0 * (l * (l / om)))))))
    else
        tmp = abs(((l * sqrt((u_42 * (2.0d0 * u)))) / (om / n)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else if (t_2 <= 5e+293) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.abs(((l * Math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1)
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	elif t_2 <= 5e+293:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.fabs(((l * math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_1))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	elseif (t_2 <= 5e+293)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = abs(Float64(Float64(l * sqrt(Float64(U_42_ * Float64(2.0 * U)))) / Float64(Om / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt((2.0 * (U * (n * t))));
	elseif (t_2 <= 5e+293)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	else
		tmp = abs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+293], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(l * N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.6%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in t around inf 46.1%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.00000000000000033e293

   1. Initial program 97.8%

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

   3. Simplified 97.8%

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

   if 5.00000000000000033e293 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 18.7%

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

   3. Simplified 27.7%

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

      1. associate-*r* 27.7%

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

      2. fma-define 31.5%

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

      3. associate-*r* 32.1%

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

   6. Applied egg-rr 32.1%

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

   7. Taylor expanded in U around 0 20.5%

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

      1. mul-1-neg 20.5%

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

      2. associate-/l* 20.6%

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

      3. unpow2 20.6%

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

      4. unpow2 20.6%

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

      5. times-frac 32.3%

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

      6. unpow2 32.3%

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

   9. Simplified 32.3%

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

   10. Taylor expanded in n around inf 24.4%

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

       1. associate-*r/ 24.4%

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

       2. associate-*r* 24.2%

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

   12. Simplified 24.2%

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

       1. pow1/2 24.3%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 33.9%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 33.9%

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

       2. unpow2 33.9%

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

       3. rem-sqrt-square 44.7%

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

       4. associate-/l* 44.0%

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

       5. *-commutative 44.0%

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

       6. associate-/l* 45.8%

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

   16. Simplified 45.8%

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

       1. associate-*r* 45.7%

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

       2. clear-num 45.7%

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

       3. un-div-inv 45.7%

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

       4. associate-*r* 45.9%

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

   18. Applied egg-rr 45.9%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot U*\right)}\\ t_2 := 2 \cdot \left(n \cdot U\right)\\ t_3 := n \cdot {\ell}^{2}\\ \mathbf{if}\;\ell \leq 1.25 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+58}:\\ \;\;\;\;{\left(\frac{\left(U \cdot -4\right) \cdot t\_3}{Om}\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+83}:\\ \;\;\;\;\left|t\_1 \cdot \frac{1}{\frac{Om}{n \cdot \ell}}\right|\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+108}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{-4 \cdot \left(U \cdot \frac{t\_3}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+132}:\\ \;\;\;\;\left|t\_1 \cdot \left(\ell \cdot \frac{n}{Om}\right)\right|\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+148}:\\ \;\;\;\;{\left(t\_2 \cdot \left({\ell}^{2} \cdot \frac{-2}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+275}:\\ \;\;\;\;\left|t\_1 \cdot \frac{\ell}{\frac{Om}{n}}\right|\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{t\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (* U U*))))
        (t_2 (* 2.0 (* n U)))
        (t_3 (* n (pow l 2.0))))
   (if (<= l 1.25e+45)
     (sqrt (* 2.0 (fabs (* U (* n t)))))
     (if (<= l 7.6e+58)
       (pow (/ (* (* U -4.0) t_3) Om) 0.5)
       (if (<= l 1.45e+83)
         (fabs (* t_1 (/ 1.0 (/ Om (* n l)))))
         (if (<= l 2.25e+108)
           (fabs (/ (* l (sqrt (* U* (* 2.0 U)))) (/ Om n)))
           (if (<= l 2.5e+111)
             (sqrt (* -4.0 (* U (/ t_3 Om))))
             (if (<= l 8.2e+132)
               (fabs (* t_1 (* l (/ n Om))))
               (if (<= l 6.6e+148)
                 (pow (* t_2 (* (pow l 2.0) (/ -2.0 Om))) 0.5)
                 (if (<= l 1.95e+275)
                   (fabs (* t_1 (/ l (/ Om n))))
                   (* l (* (sqrt (/ -2.0 Om)) (sqrt t_2)))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * (U * U_42_)));
	double t_2 = 2.0 * (n * U);
	double t_3 = n * pow(l, 2.0);
	double tmp;
	if (l <= 1.25e+45) {
		tmp = sqrt((2.0 * fabs((U * (n * t)))));
	} else if (l <= 7.6e+58) {
		tmp = pow((((U * -4.0) * t_3) / Om), 0.5);
	} else if (l <= 1.45e+83) {
		tmp = fabs((t_1 * (1.0 / (Om / (n * l)))));
	} else if (l <= 2.25e+108) {
		tmp = fabs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	} else if (l <= 2.5e+111) {
		tmp = sqrt((-4.0 * (U * (t_3 / Om))));
	} else if (l <= 8.2e+132) {
		tmp = fabs((t_1 * (l * (n / Om))));
	} else if (l <= 6.6e+148) {
		tmp = pow((t_2 * (pow(l, 2.0) * (-2.0 / Om))), 0.5);
	} else if (l <= 1.95e+275) {
		tmp = fabs((t_1 * (l / (Om / n))));
	} else {
		tmp = l * (sqrt((-2.0 / Om)) * sqrt(t_2));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * (u * u_42)))
    t_2 = 2.0d0 * (n * u)
    t_3 = n * (l ** 2.0d0)
    if (l <= 1.25d+45) then
        tmp = sqrt((2.0d0 * abs((u * (n * t)))))
    else if (l <= 7.6d+58) then
        tmp = (((u * (-4.0d0)) * t_3) / om) ** 0.5d0
    else if (l <= 1.45d+83) then
        tmp = abs((t_1 * (1.0d0 / (om / (n * l)))))
    else if (l <= 2.25d+108) then
        tmp = abs(((l * sqrt((u_42 * (2.0d0 * u)))) / (om / n)))
    else if (l <= 2.5d+111) then
        tmp = sqrt(((-4.0d0) * (u * (t_3 / om))))
    else if (l <= 8.2d+132) then
        tmp = abs((t_1 * (l * (n / om))))
    else if (l <= 6.6d+148) then
        tmp = (t_2 * ((l ** 2.0d0) * ((-2.0d0) / om))) ** 0.5d0
    else if (l <= 1.95d+275) then
        tmp = abs((t_1 * (l / (om / n))))
    else
        tmp = l * (sqrt(((-2.0d0) / om)) * sqrt(t_2))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((2.0 * (U * U_42_)));
	double t_2 = 2.0 * (n * U);
	double t_3 = n * Math.pow(l, 2.0);
	double tmp;
	if (l <= 1.25e+45) {
		tmp = Math.sqrt((2.0 * Math.abs((U * (n * t)))));
	} else if (l <= 7.6e+58) {
		tmp = Math.pow((((U * -4.0) * t_3) / Om), 0.5);
	} else if (l <= 1.45e+83) {
		tmp = Math.abs((t_1 * (1.0 / (Om / (n * l)))));
	} else if (l <= 2.25e+108) {
		tmp = Math.abs(((l * Math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	} else if (l <= 2.5e+111) {
		tmp = Math.sqrt((-4.0 * (U * (t_3 / Om))));
	} else if (l <= 8.2e+132) {
		tmp = Math.abs((t_1 * (l * (n / Om))));
	} else if (l <= 6.6e+148) {
		tmp = Math.pow((t_2 * (Math.pow(l, 2.0) * (-2.0 / Om))), 0.5);
	} else if (l <= 1.95e+275) {
		tmp = Math.abs((t_1 * (l / (Om / n))));
	} else {
		tmp = l * (Math.sqrt((-2.0 / Om)) * Math.sqrt(t_2));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((2.0 * (U * U_42_)))
	t_2 = 2.0 * (n * U)
	t_3 = n * math.pow(l, 2.0)
	tmp = 0
	if l <= 1.25e+45:
		tmp = math.sqrt((2.0 * math.fabs((U * (n * t)))))
	elif l <= 7.6e+58:
		tmp = math.pow((((U * -4.0) * t_3) / Om), 0.5)
	elif l <= 1.45e+83:
		tmp = math.fabs((t_1 * (1.0 / (Om / (n * l)))))
	elif l <= 2.25e+108:
		tmp = math.fabs(((l * math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)))
	elif l <= 2.5e+111:
		tmp = math.sqrt((-4.0 * (U * (t_3 / Om))))
	elif l <= 8.2e+132:
		tmp = math.fabs((t_1 * (l * (n / Om))))
	elif l <= 6.6e+148:
		tmp = math.pow((t_2 * (math.pow(l, 2.0) * (-2.0 / Om))), 0.5)
	elif l <= 1.95e+275:
		tmp = math.fabs((t_1 * (l / (Om / n))))
	else:
		tmp = l * (math.sqrt((-2.0 / Om)) * math.sqrt(t_2))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * Float64(U * U_42_)))
	t_2 = Float64(2.0 * Float64(n * U))
	t_3 = Float64(n * (l ^ 2.0))
	tmp = 0.0
	if (l <= 1.25e+45)
		tmp = sqrt(Float64(2.0 * abs(Float64(U * Float64(n * t)))));
	elseif (l <= 7.6e+58)
		tmp = Float64(Float64(Float64(U * -4.0) * t_3) / Om) ^ 0.5;
	elseif (l <= 1.45e+83)
		tmp = abs(Float64(t_1 * Float64(1.0 / Float64(Om / Float64(n * l)))));
	elseif (l <= 2.25e+108)
		tmp = abs(Float64(Float64(l * sqrt(Float64(U_42_ * Float64(2.0 * U)))) / Float64(Om / n)));
	elseif (l <= 2.5e+111)
		tmp = sqrt(Float64(-4.0 * Float64(U * Float64(t_3 / Om))));
	elseif (l <= 8.2e+132)
		tmp = abs(Float64(t_1 * Float64(l * Float64(n / Om))));
	elseif (l <= 6.6e+148)
		tmp = Float64(t_2 * Float64((l ^ 2.0) * Float64(-2.0 / Om))) ^ 0.5;
	elseif (l <= 1.95e+275)
		tmp = abs(Float64(t_1 * Float64(l / Float64(Om / n))));
	else
		tmp = Float64(l * Float64(sqrt(Float64(-2.0 / Om)) * sqrt(t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((2.0 * (U * U_42_)));
	t_2 = 2.0 * (n * U);
	t_3 = n * (l ^ 2.0);
	tmp = 0.0;
	if (l <= 1.25e+45)
		tmp = sqrt((2.0 * abs((U * (n * t)))));
	elseif (l <= 7.6e+58)
		tmp = (((U * -4.0) * t_3) / Om) ^ 0.5;
	elseif (l <= 1.45e+83)
		tmp = abs((t_1 * (1.0 / (Om / (n * l)))));
	elseif (l <= 2.25e+108)
		tmp = abs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	elseif (l <= 2.5e+111)
		tmp = sqrt((-4.0 * (U * (t_3 / Om))));
	elseif (l <= 8.2e+132)
		tmp = abs((t_1 * (l * (n / Om))));
	elseif (l <= 6.6e+148)
		tmp = (t_2 * ((l ^ 2.0) * (-2.0 / Om))) ^ 0.5;
	elseif (l <= 1.95e+275)
		tmp = abs((t_1 * (l / (Om / n))));
	else
		tmp = l * (sqrt((-2.0 / Om)) * sqrt(t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.25e+45], N[Sqrt[N[(2.0 * N[Abs[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 7.6e+58], N[Power[N[(N[(N[(U * -4.0), $MachinePrecision] * t$95$3), $MachinePrecision] / Om), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.45e+83], N[Abs[N[(t$95$1 * N[(1.0 / N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.25e+108], N[Abs[N[(N[(l * N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.5e+111], N[Sqrt[N[(-4.0 * N[(U * N[(t$95$3 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 8.2e+132], N[Abs[N[(t$95$1 * N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6.6e+148], N[Power[N[(t$95$2 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.95e+275], N[Abs[N[(t$95$1 * N[(l / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l * N[(N[Sqrt[N[(-2.0 / Om), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 9 regimes

2. if l < 1.25e45

   1. Initial program 53.7%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in t around inf 42.9%

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

      1. add-cube-cbrt 42.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{U \cdot \left(n \cdot t\right)} \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right) \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}} \]

      2. pow3 42.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}^{3}}} \]

      3. associate-*r* 42.1%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(U \cdot n\right) \cdot t}}\right)}^{3}} \]

      4. *-commutative 42.1%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(n \cdot U\right)} \cdot t}\right)}^{3}} \]

   7. Applied egg-rr 42.1%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\left(n \cdot U\right) \cdot t}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 42.5%

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

      2. add-sqr-sqrt 42.4%

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

      3. sqrt-unprod 28.4%

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

      4. pow2 28.4%

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

      5. associate-*l* 27.9%

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

   9. Applied egg-rr 27.9%

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

       1. unpow2 27.9%

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

       2. rem-sqrt-square 41.5%

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

       3. associate-*r* 44.1%

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

       4. *-commutative 44.1%

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

       5. associate-*r* 44.6%

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

   11. Simplified 44.6%

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

   if 1.25e45 < l < 7.5999999999999997e58

   1. Initial program 76.7%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in n around 0 57.3%

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

      1. associate-*r* 62.4%

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

      2. cancel-sign-sub-inv 62.4%

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

      3. metadata-eval 62.4%

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

      4. *-commutative 62.4%

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

      5. associate-*l/ 62.4%

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

   7. Simplified 62.4%

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

      1. pow1/2 77.6%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 77.6%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 77.6%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 77.6%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 77.6%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 77.6%

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

   9. Applied egg-rr 77.6%

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

   10. Taylor expanded in l around inf 72.1%

       \[\leadsto {\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}}^{0.5} \]
   11. Step-by-step derivation

       1. associate-*r/ 72.1%

          \[\leadsto {\color{blue}{\left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)}}^{0.5} \]

       2. associate-*r* 72.1%

          \[\leadsto {\left(\frac{\color{blue}{\left(-4 \cdot U\right) \cdot \left({\ell}^{2} \cdot n\right)}}{Om}\right)}^{0.5} \]

       3. *-commutative 72.1%

          \[\leadsto {\left(\frac{\left(-4 \cdot U\right) \cdot \color{blue}{\left(n \cdot {\ell}^{2}\right)}}{Om}\right)}^{0.5} \]

   12. Simplified 72.1%

       \[\leadsto {\color{blue}{\left(\frac{\left(-4 \cdot U\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)}}^{0.5} \]

   if 7.5999999999999997e58 < l < 1.45e83

   1. Initial program 36.8%

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

   3. Simplified 36.8%

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

      1. associate-*r* 36.8%

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

      2. fma-define 36.8%

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

      3. associate-*r* 36.8%

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

   6. Applied egg-rr 36.8%

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

   7. Taylor expanded in U around 0 4.8%

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

      1. mul-1-neg 4.8%

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

      2. associate-/l* 36.8%

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

      3. unpow2 36.8%

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

      4. unpow2 36.8%

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

      5. times-frac 36.8%

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

      6. unpow2 36.8%

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

   9. Simplified 36.8%

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

   10. Taylor expanded in n around inf 3.4%

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

       1. associate-*r/ 3.4%

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

       2. associate-*r* 2.8%

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

   12. Simplified 2.8%

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

       1. pow1/2 2.8%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 36.7%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 36.7%

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

       2. unpow2 36.7%

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

       3. rem-sqrt-square 40.7%

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

       4. associate-/l* 40.7%

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

       5. *-commutative 40.7%

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

       6. associate-/l* 40.7%

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

   16. Simplified 40.7%

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

       1. associate-*r/ 40.7%

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

       2. clear-num 40.7%

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

   18. Applied egg-rr 40.7%

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

   if 1.45e83 < l < 2.25e108

   1. Initial program 35.9%

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

   3. Simplified 35.9%

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

      1. associate-*r* 35.9%

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

      2. fma-define 52.6%

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

      3. associate-*r* 52.6%

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

   6. Applied egg-rr 52.6%

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

   7. Taylor expanded in U around 0 35.9%

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

      1. mul-1-neg 35.9%

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

      2. associate-/l* 52.6%

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

      3. unpow2 52.6%

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

      4. unpow2 52.6%

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

      5. times-frac 52.6%

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

      6. unpow2 52.6%

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

   9. Simplified 52.6%

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

   10. Taylor expanded in n around inf 35.6%

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

       1. associate-*r/ 35.6%

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

       2. associate-*r* 35.6%

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

   12. Simplified 35.6%

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

       1. pow1/2 35.6%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 35.6%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 35.6%

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

       2. unpow2 35.6%

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

       3. rem-sqrt-square 51.3%

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

       4. associate-/l* 51.3%

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

       5. *-commutative 51.3%

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

       6. associate-/l* 51.1%

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

   16. Simplified 51.1%

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

       1. associate-*r* 50.9%

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

       2. clear-num 50.9%

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

       3. un-div-inv 51.1%

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

       4. associate-*r* 51.1%

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

   18. Applied egg-rr 51.1%

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

   if 2.25e108 < l < 2.4999999999999998e111

   1. Initial program 98.4%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Om around inf 100.0%

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

      1. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 100.0%

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

      1. associate-/l* 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 2.4999999999999998e111 < l < 8.19999999999999983e132

   1. Initial program 44.2%

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

   3. Simplified 44.2%

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

      1. associate-*r* 44.2%

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

      2. fma-define 58.5%

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

      3. associate-*r* 58.5%

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

   6. Applied egg-rr 58.5%

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

   7. Taylor expanded in U around 0 44.2%

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

      1. mul-1-neg 44.2%

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

      2. associate-/l* 58.5%

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

      3. unpow2 58.5%

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

      4. unpow2 58.5%

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

      5. times-frac 58.5%

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

      6. unpow2 58.5%

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

   9. Simplified 58.5%

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

   10. Taylor expanded in n around inf 30.7%

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

       1. associate-*r/ 30.7%

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

       2. associate-*r* 30.2%

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

   12. Simplified 30.2%

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

       1. pow1/2 30.2%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 45.3%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 45.3%

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

       2. unpow2 45.3%

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

       3. rem-sqrt-square 58.7%

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

       4. associate-/l* 58.3%

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

       5. *-commutative 58.3%

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

       6. associate-/l* 71.9%

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

   16. Simplified 71.9%

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

   if 8.19999999999999983e132 < l < 6.60000000000000021e148

   1. Initial program 50.0%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in n around 0 51.7%

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

      1. associate-*r* 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. metadata-eval 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*l/ 99.2%

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

   7. Simplified 99.2%

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

      1. pow1/2 99.2%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 99.2%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 99.2%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 99.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 100.0%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in l around inf 99.2%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)}^{0.5} \]
   11. Step-by-step derivation

       1. associate-*r/ 99.2%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)}^{0.5} \]

       2. *-commutative 99.2%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om}\right)}^{0.5} \]

       3. associate-*r/ 100.0%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]

   12. Simplified 100.0%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]

   if 6.60000000000000021e148 < l < 1.95e275

   1. Initial program 15.0%

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

   3. Simplified 31.2%

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

      1. associate-*r* 31.2%

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

      2. fma-define 34.4%

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

      3. associate-*r* 34.1%

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

   6. Applied egg-rr 34.1%

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

   7. Taylor expanded in U around 0 17.1%

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

      1. mul-1-neg 17.1%

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

      2. associate-/l* 17.1%

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

      3. unpow2 17.1%

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

      4. unpow2 17.1%

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

      5. times-frac 34.2%

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

      6. unpow2 34.2%

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

   9. Simplified 34.2%

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

   10. Taylor expanded in n around inf 30.1%

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

       1. associate-*r/ 30.1%

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

       2. associate-*r* 29.9%

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

   12. Simplified 29.9%

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

       1. pow1/2 29.9%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 34.5%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 34.5%

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

       2. unpow2 34.5%

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

       3. rem-sqrt-square 34.6%

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

       4. associate-/l* 37.5%

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

       5. *-commutative 37.5%

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

       6. associate-/l* 46.5%

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

   16. Simplified 46.5%

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

       1. clear-num 46.6%

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

       2. un-div-inv 46.6%

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

   18. Applied egg-rr 46.6%

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

   if 1.95e275 < l

   1. Initial program 16.7%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in n around 0 18.4%

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

      1. associate-*r* 18.4%

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

      2. cancel-sign-sub-inv 18.4%

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

      3. metadata-eval 18.4%

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

      4. *-commutative 18.4%

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

      5. associate-*l/ 18.4%

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

   7. Simplified 18.4%

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

      1. pow1/2 18.4%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 18.4%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 18.4%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 18.4%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 18.4%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 18.4%

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

   9. Applied egg-rr 18.4%

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

   10. Taylor expanded in l around inf 18.4%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)}^{0.5} \]
   11. Step-by-step derivation

       1. associate-*r/ 18.4%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)}^{0.5} \]

       2. *-commutative 18.4%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om}\right)}^{0.5} \]

       3. associate-*r/ 18.4%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]

   12. Simplified 18.4%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 18.4%

          \[\leadsto \color{blue}{1 \cdot {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left({\ell}^{2} \cdot \frac{-2}{Om}\right)\right)}^{0.5}} \]

       2. unpow1/2 18.4%

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

       3. *-commutative 18.4%

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

       4. sqrt-prod 17.3%

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

       5. sqrt-prod 17.3%

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

       6. sqrt-pow1 44.4%

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

       7. metadata-eval 44.4%

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

       8. pow1 44.4%

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

   14. Applied egg-rr 44.4%

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

       1. *-lft-identity 44.4%

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

       2. associate-*l* 57.5%

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

       3. *-commutative 57.5%

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

   16. Simplified 57.5%

       \[\leadsto \color{blue}{\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{2 \cdot \left(U \cdot n\right)}\right)} \]
3. Recombined 9 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+58}:\\ \;\;\;\;{\left(\frac{\left(U \cdot -4\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+83}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{1}{\frac{Om}{n \cdot \ell}}\right|\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+108}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{-4 \cdot \left(U \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+132}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)\right|\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+148}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left({\ell}^{2} \cdot \frac{-2}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+275}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\ell}{\frac{Om}{n}}\right|\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left(U \cdot U*\right)}\\ t_2 := n \cdot {\ell}^{2}\\ \mathbf{if}\;\ell \leq 10^{+45}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+58}:\\ \;\;\;\;{\left(\frac{\left(U \cdot -4\right) \cdot t\_2}{Om}\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{+81}:\\ \;\;\;\;\left|t\_1 \cdot \frac{1}{\frac{Om}{n \cdot \ell}}\right|\\ \mathbf{elif}\;\ell \leq 10^{+108}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{-4 \cdot \left(U \cdot \frac{t\_2}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{+133}:\\ \;\;\;\;\left|t\_1 \cdot \left(\ell \cdot \frac{n}{Om}\right)\right|\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+150}:\\ \;\;\;\;{\left(-4 \cdot \frac{{\ell}^{2} \cdot \left(n \cdot U\right)}{Om}\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+275}:\\ \;\;\;\;\left|t\_1 \cdot \frac{\ell}{\frac{Om}{n}}\right|\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (* U U*)))) (t_2 (* n (pow l 2.0))))
   (if (<= l 1e+45)
     (sqrt (* 2.0 (fabs (* U (* n t)))))
     (if (<= l 8.5e+58)
       (pow (/ (* (* U -4.0) t_2) Om) 0.5)
       (if (<= l 9.6e+81)
         (fabs (* t_1 (/ 1.0 (/ Om (* n l)))))
         (if (<= l 1e+108)
           (fabs (/ (* l (sqrt (* U* (* 2.0 U)))) (/ Om n)))
           (if (<= l 2.05e+110)
             (sqrt (* -4.0 (* U (/ t_2 Om))))
             (if (<= l 1.12e+133)
               (fabs (* t_1 (* l (/ n Om))))
               (if (<= l 2.7e+150)
                 (pow (* -4.0 (/ (* (pow l 2.0) (* n U)) Om)) 0.5)
                 (if (<= l 1.8e+275)
                   (fabs (* t_1 (/ l (/ Om n))))
                   (*
                    l
                    (* (sqrt (/ -2.0 Om)) (sqrt (* 2.0 (* n U)))))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * (U * U_42_)));
	double t_2 = n * pow(l, 2.0);
	double tmp;
	if (l <= 1e+45) {
		tmp = sqrt((2.0 * fabs((U * (n * t)))));
	} else if (l <= 8.5e+58) {
		tmp = pow((((U * -4.0) * t_2) / Om), 0.5);
	} else if (l <= 9.6e+81) {
		tmp = fabs((t_1 * (1.0 / (Om / (n * l)))));
	} else if (l <= 1e+108) {
		tmp = fabs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	} else if (l <= 2.05e+110) {
		tmp = sqrt((-4.0 * (U * (t_2 / Om))));
	} else if (l <= 1.12e+133) {
		tmp = fabs((t_1 * (l * (n / Om))));
	} else if (l <= 2.7e+150) {
		tmp = pow((-4.0 * ((pow(l, 2.0) * (n * U)) / Om)), 0.5);
	} else if (l <= 1.8e+275) {
		tmp = fabs((t_1 * (l / (Om / n))));
	} else {
		tmp = l * (sqrt((-2.0 / Om)) * sqrt((2.0 * (n * U))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * (u * u_42)))
    t_2 = n * (l ** 2.0d0)
    if (l <= 1d+45) then
        tmp = sqrt((2.0d0 * abs((u * (n * t)))))
    else if (l <= 8.5d+58) then
        tmp = (((u * (-4.0d0)) * t_2) / om) ** 0.5d0
    else if (l <= 9.6d+81) then
        tmp = abs((t_1 * (1.0d0 / (om / (n * l)))))
    else if (l <= 1d+108) then
        tmp = abs(((l * sqrt((u_42 * (2.0d0 * u)))) / (om / n)))
    else if (l <= 2.05d+110) then
        tmp = sqrt(((-4.0d0) * (u * (t_2 / om))))
    else if (l <= 1.12d+133) then
        tmp = abs((t_1 * (l * (n / om))))
    else if (l <= 2.7d+150) then
        tmp = ((-4.0d0) * (((l ** 2.0d0) * (n * u)) / om)) ** 0.5d0
    else if (l <= 1.8d+275) then
        tmp = abs((t_1 * (l / (om / n))))
    else
        tmp = l * (sqrt(((-2.0d0) / om)) * sqrt((2.0d0 * (n * u))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((2.0 * (U * U_42_)));
	double t_2 = n * Math.pow(l, 2.0);
	double tmp;
	if (l <= 1e+45) {
		tmp = Math.sqrt((2.0 * Math.abs((U * (n * t)))));
	} else if (l <= 8.5e+58) {
		tmp = Math.pow((((U * -4.0) * t_2) / Om), 0.5);
	} else if (l <= 9.6e+81) {
		tmp = Math.abs((t_1 * (1.0 / (Om / (n * l)))));
	} else if (l <= 1e+108) {
		tmp = Math.abs(((l * Math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	} else if (l <= 2.05e+110) {
		tmp = Math.sqrt((-4.0 * (U * (t_2 / Om))));
	} else if (l <= 1.12e+133) {
		tmp = Math.abs((t_1 * (l * (n / Om))));
	} else if (l <= 2.7e+150) {
		tmp = Math.pow((-4.0 * ((Math.pow(l, 2.0) * (n * U)) / Om)), 0.5);
	} else if (l <= 1.8e+275) {
		tmp = Math.abs((t_1 * (l / (Om / n))));
	} else {
		tmp = l * (Math.sqrt((-2.0 / Om)) * Math.sqrt((2.0 * (n * U))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((2.0 * (U * U_42_)))
	t_2 = n * math.pow(l, 2.0)
	tmp = 0
	if l <= 1e+45:
		tmp = math.sqrt((2.0 * math.fabs((U * (n * t)))))
	elif l <= 8.5e+58:
		tmp = math.pow((((U * -4.0) * t_2) / Om), 0.5)
	elif l <= 9.6e+81:
		tmp = math.fabs((t_1 * (1.0 / (Om / (n * l)))))
	elif l <= 1e+108:
		tmp = math.fabs(((l * math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)))
	elif l <= 2.05e+110:
		tmp = math.sqrt((-4.0 * (U * (t_2 / Om))))
	elif l <= 1.12e+133:
		tmp = math.fabs((t_1 * (l * (n / Om))))
	elif l <= 2.7e+150:
		tmp = math.pow((-4.0 * ((math.pow(l, 2.0) * (n * U)) / Om)), 0.5)
	elif l <= 1.8e+275:
		tmp = math.fabs((t_1 * (l / (Om / n))))
	else:
		tmp = l * (math.sqrt((-2.0 / Om)) * math.sqrt((2.0 * (n * U))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * Float64(U * U_42_)))
	t_2 = Float64(n * (l ^ 2.0))
	tmp = 0.0
	if (l <= 1e+45)
		tmp = sqrt(Float64(2.0 * abs(Float64(U * Float64(n * t)))));
	elseif (l <= 8.5e+58)
		tmp = Float64(Float64(Float64(U * -4.0) * t_2) / Om) ^ 0.5;
	elseif (l <= 9.6e+81)
		tmp = abs(Float64(t_1 * Float64(1.0 / Float64(Om / Float64(n * l)))));
	elseif (l <= 1e+108)
		tmp = abs(Float64(Float64(l * sqrt(Float64(U_42_ * Float64(2.0 * U)))) / Float64(Om / n)));
	elseif (l <= 2.05e+110)
		tmp = sqrt(Float64(-4.0 * Float64(U * Float64(t_2 / Om))));
	elseif (l <= 1.12e+133)
		tmp = abs(Float64(t_1 * Float64(l * Float64(n / Om))));
	elseif (l <= 2.7e+150)
		tmp = Float64(-4.0 * Float64(Float64((l ^ 2.0) * Float64(n * U)) / Om)) ^ 0.5;
	elseif (l <= 1.8e+275)
		tmp = abs(Float64(t_1 * Float64(l / Float64(Om / n))));
	else
		tmp = Float64(l * Float64(sqrt(Float64(-2.0 / Om)) * sqrt(Float64(2.0 * Float64(n * U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((2.0 * (U * U_42_)));
	t_2 = n * (l ^ 2.0);
	tmp = 0.0;
	if (l <= 1e+45)
		tmp = sqrt((2.0 * abs((U * (n * t)))));
	elseif (l <= 8.5e+58)
		tmp = (((U * -4.0) * t_2) / Om) ^ 0.5;
	elseif (l <= 9.6e+81)
		tmp = abs((t_1 * (1.0 / (Om / (n * l)))));
	elseif (l <= 1e+108)
		tmp = abs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	elseif (l <= 2.05e+110)
		tmp = sqrt((-4.0 * (U * (t_2 / Om))));
	elseif (l <= 1.12e+133)
		tmp = abs((t_1 * (l * (n / Om))));
	elseif (l <= 2.7e+150)
		tmp = (-4.0 * (((l ^ 2.0) * (n * U)) / Om)) ^ 0.5;
	elseif (l <= 1.8e+275)
		tmp = abs((t_1 * (l / (Om / n))));
	else
		tmp = l * (sqrt((-2.0 / Om)) * sqrt((2.0 * (n * U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1e+45], N[Sqrt[N[(2.0 * N[Abs[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 8.5e+58], N[Power[N[(N[(N[(U * -4.0), $MachinePrecision] * t$95$2), $MachinePrecision] / Om), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 9.6e+81], N[Abs[N[(t$95$1 * N[(1.0 / N[(Om / N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1e+108], N[Abs[N[(N[(l * N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.05e+110], N[Sqrt[N[(-4.0 * N[(U * N[(t$95$2 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.12e+133], N[Abs[N[(t$95$1 * N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.7e+150], N[Power[N[(-4.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.8e+275], N[Abs[N[(t$95$1 * N[(l / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l * N[(N[Sqrt[N[(-2.0 / Om), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 9 regimes

2. if l < 9.9999999999999993e44

   1. Initial program 53.7%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in t around inf 42.9%

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

      1. add-cube-cbrt 42.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{U \cdot \left(n \cdot t\right)} \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right) \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}} \]

      2. pow3 42.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}^{3}}} \]

      3. associate-*r* 42.1%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(U \cdot n\right) \cdot t}}\right)}^{3}} \]

      4. *-commutative 42.1%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(n \cdot U\right)} \cdot t}\right)}^{3}} \]

   7. Applied egg-rr 42.1%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\left(n \cdot U\right) \cdot t}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 42.5%

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

      2. add-sqr-sqrt 42.4%

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

      3. sqrt-unprod 28.4%

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

      4. pow2 28.4%

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

      5. associate-*l* 27.9%

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

   9. Applied egg-rr 27.9%

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

       1. unpow2 27.9%

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

       2. rem-sqrt-square 41.5%

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

       3. associate-*r* 44.1%

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

       4. *-commutative 44.1%

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

       5. associate-*r* 44.6%

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

   11. Simplified 44.6%

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

   if 9.9999999999999993e44 < l < 8.50000000000000015e58

   1. Initial program 76.7%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in n around 0 57.3%

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

      1. associate-*r* 62.4%

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

      2. cancel-sign-sub-inv 62.4%

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

      3. metadata-eval 62.4%

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

      4. *-commutative 62.4%

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

      5. associate-*l/ 62.4%

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

   7. Simplified 62.4%

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

      1. pow1/2 77.6%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 77.6%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 77.6%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 77.6%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 77.6%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 77.6%

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

   9. Applied egg-rr 77.6%

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

   10. Taylor expanded in l around inf 72.1%

       \[\leadsto {\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}}^{0.5} \]
   11. Step-by-step derivation

       1. associate-*r/ 72.1%

          \[\leadsto {\color{blue}{\left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)}}^{0.5} \]

       2. associate-*r* 72.1%

          \[\leadsto {\left(\frac{\color{blue}{\left(-4 \cdot U\right) \cdot \left({\ell}^{2} \cdot n\right)}}{Om}\right)}^{0.5} \]

       3. *-commutative 72.1%

          \[\leadsto {\left(\frac{\left(-4 \cdot U\right) \cdot \color{blue}{\left(n \cdot {\ell}^{2}\right)}}{Om}\right)}^{0.5} \]

   12. Simplified 72.1%

       \[\leadsto {\color{blue}{\left(\frac{\left(-4 \cdot U\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)}}^{0.5} \]

   if 8.50000000000000015e58 < l < 9.59999999999999958e81

   1. Initial program 36.8%

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

   3. Simplified 36.8%

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

      1. associate-*r* 36.8%

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

      2. fma-define 36.8%

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

      3. associate-*r* 36.8%

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

   6. Applied egg-rr 36.8%

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

   7. Taylor expanded in U around 0 4.8%

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

      1. mul-1-neg 4.8%

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

      2. associate-/l* 36.8%

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

      3. unpow2 36.8%

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

      4. unpow2 36.8%

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

      5. times-frac 36.8%

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

      6. unpow2 36.8%

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

   9. Simplified 36.8%

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

   10. Taylor expanded in n around inf 3.4%

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

       1. associate-*r/ 3.4%

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

       2. associate-*r* 2.8%

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

   12. Simplified 2.8%

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

       1. pow1/2 2.8%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 36.7%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 36.7%

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

       2. unpow2 36.7%

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

       3. rem-sqrt-square 40.7%

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

       4. associate-/l* 40.7%

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

       5. *-commutative 40.7%

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

       6. associate-/l* 40.7%

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

   16. Simplified 40.7%

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

       1. associate-*r/ 40.7%

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

       2. clear-num 40.7%

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

   18. Applied egg-rr 40.7%

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

   if 9.59999999999999958e81 < l < 1e108

   1. Initial program 35.9%

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

   3. Simplified 35.9%

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

      1. associate-*r* 35.9%

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

      2. fma-define 52.6%

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

      3. associate-*r* 52.6%

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

   6. Applied egg-rr 52.6%

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

   7. Taylor expanded in U around 0 35.9%

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

      1. mul-1-neg 35.9%

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

      2. associate-/l* 52.6%

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

      3. unpow2 52.6%

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

      4. unpow2 52.6%

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

      5. times-frac 52.6%

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

      6. unpow2 52.6%

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

   9. Simplified 52.6%

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

   10. Taylor expanded in n around inf 35.6%

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

       1. associate-*r/ 35.6%

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

       2. associate-*r* 35.6%

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

   12. Simplified 35.6%

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

       1. pow1/2 35.6%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 35.6%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 35.6%

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

       2. unpow2 35.6%

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

       3. rem-sqrt-square 51.3%

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

       4. associate-/l* 51.3%

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

       5. *-commutative 51.3%

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

       6. associate-/l* 51.1%

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

   16. Simplified 51.1%

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

       1. associate-*r* 50.9%

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

       2. clear-num 50.9%

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

       3. un-div-inv 51.1%

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

       4. associate-*r* 51.1%

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

   18. Applied egg-rr 51.1%

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

   if 1e108 < l < 2.0499999999999999e110

   1. Initial program 98.4%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in Om around inf 100.0%

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

      1. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 100.0%

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

      1. associate-/l* 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if 2.0499999999999999e110 < l < 1.12e133

   1. Initial program 44.2%

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

   3. Simplified 44.2%

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

      1. associate-*r* 44.2%

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

      2. fma-define 58.5%

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

      3. associate-*r* 58.5%

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

   6. Applied egg-rr 58.5%

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

   7. Taylor expanded in U around 0 44.2%

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

      1. mul-1-neg 44.2%

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

      2. associate-/l* 58.5%

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

      3. unpow2 58.5%

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

      4. unpow2 58.5%

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

      5. times-frac 58.5%

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

      6. unpow2 58.5%

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

   9. Simplified 58.5%

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

   10. Taylor expanded in n around inf 30.7%

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

       1. associate-*r/ 30.7%

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

       2. associate-*r* 30.2%

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

   12. Simplified 30.2%

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

       1. pow1/2 30.2%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 45.3%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 45.3%

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

       2. unpow2 45.3%

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

       3. rem-sqrt-square 58.7%

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

       4. associate-/l* 58.3%

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

       5. *-commutative 58.3%

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

       6. associate-/l* 71.9%

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

   16. Simplified 71.9%

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

   if 1.12e133 < l < 2.70000000000000008e150

   1. Initial program 50.0%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in n around 0 51.7%

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

      1. associate-*r* 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. metadata-eval 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*l/ 99.2%

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

   7. Simplified 99.2%

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

      1. pow1/2 99.2%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 99.2%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 99.2%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 99.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 100.0%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in l around inf 3.3%

       \[\leadsto {\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}}^{0.5} \]
   11. Step-by-step derivation

       1. *-commutative 3.3%

          \[\leadsto {\left(-4 \cdot \frac{U \cdot \color{blue}{\left(n \cdot {\ell}^{2}\right)}}{Om}\right)}^{0.5} \]

       2. associate-*r* 99.2%

          \[\leadsto {\left(-4 \cdot \frac{\color{blue}{\left(U \cdot n\right) \cdot {\ell}^{2}}}{Om}\right)}^{0.5} \]

   12. Simplified 99.2%

       \[\leadsto {\color{blue}{\left(-4 \cdot \frac{\left(U \cdot n\right) \cdot {\ell}^{2}}{Om}\right)}}^{0.5} \]

   if 2.70000000000000008e150 < l < 1.7999999999999999e275

   1. Initial program 15.0%

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

   3. Simplified 31.2%

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

      1. associate-*r* 31.2%

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

      2. fma-define 34.4%

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

      3. associate-*r* 34.1%

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

   6. Applied egg-rr 34.1%

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

   7. Taylor expanded in U around 0 17.1%

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

      1. mul-1-neg 17.1%

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

      2. associate-/l* 17.1%

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

      3. unpow2 17.1%

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

      4. unpow2 17.1%

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

      5. times-frac 34.2%

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

      6. unpow2 34.2%

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

   9. Simplified 34.2%

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

   10. Taylor expanded in n around inf 30.1%

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

       1. associate-*r/ 30.1%

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

       2. associate-*r* 29.9%

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

   12. Simplified 29.9%

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

       1. pow1/2 29.9%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 34.5%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 34.5%

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

       2. unpow2 34.5%

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

       3. rem-sqrt-square 34.6%

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

       4. associate-/l* 37.5%

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

       5. *-commutative 37.5%

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

       6. associate-/l* 46.5%

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

   16. Simplified 46.5%

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

       1. clear-num 46.6%

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

       2. un-div-inv 46.6%

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

   18. Applied egg-rr 46.6%

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

   if 1.7999999999999999e275 < l

   1. Initial program 16.7%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in n around 0 18.4%

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

      1. associate-*r* 18.4%

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

      2. cancel-sign-sub-inv 18.4%

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

      3. metadata-eval 18.4%

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

      4. *-commutative 18.4%

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

      5. associate-*l/ 18.4%

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

   7. Simplified 18.4%

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

      1. pow1/2 18.4%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 18.4%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 18.4%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 18.4%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 18.4%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 18.4%

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

   9. Applied egg-rr 18.4%

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

   10. Taylor expanded in l around inf 18.4%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)}^{0.5} \]
   11. Step-by-step derivation

       1. associate-*r/ 18.4%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)}^{0.5} \]

       2. *-commutative 18.4%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om}\right)}^{0.5} \]

       3. associate-*r/ 18.4%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]

   12. Simplified 18.4%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 18.4%

          \[\leadsto \color{blue}{1 \cdot {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left({\ell}^{2} \cdot \frac{-2}{Om}\right)\right)}^{0.5}} \]

       2. unpow1/2 18.4%

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

       3. *-commutative 18.4%

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

       4. sqrt-prod 17.3%

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

       5. sqrt-prod 17.3%

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

       6. sqrt-pow1 44.4%

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

       7. metadata-eval 44.4%

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

       8. pow1 44.4%

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

   14. Applied egg-rr 44.4%

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

       1. *-lft-identity 44.4%

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

       2. associate-*l* 57.5%

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

       3. *-commutative 57.5%

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

   16. Simplified 57.5%

       \[\leadsto \color{blue}{\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{2 \cdot \left(U \cdot n\right)}\right)} \]
3. Recombined 9 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+45}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+58}:\\ \;\;\;\;{\left(\frac{\left(U \cdot -4\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 9.6 \cdot 10^{+81}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{1}{\frac{Om}{n \cdot \ell}}\right|\\ \mathbf{elif}\;\ell \leq 10^{+108}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{-4 \cdot \left(U \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 1.12 \cdot 10^{+133}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)\right|\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+150}:\\ \;\;\;\;{\left(-4 \cdot \frac{{\ell}^{2} \cdot \left(n \cdot U\right)}{Om}\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+275}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\ell}{\frac{Om}{n}}\right|\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ t_2 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;\ell \leq 2.02 \cdot 10^{-128}:\\ \;\;\;\;{\left(t \cdot t\_2\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+155}:\\ \;\;\;\;{\left(t\_2 \cdot \left({\ell}^{2} \cdot \frac{-2}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+211}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{t\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fabs (/ (* l (sqrt (* U* (* 2.0 U)))) (/ Om n))))
        (t_2 (* 2.0 (* n U))))
   (if (<= l 2.02e-128)
     (pow (* t t_2) 0.5)
     (if (<= l 1.35e+55)
       (sqrt (* (* 2.0 n) (* U (- t (/ (* 2.0 (pow l 2.0)) Om)))))
       (if (<= l 2.3e+124)
         t_1
         (if (<= l 7.2e+155)
           (pow (* t_2 (* (pow l 2.0) (/ -2.0 Om))) 0.5)
           (if (<= l 6.2e+211)
             t_1
             (* l (* (sqrt (/ -2.0 Om)) (sqrt t_2))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fabs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	double t_2 = 2.0 * (n * U);
	double tmp;
	if (l <= 2.02e-128) {
		tmp = pow((t * t_2), 0.5);
	} else if (l <= 1.35e+55) {
		tmp = sqrt(((2.0 * n) * (U * (t - ((2.0 * pow(l, 2.0)) / Om)))));
	} else if (l <= 2.3e+124) {
		tmp = t_1;
	} else if (l <= 7.2e+155) {
		tmp = pow((t_2 * (pow(l, 2.0) * (-2.0 / Om))), 0.5);
	} else if (l <= 6.2e+211) {
		tmp = t_1;
	} else {
		tmp = l * (sqrt((-2.0 / Om)) * sqrt(t_2));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = abs(((l * sqrt((u_42 * (2.0d0 * u)))) / (om / n)))
    t_2 = 2.0d0 * (n * u)
    if (l <= 2.02d-128) then
        tmp = (t * t_2) ** 0.5d0
    else if (l <= 1.35d+55) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - ((2.0d0 * (l ** 2.0d0)) / om)))))
    else if (l <= 2.3d+124) then
        tmp = t_1
    else if (l <= 7.2d+155) then
        tmp = (t_2 * ((l ** 2.0d0) * ((-2.0d0) / om))) ** 0.5d0
    else if (l <= 6.2d+211) then
        tmp = t_1
    else
        tmp = l * (sqrt(((-2.0d0) / om)) * sqrt(t_2))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.abs(((l * Math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	double t_2 = 2.0 * (n * U);
	double tmp;
	if (l <= 2.02e-128) {
		tmp = Math.pow((t * t_2), 0.5);
	} else if (l <= 1.35e+55) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - ((2.0 * Math.pow(l, 2.0)) / Om)))));
	} else if (l <= 2.3e+124) {
		tmp = t_1;
	} else if (l <= 7.2e+155) {
		tmp = Math.pow((t_2 * (Math.pow(l, 2.0) * (-2.0 / Om))), 0.5);
	} else if (l <= 6.2e+211) {
		tmp = t_1;
	} else {
		tmp = l * (Math.sqrt((-2.0 / Om)) * Math.sqrt(t_2));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.fabs(((l * math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)))
	t_2 = 2.0 * (n * U)
	tmp = 0
	if l <= 2.02e-128:
		tmp = math.pow((t * t_2), 0.5)
	elif l <= 1.35e+55:
		tmp = math.sqrt(((2.0 * n) * (U * (t - ((2.0 * math.pow(l, 2.0)) / Om)))))
	elif l <= 2.3e+124:
		tmp = t_1
	elif l <= 7.2e+155:
		tmp = math.pow((t_2 * (math.pow(l, 2.0) * (-2.0 / Om))), 0.5)
	elif l <= 6.2e+211:
		tmp = t_1
	else:
		tmp = l * (math.sqrt((-2.0 / Om)) * math.sqrt(t_2))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = abs(Float64(Float64(l * sqrt(Float64(U_42_ * Float64(2.0 * U)))) / Float64(Om / n)))
	t_2 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (l <= 2.02e-128)
		tmp = Float64(t * t_2) ^ 0.5;
	elseif (l <= 1.35e+55)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(Float64(2.0 * (l ^ 2.0)) / Om)))));
	elseif (l <= 2.3e+124)
		tmp = t_1;
	elseif (l <= 7.2e+155)
		tmp = Float64(t_2 * Float64((l ^ 2.0) * Float64(-2.0 / Om))) ^ 0.5;
	elseif (l <= 6.2e+211)
		tmp = t_1;
	else
		tmp = Float64(l * Float64(sqrt(Float64(-2.0 / Om)) * sqrt(t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = abs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	t_2 = 2.0 * (n * U);
	tmp = 0.0;
	if (l <= 2.02e-128)
		tmp = (t * t_2) ^ 0.5;
	elseif (l <= 1.35e+55)
		tmp = sqrt(((2.0 * n) * (U * (t - ((2.0 * (l ^ 2.0)) / Om)))));
	elseif (l <= 2.3e+124)
		tmp = t_1;
	elseif (l <= 7.2e+155)
		tmp = (t_2 * ((l ^ 2.0) * (-2.0 / Om))) ^ 0.5;
	elseif (l <= 6.2e+211)
		tmp = t_1;
	else
		tmp = l * (sqrt((-2.0 / Om)) * sqrt(t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Abs[N[(N[(l * N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 2.02e-128], N[Power[N[(t * t$95$2), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.35e+55], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.3e+124], t$95$1, If[LessEqual[l, 7.2e+155], N[Power[N[(t$95$2 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 6.2e+211], t$95$1, N[(l * N[(N[Sqrt[N[(-2.0 / Om), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < 2.0199999999999999e-128

   1. Initial program 51.6%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in n around 0 42.1%

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

      1. associate-*r* 41.6%

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

      2. cancel-sign-sub-inv 41.6%

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

      3. metadata-eval 41.6%

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

      4. *-commutative 41.6%

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

      5. associate-*l/ 41.6%

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

   7. Simplified 41.6%

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

      1. pow1/2 46.1%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 46.1%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 46.1%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 46.1%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 46.1%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 46.1%

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

   9. Applied egg-rr 46.1%

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

   10. Taylor expanded in l around 0 40.8%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{t}\right)}^{0.5} \]

   if 2.0199999999999999e-128 < l < 1.34999999999999988e55

   1. Initial program 66.0%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in Om around inf 63.4%

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

      1. associate-*r/ 63.4%

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

   7. Simplified 63.4%

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

   if 1.34999999999999988e55 < l < 2.29999999999999985e124 or 7.20000000000000015e155 < l < 6.2000000000000003e211

   1. Initial program 36.5%

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

   3. Simplified 43.3%

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

      1. associate-*r* 43.3%

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

      2. fma-define 49.9%

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

      3. associate-*r* 49.8%

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

   6. Applied egg-rr 49.8%

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

   7. Taylor expanded in U around 0 29.4%

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

      1. mul-1-neg 29.4%

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

      2. associate-/l* 42.6%

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

      3. unpow2 42.6%

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

      4. unpow2 42.6%

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

      5. times-frac 49.8%

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

      6. unpow2 49.8%

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

   9. Simplified 49.8%

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

   10. Taylor expanded in n around inf 28.3%

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

       1. associate-*r/ 28.3%

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

       2. associate-*r* 28.1%

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

   12. Simplified 28.1%

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

       1. pow1/2 28.1%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 38.6%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 38.6%

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

       2. unpow2 38.6%

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

       3. rem-sqrt-square 42.2%

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

       4. associate-/l* 42.1%

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

       5. *-commutative 42.1%

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

       6. associate-/l* 48.2%

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

   16. Simplified 48.2%

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

       1. associate-*r* 48.2%

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

       2. clear-num 48.2%

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

       3. un-div-inv 48.3%

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

       4. associate-*r* 48.3%

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

   18. Applied egg-rr 48.3%

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

   if 2.29999999999999985e124 < l < 7.20000000000000015e155

   1. Initial program 43.5%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in n around 0 44.0%

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

      1. associate-*r* 57.6%

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

      2. cancel-sign-sub-inv 57.6%

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

      3. metadata-eval 57.6%

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

      4. *-commutative 57.6%

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

      5. associate-*l/ 57.6%

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

   7. Simplified 57.6%

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

      1. pow1/2 86.2%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 86.2%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 86.2%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 86.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 86.4%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 86.4%

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

   9. Applied egg-rr 86.4%

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

   10. Taylor expanded in l around inf 86.2%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)}^{0.5} \]
   11. Step-by-step derivation

       1. associate-*r/ 86.2%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)}^{0.5} \]

       2. *-commutative 86.2%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om}\right)}^{0.5} \]

       3. associate-*r/ 86.4%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]

   12. Simplified 86.4%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]

   if 6.2000000000000003e211 < l

   1. Initial program 7.0%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in n around 0 12.8%

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

      1. associate-*r* 7.8%

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

      2. cancel-sign-sub-inv 7.8%

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

      3. metadata-eval 7.8%

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

      4. *-commutative 7.8%

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

      5. associate-*l/ 7.8%

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

   7. Simplified 7.8%

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

      1. pow1/2 12.7%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 12.7%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 12.7%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 12.7%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 12.7%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 12.7%

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

   9. Applied egg-rr 12.7%

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

   10. Taylor expanded in l around inf 12.7%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)}^{0.5} \]
   11. Step-by-step derivation

       1. associate-*r/ 12.7%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)}^{0.5} \]

       2. *-commutative 12.7%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om}\right)}^{0.5} \]

       3. associate-*r/ 12.7%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]

   12. Simplified 12.7%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 12.7%

          \[\leadsto \color{blue}{1 \cdot {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left({\ell}^{2} \cdot \frac{-2}{Om}\right)\right)}^{0.5}} \]

       2. unpow1/2 7.8%

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

       3. *-commutative 7.8%

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

       4. sqrt-prod 6.5%

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

       5. sqrt-prod 6.5%

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

       6. sqrt-pow1 32.8%

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

       7. metadata-eval 32.8%

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

       8. pow1 32.8%

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

   14. Applied egg-rr 32.8%

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

       1. *-lft-identity 32.8%

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

       2. associate-*l* 37.0%

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

       3. *-commutative 37.0%

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

   16. Simplified 37.0%

       \[\leadsto \color{blue}{\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{2 \cdot \left(U \cdot n\right)}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.02 \cdot 10^{-128}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+124}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{+155}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left({\ell}^{2} \cdot \frac{-2}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+211}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ t_2 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;\ell \leq 7.5 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+126}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+151}:\\ \;\;\;\;{\left(t\_2 \cdot \left({\ell}^{2} \cdot \frac{-2}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+156}:\\ \;\;\;\;{\left(\left(t\_1 \cdot t\_1\right) \cdot 4\right)}^{0.25}\\ \mathbf{elif}\;\ell \leq 1.36 \cdot 10^{+206}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\ell}{\frac{Om}{n}}\right|\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{t\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* U t))) (t_2 (* 2.0 (* n U))))
   (if (<= l 7.5e+64)
     (sqrt (* 2.0 (* (* n U) (+ t (/ (* -2.0 (pow l 2.0)) Om)))))
     (if (<= l 1.75e+126)
       (fabs (/ (* l (sqrt (* U* (* 2.0 U)))) (/ Om n)))
       (if (<= l 3.6e+151)
         (pow (* t_2 (* (pow l 2.0) (/ -2.0 Om))) 0.5)
         (if (<= l 2.7e+156)
           (pow (* (* t_1 t_1) 4.0) 0.25)
           (if (<= l 1.36e+206)
             (fabs (* (sqrt (* 2.0 (* U U*))) (/ l (/ Om n))))
             (* l (* (sqrt (/ -2.0 Om)) (sqrt t_2))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * t);
	double t_2 = 2.0 * (n * U);
	double tmp;
	if (l <= 7.5e+64) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((-2.0 * pow(l, 2.0)) / Om)))));
	} else if (l <= 1.75e+126) {
		tmp = fabs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	} else if (l <= 3.6e+151) {
		tmp = pow((t_2 * (pow(l, 2.0) * (-2.0 / Om))), 0.5);
	} else if (l <= 2.7e+156) {
		tmp = pow(((t_1 * t_1) * 4.0), 0.25);
	} else if (l <= 1.36e+206) {
		tmp = fabs((sqrt((2.0 * (U * U_42_))) * (l / (Om / n))));
	} else {
		tmp = l * (sqrt((-2.0 / Om)) * sqrt(t_2));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = n * (u * t)
    t_2 = 2.0d0 * (n * u)
    if (l <= 7.5d+64) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (((-2.0d0) * (l ** 2.0d0)) / om)))))
    else if (l <= 1.75d+126) then
        tmp = abs(((l * sqrt((u_42 * (2.0d0 * u)))) / (om / n)))
    else if (l <= 3.6d+151) then
        tmp = (t_2 * ((l ** 2.0d0) * ((-2.0d0) / om))) ** 0.5d0
    else if (l <= 2.7d+156) then
        tmp = ((t_1 * t_1) * 4.0d0) ** 0.25d0
    else if (l <= 1.36d+206) then
        tmp = abs((sqrt((2.0d0 * (u * u_42))) * (l / (om / n))))
    else
        tmp = l * (sqrt(((-2.0d0) / om)) * sqrt(t_2))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * t);
	double t_2 = 2.0 * (n * U);
	double tmp;
	if (l <= 7.5e+64) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((-2.0 * Math.pow(l, 2.0)) / Om)))));
	} else if (l <= 1.75e+126) {
		tmp = Math.abs(((l * Math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	} else if (l <= 3.6e+151) {
		tmp = Math.pow((t_2 * (Math.pow(l, 2.0) * (-2.0 / Om))), 0.5);
	} else if (l <= 2.7e+156) {
		tmp = Math.pow(((t_1 * t_1) * 4.0), 0.25);
	} else if (l <= 1.36e+206) {
		tmp = Math.abs((Math.sqrt((2.0 * (U * U_42_))) * (l / (Om / n))));
	} else {
		tmp = l * (Math.sqrt((-2.0 / Om)) * Math.sqrt(t_2));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * (U * t)
	t_2 = 2.0 * (n * U)
	tmp = 0
	if l <= 7.5e+64:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((-2.0 * math.pow(l, 2.0)) / Om)))))
	elif l <= 1.75e+126:
		tmp = math.fabs(((l * math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)))
	elif l <= 3.6e+151:
		tmp = math.pow((t_2 * (math.pow(l, 2.0) * (-2.0 / Om))), 0.5)
	elif l <= 2.7e+156:
		tmp = math.pow(((t_1 * t_1) * 4.0), 0.25)
	elif l <= 1.36e+206:
		tmp = math.fabs((math.sqrt((2.0 * (U * U_42_))) * (l / (Om / n))))
	else:
		tmp = l * (math.sqrt((-2.0 / Om)) * math.sqrt(t_2))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U * t))
	t_2 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (l <= 7.5e+64)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(-2.0 * (l ^ 2.0)) / Om)))));
	elseif (l <= 1.75e+126)
		tmp = abs(Float64(Float64(l * sqrt(Float64(U_42_ * Float64(2.0 * U)))) / Float64(Om / n)));
	elseif (l <= 3.6e+151)
		tmp = Float64(t_2 * Float64((l ^ 2.0) * Float64(-2.0 / Om))) ^ 0.5;
	elseif (l <= 2.7e+156)
		tmp = Float64(Float64(t_1 * t_1) * 4.0) ^ 0.25;
	elseif (l <= 1.36e+206)
		tmp = abs(Float64(sqrt(Float64(2.0 * Float64(U * U_42_))) * Float64(l / Float64(Om / n))));
	else
		tmp = Float64(l * Float64(sqrt(Float64(-2.0 / Om)) * sqrt(t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * (U * t);
	t_2 = 2.0 * (n * U);
	tmp = 0.0;
	if (l <= 7.5e+64)
		tmp = sqrt((2.0 * ((n * U) * (t + ((-2.0 * (l ^ 2.0)) / Om)))));
	elseif (l <= 1.75e+126)
		tmp = abs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	elseif (l <= 3.6e+151)
		tmp = (t_2 * ((l ^ 2.0) * (-2.0 / Om))) ^ 0.5;
	elseif (l <= 2.7e+156)
		tmp = ((t_1 * t_1) * 4.0) ^ 0.25;
	elseif (l <= 1.36e+206)
		tmp = abs((sqrt((2.0 * (U * U_42_))) * (l / (Om / n))));
	else
		tmp = l * (sqrt((-2.0 / Om)) * sqrt(t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 7.5e+64], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(-2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.75e+126], N[Abs[N[(N[(l * N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.6e+151], N[Power[N[(t$95$2 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 2.7e+156], N[Power[N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 4.0), $MachinePrecision], 0.25], $MachinePrecision], If[LessEqual[l, 1.36e+206], N[Abs[N[(N[Sqrt[N[(2.0 * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l * N[(N[Sqrt[N[(-2.0 / Om), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if l < 7.5000000000000005e64

   1. Initial program 54.5%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in n around 0 47.1%

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

      1. associate-*r* 45.9%

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

      2. cancel-sign-sub-inv 45.9%

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

      3. metadata-eval 45.9%

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

      4. *-commutative 45.9%

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

      5. associate-*l/ 45.9%

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

   7. Simplified 45.9%

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

   if 7.5000000000000005e64 < l < 1.7500000000000001e126

   1. Initial program 35.7%

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

   3. Simplified 35.8%

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

      1. associate-*r* 35.8%

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

      2. fma-define 49.2%

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

      3. associate-*r* 49.2%

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

   6. Applied egg-rr 49.2%

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

   7. Taylor expanded in U around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. associate-/l* 49.2%

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

      3. unpow2 49.2%

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

      4. unpow2 49.2%

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

      5. times-frac 49.2%

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

      6. unpow2 49.2%

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

   9. Simplified 49.2%

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

   10. Taylor expanded in n around inf 22.6%

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

       1. associate-*r/ 22.6%

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

       2. associate-*r* 22.2%

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

   12. Simplified 22.2%

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

       1. pow1/2 22.2%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 36.0%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 36.0%

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

       2. unpow2 36.0%

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

       3. rem-sqrt-square 49.4%

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

       4. associate-/l* 49.2%

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

       5. *-commutative 49.2%

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

       6. associate-/l* 55.2%

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

   16. Simplified 55.2%

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

       1. associate-*r* 55.2%

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

       2. clear-num 55.2%

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

       3. un-div-inv 55.1%

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

       4. associate-*r* 55.1%

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

   18. Applied egg-rr 55.1%

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

   if 1.7500000000000001e126 < l < 3.6e151

   1. Initial program 74.6%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in n around 0 75.5%

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

      1. associate-*r* 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. metadata-eval 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*l/ 99.2%

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

   7. Simplified 99.2%

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

      1. pow1/2 99.2%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 99.2%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 99.2%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 99.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 99.6%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in l around inf 99.2%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)}^{0.5} \]
   11. Step-by-step derivation

       1. associate-*r/ 99.2%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)}^{0.5} \]

       2. *-commutative 99.2%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om}\right)}^{0.5} \]

       3. associate-*r/ 99.6%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]

   12. Simplified 99.6%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]

   if 3.6e151 < l < 2.7e156

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{U \cdot \left(n \cdot t\right)} \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right) \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}} \]

      2. pow3 100.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}^{3}}} \]

      3. associate-*r* 100.0%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(U \cdot n\right) \cdot t}}\right)}^{3}} \]

      4. *-commutative 100.0%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(n \cdot U\right)} \cdot t}\right)}^{3}} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\left(n \cdot U\right) \cdot t}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. pow1/2 100.0%

         \[\leadsto \color{blue}{{\left(2 \cdot {\left(\sqrt[3]{\left(n \cdot U\right) \cdot t}\right)}^{3}\right)}^{0.5}} \]

      2. rem-cube-cbrt 100.0%

         \[\leadsto {\left(2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}\right)}^{0.5} \]

      3. associate-*l* 100.0%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}}^{0.5} \]

      4. sqr-pow 100.0%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}^{\left(\frac{0.5}{2}\right)}} \]

      5. pow-prod-down 100.0%

         \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right) \cdot \left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right)}^{\left(\frac{0.5}{2}\right)}} \]

      6. associate-*l* 100.0%

         \[\leadsto {\left(\color{blue}{\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)} \cdot \left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      7. associate-*l* 100.0%

         \[\leadsto {\left(\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right) \cdot \color{blue}{\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. *-commutative 100.0%

         \[\leadsto {\left(\color{blue}{\left(\left(\left(n \cdot U\right) \cdot t\right) \cdot 2\right)} \cdot \left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. *-commutative 100.0%

         \[\leadsto {\left(\left(\left(\left(n \cdot U\right) \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(\left(\left(n \cdot U\right) \cdot t\right) \cdot 2\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. swap-sqr 100.0%

          \[\leadsto {\color{blue}{\left(\left(\left(\left(n \cdot U\right) \cdot t\right) \cdot \left(\left(n \cdot U\right) \cdot t\right)\right) \cdot \left(2 \cdot 2\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      11. pow2 100.0%

          \[\leadsto {\left(\color{blue}{{\left(\left(n \cdot U\right) \cdot t\right)}^{2}} \cdot \left(2 \cdot 2\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. associate-*l* 100.0%

          \[\leadsto {\left({\color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}}^{2} \cdot \left(2 \cdot 2\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. metadata-eval 100.0%

          \[\leadsto {\left({\left(n \cdot \left(U \cdot t\right)\right)}^{2} \cdot \color{blue}{4}\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. metadata-eval 100.0%

          \[\leadsto {\left({\left(n \cdot \left(U \cdot t\right)\right)}^{2} \cdot 4\right)}^{\color{blue}{0.25}} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left({\left(n \cdot \left(U \cdot t\right)\right)}^{2} \cdot 4\right)}^{0.25}} \]
   10. Step-by-step derivation

       1. unpow2 100.0%

          \[\leadsto {\left(\color{blue}{\left(\left(n \cdot \left(U \cdot t\right)\right) \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)} \cdot 4\right)}^{0.25} \]

   11. Applied egg-rr 100.0%

       \[\leadsto {\left(\color{blue}{\left(\left(n \cdot \left(U \cdot t\right)\right) \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)} \cdot 4\right)}^{0.25} \]

   if 2.7e156 < l < 1.36e206

   1. Initial program 23.4%

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

   3. Simplified 37.7%

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

      1. associate-*r* 37.7%

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

      2. fma-define 44.9%

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

      3. associate-*r* 44.5%

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

   6. Applied egg-rr 44.5%

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

   7. Taylor expanded in U around 0 29.3%

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

      1. mul-1-neg 29.3%

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

      2. associate-/l* 29.3%

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

      3. unpow2 29.3%

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

      4. unpow2 29.3%

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

      5. times-frac 44.5%

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

      6. unpow2 44.5%

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

   9. Simplified 44.5%

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

   10. Taylor expanded in n around inf 36.4%

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

       1. associate-*r/ 36.4%

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

       2. associate-*r* 36.4%

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

   12. Simplified 36.4%

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

       1. pow1/2 36.4%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 44.1%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 44.1%

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

       2. unpow2 44.1%

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

       3. rem-sqrt-square 44.1%

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

       4. associate-/l* 44.1%

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

       5. *-commutative 44.1%

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

       6. associate-/l* 50.7%

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

   16. Simplified 50.7%

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

       1. clear-num 50.9%

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

       2. un-div-inv 50.9%

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

   18. Applied egg-rr 50.9%

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

   if 1.36e206 < l

   1. Initial program 11.1%

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

   3. Simplified 36.3%

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

   5. Taylor expanded in n around 0 16.6%

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

      1. associate-*r* 11.8%

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

      2. cancel-sign-sub-inv 11.8%

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

      3. metadata-eval 11.8%

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

      4. *-commutative 11.8%

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

      5. associate-*l/ 11.8%

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

   7. Simplified 11.8%

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

      1. pow1/2 16.5%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 16.5%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 16.5%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 16.5%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 16.5%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 16.5%

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

   9. Applied egg-rr 16.5%

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

   10. Taylor expanded in l around inf 16.5%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)}^{0.5} \]
   11. Step-by-step derivation

       1. associate-*r/ 16.5%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)}^{0.5} \]

       2. *-commutative 16.5%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om}\right)}^{0.5} \]

       3. associate-*r/ 16.5%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]

   12. Simplified 16.5%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 16.5%

          \[\leadsto \color{blue}{1 \cdot {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left({\ell}^{2} \cdot \frac{-2}{Om}\right)\right)}^{0.5}} \]

       2. unpow1/2 11.8%

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

       3. *-commutative 11.8%

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

       4. sqrt-prod 6.2%

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

       5. sqrt-prod 6.2%

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

       6. sqrt-pow1 31.4%

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

       7. metadata-eval 31.4%

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

       8. pow1 31.4%

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

   14. Applied egg-rr 31.4%

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

       1. *-lft-identity 31.4%

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

       2. associate-*l* 35.4%

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

       3. *-commutative 35.4%

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

   16. Simplified 35.4%

       \[\leadsto \color{blue}{\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{2 \cdot \left(U \cdot n\right)}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+126}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+151}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left({\ell}^{2} \cdot \frac{-2}{Om}\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+156}:\\ \;\;\;\;{\left(\left(\left(n \cdot \left(U \cdot t\right)\right) \cdot \left(n \cdot \left(U \cdot t\right)\right)\right) \cdot 4\right)}^{0.25}\\ \mathbf{elif}\;\ell \leq 1.36 \cdot 10^{+206}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\ell}{\frac{Om}{n}}\right|\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{if}\;\ell \leq 1.12 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+108}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+275}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\ell}{\frac{Om}{n}}\right|\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (* 2.0 (* U (* n (+ t (* -2.0 (/ (pow l 2.0) Om)))))) 0.5)))
   (if (<= l 1.12e+61)
     t_1
     (if (<= l 1.6e+108)
       (fabs (/ (* l (sqrt (* U* (* 2.0 U)))) (/ Om n)))
       (if (<= l 7.5e+136)
         t_1
         (if (<= l 1.95e+275)
           (fabs (* (sqrt (* 2.0 (* U U*))) (/ l (/ Om n))))
           (* l (* (sqrt (/ -2.0 Om)) (sqrt (* 2.0 (* n U)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((2.0 * (U * (n * (t + (-2.0 * (pow(l, 2.0) / Om)))))), 0.5);
	double tmp;
	if (l <= 1.12e+61) {
		tmp = t_1;
	} else if (l <= 1.6e+108) {
		tmp = fabs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	} else if (l <= 7.5e+136) {
		tmp = t_1;
	} else if (l <= 1.95e+275) {
		tmp = fabs((sqrt((2.0 * (U * U_42_))) * (l / (Om / n))));
	} else {
		tmp = l * (sqrt((-2.0 / Om)) * sqrt((2.0 * (n * U))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 * (u * (n * (t + ((-2.0d0) * ((l ** 2.0d0) / om)))))) ** 0.5d0
    if (l <= 1.12d+61) then
        tmp = t_1
    else if (l <= 1.6d+108) then
        tmp = abs(((l * sqrt((u_42 * (2.0d0 * u)))) / (om / n)))
    else if (l <= 7.5d+136) then
        tmp = t_1
    else if (l <= 1.95d+275) then
        tmp = abs((sqrt((2.0d0 * (u * u_42))) * (l / (om / n))))
    else
        tmp = l * (sqrt(((-2.0d0) / om)) * sqrt((2.0d0 * (n * u))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.pow((2.0 * (U * (n * (t + (-2.0 * (Math.pow(l, 2.0) / Om)))))), 0.5);
	double tmp;
	if (l <= 1.12e+61) {
		tmp = t_1;
	} else if (l <= 1.6e+108) {
		tmp = Math.abs(((l * Math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	} else if (l <= 7.5e+136) {
		tmp = t_1;
	} else if (l <= 1.95e+275) {
		tmp = Math.abs((Math.sqrt((2.0 * (U * U_42_))) * (l / (Om / n))));
	} else {
		tmp = l * (Math.sqrt((-2.0 / Om)) * Math.sqrt((2.0 * (n * U))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.pow((2.0 * (U * (n * (t + (-2.0 * (math.pow(l, 2.0) / Om)))))), 0.5)
	tmp = 0
	if l <= 1.12e+61:
		tmp = t_1
	elif l <= 1.6e+108:
		tmp = math.fabs(((l * math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)))
	elif l <= 7.5e+136:
		tmp = t_1
	elif l <= 1.95e+275:
		tmp = math.fabs((math.sqrt((2.0 * (U * U_42_))) * (l / (Om / n))))
	else:
		tmp = l * (math.sqrt((-2.0 / Om)) * math.sqrt((2.0 * (n * U))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(U * Float64(n * Float64(t + Float64(-2.0 * Float64((l ^ 2.0) / Om)))))) ^ 0.5
	tmp = 0.0
	if (l <= 1.12e+61)
		tmp = t_1;
	elseif (l <= 1.6e+108)
		tmp = abs(Float64(Float64(l * sqrt(Float64(U_42_ * Float64(2.0 * U)))) / Float64(Om / n)));
	elseif (l <= 7.5e+136)
		tmp = t_1;
	elseif (l <= 1.95e+275)
		tmp = abs(Float64(sqrt(Float64(2.0 * Float64(U * U_42_))) * Float64(l / Float64(Om / n))));
	else
		tmp = Float64(l * Float64(sqrt(Float64(-2.0 / Om)) * sqrt(Float64(2.0 * Float64(n * U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * (U * (n * (t + (-2.0 * ((l ^ 2.0) / Om)))))) ^ 0.5;
	tmp = 0.0;
	if (l <= 1.12e+61)
		tmp = t_1;
	elseif (l <= 1.6e+108)
		tmp = abs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	elseif (l <= 7.5e+136)
		tmp = t_1;
	elseif (l <= 1.95e+275)
		tmp = abs((sqrt((2.0 * (U * U_42_))) * (l / (Om / n))));
	else
		tmp = l * (sqrt((-2.0 / Om)) * sqrt((2.0 * (n * U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(2.0 * N[(U * N[(n * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[l, 1.12e+61], t$95$1, If[LessEqual[l, 1.6e+108], N[Abs[N[(N[(l * N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 7.5e+136], t$95$1, If[LessEqual[l, 1.95e+275], N[Abs[N[(N[Sqrt[N[(2.0 * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l * N[(N[Sqrt[N[(-2.0 / Om), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < 1.12e61 or 1.6e108 < l < 7.5000000000000002e136

   1. Initial program 54.3%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in n around 0 47.1%

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

      1. associate-*r* 46.1%

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

      2. cancel-sign-sub-inv 46.1%

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

      3. metadata-eval 46.1%

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

      4. *-commutative 46.1%

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

      5. associate-*l/ 46.1%

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

   7. Simplified 46.1%

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

      1. pow1/2 50.6%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 50.6%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 50.6%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 50.6%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 50.6%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 50.6%

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

   9. Applied egg-rr 50.6%

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

   10. Taylor expanded in n around 0 51.7%

       \[\leadsto {\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}}^{0.5} \]

   if 1.12e61 < l < 1.6e108

   1. Initial program 36.2%

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

   3. Simplified 36.2%

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

      1. associate-*r* 36.2%

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

      2. fma-define 47.3%

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

      3. associate-*r* 47.3%

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

   6. Applied egg-rr 47.3%

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

   7. Taylor expanded in U around 0 25.5%

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

      1. mul-1-neg 25.5%

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

      2. associate-/l* 47.3%

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

      3. unpow2 47.3%

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

      4. unpow2 47.3%

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

      5. times-frac 47.3%

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

      6. unpow2 47.3%

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

   9. Simplified 47.3%

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

   10. Taylor expanded in n around inf 24.9%

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

       1. associate-*r/ 24.9%

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

       2. associate-*r* 24.7%

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

   12. Simplified 24.7%

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

       1. pow1/2 24.7%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 36.0%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 36.0%

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

       2. unpow2 36.0%

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

       3. rem-sqrt-square 47.8%

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

       4. associate-/l* 47.8%

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

       5. *-commutative 47.8%

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

       6. associate-/l* 47.6%

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

   16. Simplified 47.6%

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

       1. associate-*r* 47.5%

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

       2. clear-num 47.5%

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

       3. un-div-inv 47.6%

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

       4. associate-*r* 47.6%

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

   18. Applied egg-rr 47.6%

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

   if 7.5000000000000002e136 < l < 1.95e275

   1. Initial program 17.1%

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

   3. Simplified 32.3%

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

      1. associate-*r* 32.3%

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

      2. fma-define 35.3%

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

      3. associate-*r* 38.0%

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

   6. Applied egg-rr 38.0%

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

   7. Taylor expanded in U around 0 22.1%

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

      1. mul-1-neg 22.1%

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

      2. associate-/l* 22.1%

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

      3. unpow2 22.1%

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

      4. unpow2 22.1%

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

      5. times-frac 38.1%

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

      6. unpow2 38.1%

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

   9. Simplified 38.1%

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

   10. Taylor expanded in n around inf 28.2%

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

       1. associate-*r/ 28.2%

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

       2. associate-*r* 28.1%

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

   12. Simplified 28.1%

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

       1. pow1/2 28.1%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 32.6%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 32.6%

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

       2. unpow2 32.6%

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

       3. rem-sqrt-square 32.7%

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

       4. associate-/l* 35.5%

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

       5. *-commutative 35.5%

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

       6. associate-/l* 44.0%

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

   16. Simplified 44.0%

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

       1. clear-num 44.0%

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

       2. un-div-inv 44.0%

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

   18. Applied egg-rr 44.0%

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

   if 1.95e275 < l

   1. Initial program 16.7%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in n around 0 18.4%

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

      1. associate-*r* 18.4%

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

      2. cancel-sign-sub-inv 18.4%

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

      3. metadata-eval 18.4%

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

      4. *-commutative 18.4%

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

      5. associate-*l/ 18.4%

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

   7. Simplified 18.4%

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

      1. pow1/2 18.4%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 18.4%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 18.4%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 18.4%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 18.4%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 18.4%

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

   9. Applied egg-rr 18.4%

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

   10. Taylor expanded in l around inf 18.4%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)}^{0.5} \]
   11. Step-by-step derivation

       1. associate-*r/ 18.4%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)}^{0.5} \]

       2. *-commutative 18.4%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om}\right)}^{0.5} \]

       3. associate-*r/ 18.4%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]

   12. Simplified 18.4%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{-2}{Om}\right)}\right)}^{0.5} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 18.4%

          \[\leadsto \color{blue}{1 \cdot {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left({\ell}^{2} \cdot \frac{-2}{Om}\right)\right)}^{0.5}} \]

       2. unpow1/2 18.4%

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

       3. *-commutative 18.4%

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

       4. sqrt-prod 17.3%

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

       5. sqrt-prod 17.3%

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

       6. sqrt-pow1 44.4%

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

       7. metadata-eval 44.4%

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

       8. pow1 44.4%

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

   14. Applied egg-rr 44.4%

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

       1. *-lft-identity 44.4%

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

       2. associate-*l* 57.5%

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

       3. *-commutative 57.5%

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

   16. Simplified 57.5%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.12 \cdot 10^{+61}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+108}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{+136}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+275}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\ell}{\frac{Om}{n}}\right|\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\frac{-2}{Om}} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{if}\;Om \leq -3.8 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 1.65 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{+242}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (fabs (* U (* n t)))))))
   (if (<= Om -3.8e-75)
     t_1
     (if (<= Om 1.65e+15)
       (fabs (/ (* l (sqrt (* U* (* 2.0 U)))) (/ Om n)))
       (if (<= Om 1.05e+242) t_1 (* (sqrt (* (* 2.0 n) U)) (sqrt t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * fabs((U * (n * t)))));
	double tmp;
	if (Om <= -3.8e-75) {
		tmp = t_1;
	} else if (Om <= 1.65e+15) {
		tmp = fabs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	} else if (Om <= 1.05e+242) {
		tmp = t_1;
	} else {
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * abs((u * (n * t)))))
    if (om <= (-3.8d-75)) then
        tmp = t_1
    else if (om <= 1.65d+15) then
        tmp = abs(((l * sqrt((u_42 * (2.0d0 * u)))) / (om / n)))
    else if (om <= 1.05d+242) then
        tmp = t_1
    else
        tmp = sqrt(((2.0d0 * n) * u)) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((2.0 * Math.abs((U * (n * t)))));
	double tmp;
	if (Om <= -3.8e-75) {
		tmp = t_1;
	} else if (Om <= 1.65e+15) {
		tmp = Math.abs(((l * Math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	} else if (Om <= 1.05e+242) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(((2.0 * n) * U)) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((2.0 * math.fabs((U * (n * t)))))
	tmp = 0
	if Om <= -3.8e-75:
		tmp = t_1
	elif Om <= 1.65e+15:
		tmp = math.fabs(((l * math.sqrt((U_42_ * (2.0 * U)))) / (Om / n)))
	elif Om <= 1.05e+242:
		tmp = t_1
	else:
		tmp = math.sqrt(((2.0 * n) * U)) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * abs(Float64(U * Float64(n * t)))))
	tmp = 0.0
	if (Om <= -3.8e-75)
		tmp = t_1;
	elseif (Om <= 1.65e+15)
		tmp = abs(Float64(Float64(l * sqrt(Float64(U_42_ * Float64(2.0 * U)))) / Float64(Om / n)));
	elseif (Om <= 1.05e+242)
		tmp = t_1;
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * U)) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((2.0 * abs((U * (n * t)))));
	tmp = 0.0;
	if (Om <= -3.8e-75)
		tmp = t_1;
	elseif (Om <= 1.65e+15)
		tmp = abs(((l * sqrt((U_42_ * (2.0 * U)))) / (Om / n)));
	elseif (Om <= 1.05e+242)
		tmp = t_1;
	else
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * N[Abs[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -3.8e-75], t$95$1, If[LessEqual[Om, 1.65e+15], N[Abs[N[(N[(l * N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1.05e+242], t$95$1, N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -3.79999999999999994e-75 or 1.65e15 < Om < 1.05e242

   1. Initial program 55.7%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in t around inf 49.6%

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

      1. add-cube-cbrt 49.1%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{U \cdot \left(n \cdot t\right)} \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right) \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}} \]

      2. pow3 49.1%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}^{3}}} \]

      3. associate-*r* 45.5%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(U \cdot n\right) \cdot t}}\right)}^{3}} \]

      4. *-commutative 45.5%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(n \cdot U\right)} \cdot t}\right)}^{3}} \]

   7. Applied egg-rr 45.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\left(n \cdot U\right) \cdot t}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 46.0%

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

      2. add-sqr-sqrt 45.9%

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

      3. sqrt-unprod 27.1%

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

      4. pow2 27.1%

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

      5. associate-*l* 27.8%

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

   9. Applied egg-rr 27.8%

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

       1. unpow2 27.8%

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

       2. rem-sqrt-square 46.1%

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

       3. associate-*r* 46.8%

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

       4. *-commutative 46.8%

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

       5. associate-*r* 50.5%

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

   11. Simplified 50.5%

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

   if -3.79999999999999994e-75 < Om < 1.65e15

   1. Initial program 38.8%

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

   3. Simplified 38.8%

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

      1. associate-*r* 38.8%

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

      2. fma-define 42.9%

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

      3. associate-*r* 41.9%

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

   6. Applied egg-rr 41.9%

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

   7. Taylor expanded in U around 0 28.2%

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

      1. mul-1-neg 28.2%

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

      2. associate-/l* 27.2%

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

      3. unpow2 27.2%

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

      4. unpow2 27.2%

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

      5. times-frac 42.1%

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

      6. unpow2 42.1%

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

   9. Simplified 42.1%

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

   10. Taylor expanded in n around inf 26.7%

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

       1. associate-*r/ 26.7%

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

       2. associate-*r* 26.4%

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

   12. Simplified 26.4%

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

       1. pow1/2 26.5%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 46.9%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 46.9%

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

       2. unpow2 46.9%

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

       3. rem-sqrt-square 59.5%

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

       4. associate-/l* 57.6%

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

       5. *-commutative 57.6%

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

       6. associate-/l* 59.1%

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

   16. Simplified 59.1%

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

       1. associate-*r* 59.2%

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

       2. clear-num 59.2%

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

       3. un-div-inv 59.2%

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

       4. associate-*r* 59.4%

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

   18. Applied egg-rr 59.4%

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

   if 1.05e242 < Om

   1. Initial program 43.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in t around inf 29.7%

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

      1. *-un-lft-identity 29.7%

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

      2. associate-*r* 35.4%

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

      3. *-commutative 35.4%

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

   7. Applied egg-rr 35.4%

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

      1. *-lft-identity 35.4%

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

      2. associate-*r* 35.4%

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

      3. associate-*r* 35.4%

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

   9. Simplified 35.4%

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

       1. sqrt-prod 40.0%

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

       2. *-commutative 40.0%

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

       3. *-commutative 40.0%

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

   11. Applied egg-rr 40.0%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -3.8 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;Om \leq 1.65 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{\ell \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\right|\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{+242}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{if}\;Om \leq -2.3 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 3200000000000:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\ell}{\frac{Om}{n}}\right|\\ \mathbf{elif}\;Om \leq 8.2 \cdot 10^{+273}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (fabs (* U (* n t)))))))
   (if (<= Om -2.3e-74)
     t_1
     (if (<= Om 3200000000000.0)
       (fabs (* (sqrt (* 2.0 (* U U*))) (/ l (/ Om n))))
       (if (<= Om 8.2e+273) t_1 (* (sqrt (* (* 2.0 n) U)) (sqrt t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * fabs((U * (n * t)))));
	double tmp;
	if (Om <= -2.3e-74) {
		tmp = t_1;
	} else if (Om <= 3200000000000.0) {
		tmp = fabs((sqrt((2.0 * (U * U_42_))) * (l / (Om / n))));
	} else if (Om <= 8.2e+273) {
		tmp = t_1;
	} else {
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * abs((u * (n * t)))))
    if (om <= (-2.3d-74)) then
        tmp = t_1
    else if (om <= 3200000000000.0d0) then
        tmp = abs((sqrt((2.0d0 * (u * u_42))) * (l / (om / n))))
    else if (om <= 8.2d+273) then
        tmp = t_1
    else
        tmp = sqrt(((2.0d0 * n) * u)) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((2.0 * Math.abs((U * (n * t)))));
	double tmp;
	if (Om <= -2.3e-74) {
		tmp = t_1;
	} else if (Om <= 3200000000000.0) {
		tmp = Math.abs((Math.sqrt((2.0 * (U * U_42_))) * (l / (Om / n))));
	} else if (Om <= 8.2e+273) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(((2.0 * n) * U)) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((2.0 * math.fabs((U * (n * t)))))
	tmp = 0
	if Om <= -2.3e-74:
		tmp = t_1
	elif Om <= 3200000000000.0:
		tmp = math.fabs((math.sqrt((2.0 * (U * U_42_))) * (l / (Om / n))))
	elif Om <= 8.2e+273:
		tmp = t_1
	else:
		tmp = math.sqrt(((2.0 * n) * U)) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * abs(Float64(U * Float64(n * t)))))
	tmp = 0.0
	if (Om <= -2.3e-74)
		tmp = t_1;
	elseif (Om <= 3200000000000.0)
		tmp = abs(Float64(sqrt(Float64(2.0 * Float64(U * U_42_))) * Float64(l / Float64(Om / n))));
	elseif (Om <= 8.2e+273)
		tmp = t_1;
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * U)) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((2.0 * abs((U * (n * t)))));
	tmp = 0.0;
	if (Om <= -2.3e-74)
		tmp = t_1;
	elseif (Om <= 3200000000000.0)
		tmp = abs((sqrt((2.0 * (U * U_42_))) * (l / (Om / n))));
	elseif (Om <= 8.2e+273)
		tmp = t_1;
	else
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * N[Abs[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -2.3e-74], t$95$1, If[LessEqual[Om, 3200000000000.0], N[Abs[N[(N[Sqrt[N[(2.0 * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 8.2e+273], t$95$1, N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -2.2999999999999998e-74 or 3.2e12 < Om < 8.19999999999999982e273

   1. Initial program 55.7%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in t around inf 47.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 47.3%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{U \cdot \left(n \cdot t\right)} \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right) \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}} \]

      2. pow3 47.3%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}^{3}}} \]

      3. associate-*r* 44.9%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(U \cdot n\right) \cdot t}}\right)}^{3}} \]

      4. *-commutative 44.9%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(n \cdot U\right)} \cdot t}\right)}^{3}} \]

   7. Applied egg-rr 44.9%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\left(n \cdot U\right) \cdot t}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 45.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}} \]

      2. add-sqr-sqrt 45.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\sqrt{\left(n \cdot U\right) \cdot t} \cdot \sqrt{\left(n \cdot U\right) \cdot t}\right)}} \]

      3. sqrt-unprod 27.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot \left(\left(n \cdot U\right) \cdot t\right)}}} \]

      4. pow2 27.4%

         \[\leadsto \sqrt{2 \cdot \sqrt{\color{blue}{{\left(\left(n \cdot U\right) \cdot t\right)}^{2}}}} \]

      5. associate-*l* 28.0%

         \[\leadsto \sqrt{2 \cdot \sqrt{{\color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}}^{2}}} \]

   9. Applied egg-rr 28.0%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\sqrt{{\left(n \cdot \left(U \cdot t\right)\right)}^{2}}}} \]
   10. Step-by-step derivation

       1. unpow2 28.0%

          \[\leadsto \sqrt{2 \cdot \sqrt{\color{blue}{\left(n \cdot \left(U \cdot t\right)\right) \cdot \left(n \cdot \left(U \cdot t\right)\right)}}} \]

       2. rem-sqrt-square 45.3%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left|n \cdot \left(U \cdot t\right)\right|}} \]

       3. associate-*r* 46.3%

          \[\leadsto \sqrt{2 \cdot \left|\color{blue}{\left(n \cdot U\right) \cdot t}\right|} \]

       4. *-commutative 46.3%

          \[\leadsto \sqrt{2 \cdot \left|\color{blue}{\left(U \cdot n\right)} \cdot t\right|} \]

       5. associate-*r* 48.7%

          \[\leadsto \sqrt{2 \cdot \left|\color{blue}{U \cdot \left(n \cdot t\right)}\right|} \]

   11. Simplified 48.7%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left|U \cdot \left(n \cdot t\right)\right|}} \]

   if -2.2999999999999998e-74 < Om < 3.2e12

   1. Initial program 38.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 38.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 38.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      2. fma-define 42.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]

      3. associate-*r* 41.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]

   6. Applied egg-rr 41.9%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]

   7. Taylor expanded in U around 0 28.2%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-1 \cdot \frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 28.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-\frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]

      2. associate-/l* 27.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-\color{blue}{U* \cdot \frac{{\ell}^{2}}{{Om}^{2}}}\right)\right)\right)} \]

      3. unpow2 27.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right)\right)\right)} \]

      4. unpow2 27.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

      5. times-frac 42.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]

      6. unpow2 42.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right)\right)\right)} \]

   9. Simplified 42.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-U* \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   10. Taylor expanded in n around inf 26.7%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 26.7%

          \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}{{Om}^{2}}}} \]

       2. associate-*r* 26.4%

          \[\leadsto \sqrt{\frac{2 \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{{Om}^{2}}} \]

   12. Simplified 26.4%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   13. Step-by-step derivation

       1. pow1/2 26.5%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 46.9%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 46.9%

          \[\leadsto \color{blue}{\sqrt{{\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}}} \]

       2. unpow2 46.9%

          \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}}} \]

       3. rem-sqrt-square 59.5%

          \[\leadsto \color{blue}{\left|\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|} \]

       4. associate-/l* 57.6%

          \[\leadsto \left|\color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{n \cdot \ell}{Om}}\right| \]

       5. *-commutative 57.6%

          \[\leadsto \left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\color{blue}{\ell \cdot n}}{Om}\right| \]

       6. associate-/l* 59.1%

          \[\leadsto \left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\ell \cdot \frac{n}{Om}\right)}\right| \]

   16. Simplified 59.1%

       \[\leadsto \color{blue}{\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)\right|} \]
   17. Step-by-step derivation

       1. clear-num 59.1%

          \[\leadsto \left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \color{blue}{\frac{1}{\frac{Om}{n}}}\right)\right| \]

       2. un-div-inv 59.2%

          \[\leadsto \left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\frac{\ell}{\frac{Om}{n}}}\right| \]

   18. Applied egg-rr 59.2%

       \[\leadsto \left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\frac{\ell}{\frac{Om}{n}}}\right| \]

   if 8.19999999999999982e273 < Om

   1. Initial program 32.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 32.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 32.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 32.6%

         \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. associate-*r* 32.6%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      3. *-commutative 32.6%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

   7. Applied egg-rr 32.6%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. *-lft-identity 32.6%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]

      2. associate-*r* 32.6%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t}} \]

      3. associate-*r* 32.6%

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

   9. Simplified 32.6%

      \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]
   10. Step-by-step derivation

       1. sqrt-prod 57.2%

          \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}} \]

       2. *-commutative 57.2%

          \[\leadsto \sqrt{\color{blue}{U \cdot \left(2 \cdot n\right)}} \cdot \sqrt{t} \]

       3. *-commutative 57.2%

          \[\leadsto \sqrt{U \cdot \color{blue}{\left(n \cdot 2\right)}} \cdot \sqrt{t} \]

   11. Applied egg-rr 57.2%

       \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot 2\right)} \cdot \sqrt{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.3 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;Om \leq 3200000000000:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\ell}{\frac{Om}{n}}\right|\\ \mathbf{elif}\;Om \leq 8.2 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{if}\;Om \leq -2.1 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 1.25 \cdot 10^{+83}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)\right|\\ \mathbf{elif}\;Om \leq 5.6 \cdot 10^{+250}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (fabs (* U (* n t)))))))
   (if (<= Om -2.1e-75)
     t_1
     (if (<= Om 1.25e+83)
       (fabs (* (sqrt (* 2.0 (* U U*))) (* l (/ n Om))))
       (if (<= Om 5.6e+250) t_1 (* (sqrt (* (* 2.0 n) U)) (sqrt t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * fabs((U * (n * t)))));
	double tmp;
	if (Om <= -2.1e-75) {
		tmp = t_1;
	} else if (Om <= 1.25e+83) {
		tmp = fabs((sqrt((2.0 * (U * U_42_))) * (l * (n / Om))));
	} else if (Om <= 5.6e+250) {
		tmp = t_1;
	} else {
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * abs((u * (n * t)))))
    if (om <= (-2.1d-75)) then
        tmp = t_1
    else if (om <= 1.25d+83) then
        tmp = abs((sqrt((2.0d0 * (u * u_42))) * (l * (n / om))))
    else if (om <= 5.6d+250) then
        tmp = t_1
    else
        tmp = sqrt(((2.0d0 * n) * u)) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((2.0 * Math.abs((U * (n * t)))));
	double tmp;
	if (Om <= -2.1e-75) {
		tmp = t_1;
	} else if (Om <= 1.25e+83) {
		tmp = Math.abs((Math.sqrt((2.0 * (U * U_42_))) * (l * (n / Om))));
	} else if (Om <= 5.6e+250) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(((2.0 * n) * U)) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((2.0 * math.fabs((U * (n * t)))))
	tmp = 0
	if Om <= -2.1e-75:
		tmp = t_1
	elif Om <= 1.25e+83:
		tmp = math.fabs((math.sqrt((2.0 * (U * U_42_))) * (l * (n / Om))))
	elif Om <= 5.6e+250:
		tmp = t_1
	else:
		tmp = math.sqrt(((2.0 * n) * U)) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * abs(Float64(U * Float64(n * t)))))
	tmp = 0.0
	if (Om <= -2.1e-75)
		tmp = t_1;
	elseif (Om <= 1.25e+83)
		tmp = abs(Float64(sqrt(Float64(2.0 * Float64(U * U_42_))) * Float64(l * Float64(n / Om))));
	elseif (Om <= 5.6e+250)
		tmp = t_1;
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * U)) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((2.0 * abs((U * (n * t)))));
	tmp = 0.0;
	if (Om <= -2.1e-75)
		tmp = t_1;
	elseif (Om <= 1.25e+83)
		tmp = abs((sqrt((2.0 * (U * U_42_))) * (l * (n / Om))));
	elseif (Om <= 5.6e+250)
		tmp = t_1;
	else
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * N[Abs[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -2.1e-75], t$95$1, If[LessEqual[Om, 1.25e+83], N[Abs[N[(N[Sqrt[N[(2.0 * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 5.6e+250], t$95$1, N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -2.1000000000000001e-75 or 1.25000000000000007e83 < Om < 5.60000000000000019e250

   1. Initial program 56.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 64.3%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 53.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 52.9%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{U \cdot \left(n \cdot t\right)} \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right) \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}} \]

      2. pow3 53.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}^{3}}} \]

      3. associate-*r* 48.6%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(U \cdot n\right) \cdot t}}\right)}^{3}} \]

      4. *-commutative 48.6%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(n \cdot U\right)} \cdot t}\right)}^{3}} \]

   7. Applied egg-rr 48.6%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\left(n \cdot U\right) \cdot t}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 49.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}} \]

      2. add-sqr-sqrt 49.1%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\sqrt{\left(n \cdot U\right) \cdot t} \cdot \sqrt{\left(n \cdot U\right) \cdot t}\right)}} \]

      3. sqrt-unprod 29.1%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot \left(\left(n \cdot U\right) \cdot t\right)}}} \]

      4. pow2 29.1%

         \[\leadsto \sqrt{2 \cdot \sqrt{\color{blue}{{\left(\left(n \cdot U\right) \cdot t\right)}^{2}}}} \]

      5. associate-*l* 29.9%

         \[\leadsto \sqrt{2 \cdot \sqrt{{\color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}}^{2}}} \]

   9. Applied egg-rr 29.9%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\sqrt{{\left(n \cdot \left(U \cdot t\right)\right)}^{2}}}} \]
   10. Step-by-step derivation

       1. unpow2 29.9%

          \[\leadsto \sqrt{2 \cdot \sqrt{\color{blue}{\left(n \cdot \left(U \cdot t\right)\right) \cdot \left(n \cdot \left(U \cdot t\right)\right)}}} \]

       2. rem-sqrt-square 49.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left|n \cdot \left(U \cdot t\right)\right|}} \]

       3. associate-*r* 49.9%

          \[\leadsto \sqrt{2 \cdot \left|\color{blue}{\left(n \cdot U\right) \cdot t}\right|} \]

       4. *-commutative 49.9%

          \[\leadsto \sqrt{2 \cdot \left|\color{blue}{\left(U \cdot n\right)} \cdot t\right|} \]

       5. associate-*r* 54.3%

          \[\leadsto \sqrt{2 \cdot \left|\color{blue}{U \cdot \left(n \cdot t\right)}\right|} \]

   11. Simplified 54.3%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left|U \cdot \left(n \cdot t\right)\right|}} \]

   if -2.1000000000000001e-75 < Om < 1.25000000000000007e83

   1. Initial program 39.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 39.9%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 39.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      2. fma-define 43.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]

      3. associate-*r* 42.6%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]

   6. Applied egg-rr 42.6%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]

   7. Taylor expanded in U around 0 29.6%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-1 \cdot \frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 29.6%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-\frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]

      2. associate-/l* 29.6%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-\color{blue}{U* \cdot \frac{{\ell}^{2}}{{Om}^{2}}}\right)\right)\right)} \]

      3. unpow2 29.6%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right)\right)\right)} \]

      4. unpow2 29.6%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

      5. times-frac 42.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]

      6. unpow2 42.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right)\right)\right)} \]

   9. Simplified 42.8%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-U* \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   10. Taylor expanded in n around inf 24.9%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 24.9%

          \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}{{Om}^{2}}}} \]

       2. associate-*r* 24.6%

          \[\leadsto \sqrt{\frac{2 \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{{Om}^{2}}} \]

   12. Simplified 24.6%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   13. Step-by-step derivation

       1. pow1/2 24.7%

          \[\leadsto \color{blue}{{\left(\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}\right)}^{0.5}} \]

   14. Applied egg-rr 43.7%

       \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}\right)}^{0.5}} \]
   15. Step-by-step derivation

       1. unpow1/2 43.7%

          \[\leadsto \color{blue}{\sqrt{{\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right)}^{2}}} \]

       2. unpow2 43.7%

          \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om} \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}}} \]

       3. rem-sqrt-square 54.8%

          \[\leadsto \color{blue}{\left|\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|} \]

       4. associate-/l* 53.1%

          \[\leadsto \left|\color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{n \cdot \ell}{Om}}\right| \]

       5. *-commutative 53.1%

          \[\leadsto \left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\color{blue}{\ell \cdot n}}{Om}\right| \]

       6. associate-/l* 55.3%

          \[\leadsto \left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\ell \cdot \frac{n}{Om}\right)}\right| \]

   16. Simplified 55.3%

       \[\leadsto \color{blue}{\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)\right|} \]

   if 5.60000000000000019e250 < Om

   1. Initial program 42.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 25.5%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 25.5%

         \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. associate-*r* 34.0%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      3. *-commutative 34.0%

         \[\leadsto 1 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

   7. Applied egg-rr 34.0%

      \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]
   8. Step-by-step derivation

      1. *-lft-identity 34.0%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]

      2. associate-*r* 34.0%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t}} \]

      3. associate-*r* 34.0%

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t} \]

   9. Simplified 34.0%

      \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}} \]
   10. Step-by-step derivation

       1. sqrt-prod 43.5%

          \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}} \]

       2. *-commutative 43.5%

          \[\leadsto \sqrt{\color{blue}{U \cdot \left(2 \cdot n\right)}} \cdot \sqrt{t} \]

       3. *-commutative 43.5%

          \[\leadsto \sqrt{U \cdot \color{blue}{\left(n \cdot 2\right)}} \cdot \sqrt{t} \]

   11. Applied egg-rr 43.5%

       \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot 2\right)} \cdot \sqrt{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.1 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{elif}\;Om \leq 1.25 \cdot 10^{+83}:\\ \;\;\;\;\left|\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)\right|\\ \mathbf{elif}\;Om \leq 5.6 \cdot 10^{+250}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{n}{Om}\\ \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U* \cdot \left(2 \cdot U\right)\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ n Om))))
   (if (<= l 5.6e+27)
     (sqrt (* 2.0 (fabs (* U (* n t)))))
     (sqrt (* (* U* (* 2.0 U)) (* t_1 t_1))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (n / Om);
	double tmp;
	if (l <= 5.6e+27) {
		tmp = sqrt((2.0 * fabs((U * (n * t)))));
	} else {
		tmp = sqrt(((U_42_ * (2.0 * U)) * (t_1 * t_1)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l * (n / om)
    if (l <= 5.6d+27) then
        tmp = sqrt((2.0d0 * abs((u * (n * t)))))
    else
        tmp = sqrt(((u_42 * (2.0d0 * u)) * (t_1 * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (n / Om);
	double tmp;
	if (l <= 5.6e+27) {
		tmp = Math.sqrt((2.0 * Math.abs((U * (n * t)))));
	} else {
		tmp = Math.sqrt(((U_42_ * (2.0 * U)) * (t_1 * t_1)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (n / Om)
	tmp = 0
	if l <= 5.6e+27:
		tmp = math.sqrt((2.0 * math.fabs((U * (n * t)))))
	else:
		tmp = math.sqrt(((U_42_ * (2.0 * U)) * (t_1 * t_1)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(n / Om))
	tmp = 0.0
	if (l <= 5.6e+27)
		tmp = sqrt(Float64(2.0 * abs(Float64(U * Float64(n * t)))));
	else
		tmp = sqrt(Float64(Float64(U_42_ * Float64(2.0 * U)) * Float64(t_1 * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (n / Om);
	tmp = 0.0;
	if (l <= 5.6e+27)
		tmp = sqrt((2.0 * abs((U * (n * t)))));
	else
		tmp = sqrt(((U_42_ * (2.0 * U)) * (t_1 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 5.6e+27], N[Sqrt[N[(2.0 * N[Abs[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 5.5999999999999999e27

   1. Initial program 53.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 53.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 42.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 42.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{U \cdot \left(n \cdot t\right)} \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right) \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}} \]

      2. pow3 42.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}^{3}}} \]

      3. associate-*r* 41.5%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(U \cdot n\right) \cdot t}}\right)}^{3}} \]

      4. *-commutative 41.5%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(n \cdot U\right)} \cdot t}\right)}^{3}} \]

   7. Applied egg-rr 41.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\left(n \cdot U\right) \cdot t}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 41.9%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}} \]

      2. add-sqr-sqrt 41.8%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\sqrt{\left(n \cdot U\right) \cdot t} \cdot \sqrt{\left(n \cdot U\right) \cdot t}\right)}} \]

      3. sqrt-unprod 28.7%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\sqrt{\left(\left(n \cdot U\right) \cdot t\right) \cdot \left(\left(n \cdot U\right) \cdot t\right)}}} \]

      4. pow2 28.7%

         \[\leadsto \sqrt{2 \cdot \sqrt{\color{blue}{{\left(\left(n \cdot U\right) \cdot t\right)}^{2}}}} \]

      5. associate-*l* 28.2%

         \[\leadsto \sqrt{2 \cdot \sqrt{{\color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}}^{2}}} \]

   9. Applied egg-rr 28.2%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\sqrt{{\left(n \cdot \left(U \cdot t\right)\right)}^{2}}}} \]
   10. Step-by-step derivation

       1. unpow2 28.2%

          \[\leadsto \sqrt{2 \cdot \sqrt{\color{blue}{\left(n \cdot \left(U \cdot t\right)\right) \cdot \left(n \cdot \left(U \cdot t\right)\right)}}} \]

       2. rem-sqrt-square 40.8%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left|n \cdot \left(U \cdot t\right)\right|}} \]

       3. associate-*r* 43.5%

          \[\leadsto \sqrt{2 \cdot \left|\color{blue}{\left(n \cdot U\right) \cdot t}\right|} \]

       4. *-commutative 43.5%

          \[\leadsto \sqrt{2 \cdot \left|\color{blue}{\left(U \cdot n\right)} \cdot t\right|} \]

       5. associate-*r* 44.5%

          \[\leadsto \sqrt{2 \cdot \left|\color{blue}{U \cdot \left(n \cdot t\right)}\right|} \]

   11. Simplified 44.5%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left|U \cdot \left(n \cdot t\right)\right|}} \]

   if 5.5999999999999999e27 < l

   1. Initial program 32.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 44.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 44.6%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      2. fma-define 49.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]

      3. associate-*r* 50.4%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]

   6. Applied egg-rr 50.4%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]

   7. Taylor expanded in U around 0 30.4%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-1 \cdot \frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 30.4%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-\frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]

      2. associate-/l* 37.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-\color{blue}{U* \cdot \frac{{\ell}^{2}}{{Om}^{2}}}\right)\right)\right)} \]

      3. unpow2 37.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right)\right)\right)} \]

      4. unpow2 37.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

      5. times-frac 50.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]

      6. unpow2 50.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right)\right)\right)} \]

   9. Simplified 50.5%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-U* \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   10. Taylor expanded in n around inf 24.2%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 24.2%

          \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}{{Om}^{2}}}} \]

       2. associate-*r* 23.9%

          \[\leadsto \sqrt{\frac{2 \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{{Om}^{2}}} \]

   12. Simplified 23.9%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 23.9%

          \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]

       2. sqrt-div 23.9%

          \[\leadsto 1 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{\sqrt{{Om}^{2}}}} \]

       3. associate-*r* 23.9%

          \[\leadsto 1 \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot \left(U \cdot U*\right)\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}}}{\sqrt{{Om}^{2}}} \]

       4. sqrt-prod 23.9%

          \[\leadsto 1 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{{\ell}^{2} \cdot {n}^{2}}}}{\sqrt{{Om}^{2}}} \]

       5. *-commutative 23.9%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{\color{blue}{{n}^{2} \cdot {\ell}^{2}}}}{\sqrt{{Om}^{2}}} \]

       6. sqrt-prod 24.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\sqrt{{n}^{2}} \cdot \sqrt{{\ell}^{2}}\right)}}{\sqrt{{Om}^{2}}} \]

       7. sqrt-pow1 13.1%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{{n}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       8. metadata-eval 13.1%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left({n}^{\color{blue}{1}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       9. pow1 13.1%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{n} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       10. sqrt-pow1 13.6%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}\right)}{\sqrt{{Om}^{2}}} \]

       11. metadata-eval 13.6%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot {\ell}^{\color{blue}{1}}\right)}{\sqrt{{Om}^{2}}} \]

       12. pow1 13.6%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{\ell}\right)}{\sqrt{{Om}^{2}}} \]

       13. sqrt-pow1 13.8%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{{Om}^{\left(\frac{2}{2}\right)}}} \]

       14. metadata-eval 13.8%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{{Om}^{\color{blue}{1}}} \]

       15. pow1 13.8%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{Om}} \]

   14. Applied egg-rr 13.8%

       \[\leadsto \color{blue}{1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]
   15. Step-by-step derivation

       1. *-lft-identity 13.8%

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]

       2. associate-/l* 13.8%

          \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{n \cdot \ell}{Om}} \]

       3. *-commutative 13.8%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\color{blue}{\ell \cdot n}}{Om} \]

       4. associate-/l* 18.1%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\ell \cdot \frac{n}{Om}\right)} \]

   16. Simplified 18.1%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)} \]
   17. Step-by-step derivation

       1. add-sqr-sqrt 17.9%

          \[\leadsto \color{blue}{\sqrt{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)} \cdot \sqrt{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)}} \]

       2. sqrt-unprod 30.8%

          \[\leadsto \color{blue}{\sqrt{\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)\right) \cdot \left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)}} \]

       3. swap-sqr 27.8%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{2 \cdot \left(U \cdot U*\right)}\right) \cdot \left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)}} \]

       4. add-sqr-sqrt 27.9%

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot U*\right)\right)} \cdot \left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)} \]

       5. associate-*r* 27.9%

          \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot U*\right)} \cdot \left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)} \]

       6. pow2 27.9%

          \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot U*\right) \cdot \color{blue}{{\left(\ell \cdot \frac{n}{Om}\right)}^{2}}} \]

   18. Applied egg-rr 27.9%

       \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot U\right) \cdot U*\right) \cdot {\left(\ell \cdot \frac{n}{Om}\right)}^{2}}} \]
   19. Step-by-step derivation

       1. unpow2 27.9%

          \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot U*\right) \cdot \color{blue}{\left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)}} \]

   20. Applied egg-rr 27.9%

       \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot U*\right) \cdot \color{blue}{\left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U* \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;\ell \leq 3.45 \cdot 10^{+92}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right)\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{+127}:\\ \;\;\;\;{\left(\left(t\_1 \cdot t\_1\right) \cdot 4\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* U t))))
   (if (<= l 3.45e+92)
     (pow (* t (* 2.0 (* n U))) 0.5)
     (if (<= l 2.1e+108)
       (* (sqrt (* 2.0 (* U U*))) (* n (/ l Om)))
       (if (<= l 1.02e+127)
         (pow (* (* t_1 t_1) 4.0) 0.25)
         (* l (/ (sqrt (* U* (* 2.0 U))) (/ Om n))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * t);
	double tmp;
	if (l <= 3.45e+92) {
		tmp = pow((t * (2.0 * (n * U))), 0.5);
	} else if (l <= 2.1e+108) {
		tmp = sqrt((2.0 * (U * U_42_))) * (n * (l / Om));
	} else if (l <= 1.02e+127) {
		tmp = pow(((t_1 * t_1) * 4.0), 0.25);
	} else {
		tmp = l * (sqrt((U_42_ * (2.0 * U))) / (Om / n));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = n * (u * t)
    if (l <= 3.45d+92) then
        tmp = (t * (2.0d0 * (n * u))) ** 0.5d0
    else if (l <= 2.1d+108) then
        tmp = sqrt((2.0d0 * (u * u_42))) * (n * (l / om))
    else if (l <= 1.02d+127) then
        tmp = ((t_1 * t_1) * 4.0d0) ** 0.25d0
    else
        tmp = l * (sqrt((u_42 * (2.0d0 * u))) / (om / n))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * t);
	double tmp;
	if (l <= 3.45e+92) {
		tmp = Math.pow((t * (2.0 * (n * U))), 0.5);
	} else if (l <= 2.1e+108) {
		tmp = Math.sqrt((2.0 * (U * U_42_))) * (n * (l / Om));
	} else if (l <= 1.02e+127) {
		tmp = Math.pow(((t_1 * t_1) * 4.0), 0.25);
	} else {
		tmp = l * (Math.sqrt((U_42_ * (2.0 * U))) / (Om / n));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * (U * t)
	tmp = 0
	if l <= 3.45e+92:
		tmp = math.pow((t * (2.0 * (n * U))), 0.5)
	elif l <= 2.1e+108:
		tmp = math.sqrt((2.0 * (U * U_42_))) * (n * (l / Om))
	elif l <= 1.02e+127:
		tmp = math.pow(((t_1 * t_1) * 4.0), 0.25)
	else:
		tmp = l * (math.sqrt((U_42_ * (2.0 * U))) / (Om / n))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U * t))
	tmp = 0.0
	if (l <= 3.45e+92)
		tmp = Float64(t * Float64(2.0 * Float64(n * U))) ^ 0.5;
	elseif (l <= 2.1e+108)
		tmp = Float64(sqrt(Float64(2.0 * Float64(U * U_42_))) * Float64(n * Float64(l / Om)));
	elseif (l <= 1.02e+127)
		tmp = Float64(Float64(t_1 * t_1) * 4.0) ^ 0.25;
	else
		tmp = Float64(l * Float64(sqrt(Float64(U_42_ * Float64(2.0 * U))) / Float64(Om / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * (U * t);
	tmp = 0.0;
	if (l <= 3.45e+92)
		tmp = (t * (2.0 * (n * U))) ^ 0.5;
	elseif (l <= 2.1e+108)
		tmp = sqrt((2.0 * (U * U_42_))) * (n * (l / Om));
	elseif (l <= 1.02e+127)
		tmp = ((t_1 * t_1) * 4.0) ^ 0.25;
	else
		tmp = l * (sqrt((U_42_ * (2.0 * U))) / (Om / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 3.45e+92], N[Power[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 2.1e+108], N[(N[Sqrt[N[(2.0 * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.02e+127], N[Power[N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 4.0), $MachinePrecision], 0.25], $MachinePrecision], N[(l * N[(N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < 3.45000000000000012e92

   1. Initial program 54.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0 47.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* 46.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. cancel-sign-sub-inv 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      3. metadata-eval 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      4. *-commutative 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{\frac{{\ell}^{2}}{Om} \cdot -2}\right)\right)} \]

      5. associate-*l/ 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{\frac{{\ell}^{2} \cdot -2}{Om}}\right)\right)} \]

   7. Simplified 46.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 50.2%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 50.2%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 50.2%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 50.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 50.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 50.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}\right)}^{0.5} \]

   9. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)\right)}^{0.5}} \]

   10. Taylor expanded in l around 0 43.3%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{t}\right)}^{0.5} \]

   if 3.45000000000000012e92 < l < 2.1000000000000001e108

   1. Initial program 23.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 23.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 23.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      2. fma-define 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]

      3. associate-*r* 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]

   6. Applied egg-rr 43.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]

   7. Taylor expanded in U around 0 43.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-1 \cdot \frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-\frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]

      2. associate-/l* 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-\color{blue}{U* \cdot \frac{{\ell}^{2}}{{Om}^{2}}}\right)\right)\right)} \]

      3. unpow2 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right)\right)\right)} \]

      4. unpow2 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

      5. times-frac 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]

      6. unpow2 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right)\right)\right)} \]

   9. Simplified 43.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-U* \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   10. Taylor expanded in n around inf 42.1%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 42.1%

          \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}{{Om}^{2}}}} \]

       2. associate-*r* 42.1%

          \[\leadsto \sqrt{\frac{2 \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{{Om}^{2}}} \]

   12. Simplified 42.1%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 42.1%

          \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]

       2. sqrt-div 42.1%

          \[\leadsto 1 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{\sqrt{{Om}^{2}}}} \]

       3. associate-*r* 42.1%

          \[\leadsto 1 \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot \left(U \cdot U*\right)\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}}}{\sqrt{{Om}^{2}}} \]

       4. sqrt-prod 42.1%

          \[\leadsto 1 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{{\ell}^{2} \cdot {n}^{2}}}}{\sqrt{{Om}^{2}}} \]

       5. *-commutative 42.1%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{\color{blue}{{n}^{2} \cdot {\ell}^{2}}}}{\sqrt{{Om}^{2}}} \]

       6. sqrt-prod 42.1%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\sqrt{{n}^{2}} \cdot \sqrt{{\ell}^{2}}\right)}}{\sqrt{{Om}^{2}}} \]

       7. sqrt-pow1 41.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{{n}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       8. metadata-eval 41.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left({n}^{\color{blue}{1}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       9. pow1 41.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{n} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       10. sqrt-pow1 41.0%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}\right)}{\sqrt{{Om}^{2}}} \]

       11. metadata-eval 41.0%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot {\ell}^{\color{blue}{1}}\right)}{\sqrt{{Om}^{2}}} \]

       12. pow1 41.0%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{\ell}\right)}{\sqrt{{Om}^{2}}} \]

       13. sqrt-pow1 40.9%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{{Om}^{\left(\frac{2}{2}\right)}}} \]

       14. metadata-eval 40.9%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{{Om}^{\color{blue}{1}}} \]

       15. pow1 40.9%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{Om}} \]

   14. Applied egg-rr 40.9%

       \[\leadsto \color{blue}{1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]
   15. Step-by-step derivation

       1. *-lft-identity 40.9%

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]

       2. associate-/l* 40.9%

          \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{n \cdot \ell}{Om}} \]

       3. *-commutative 40.9%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\color{blue}{\ell \cdot n}}{Om} \]

       4. associate-/l* 40.6%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\ell \cdot \frac{n}{Om}\right)} \]

   16. Simplified 40.6%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)} \]
   17. Step-by-step derivation

       1. associate-*r* 60.4%

          \[\leadsto \left|\color{blue}{\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell\right) \cdot \frac{n}{Om}}\right| \]

       2. clear-num 60.4%

          \[\leadsto \left|\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{Om}{n}}}\right| \]

       3. un-div-inv 60.6%

          \[\leadsto \left|\color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell}{\frac{Om}{n}}}\right| \]

       4. associate-*r* 60.6%

          \[\leadsto \left|\frac{\sqrt{\color{blue}{\left(2 \cdot U\right) \cdot U*}} \cdot \ell}{\frac{Om}{n}}\right| \]

   18. Applied egg-rr 40.6%

       \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot U\right) \cdot U*} \cdot \ell}{\frac{Om}{n}}} \]
   19. Step-by-step derivation

       1. associate-/l* 40.6%

          \[\leadsto \color{blue}{\sqrt{\left(2 \cdot U\right) \cdot U*} \cdot \frac{\ell}{\frac{Om}{n}}} \]

       2. associate-*l* 40.6%

          \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot U*\right)}} \cdot \frac{\ell}{\frac{Om}{n}} \]

       3. associate-/r/ 40.9%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot n\right)} \]

   20. Simplified 40.9%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\frac{\ell}{Om} \cdot n\right)} \]

   if 2.1000000000000001e108 < l < 1.02e127

   1. Initial program 35.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 35.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 3.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 3.1%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{U \cdot \left(n \cdot t\right)} \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right) \cdot \sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}} \]

      2. pow3 3.1%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{U \cdot \left(n \cdot t\right)}\right)}^{3}}} \]

      3. associate-*r* 3.5%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(U \cdot n\right) \cdot t}}\right)}^{3}} \]

      4. *-commutative 3.5%

         \[\leadsto \sqrt{2 \cdot {\left(\sqrt[3]{\color{blue}{\left(n \cdot U\right)} \cdot t}\right)}^{3}} \]

   7. Applied egg-rr 3.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\left(n \cdot U\right) \cdot t}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. pow1/2 3.5%

         \[\leadsto \color{blue}{{\left(2 \cdot {\left(\sqrt[3]{\left(n \cdot U\right) \cdot t}\right)}^{3}\right)}^{0.5}} \]

      2. rem-cube-cbrt 3.5%

         \[\leadsto {\left(2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}\right)}^{0.5} \]

      3. associate-*l* 3.5%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}}^{0.5} \]

      4. sqr-pow 3.5%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}^{\left(\frac{0.5}{2}\right)}} \]

      5. pow-prod-down 20.1%

         \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right) \cdot \left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right)}^{\left(\frac{0.5}{2}\right)}} \]

      6. associate-*l* 20.1%

         \[\leadsto {\left(\color{blue}{\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)} \cdot \left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      7. associate-*l* 20.1%

         \[\leadsto {\left(\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right) \cdot \color{blue}{\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      8. *-commutative 20.1%

         \[\leadsto {\left(\color{blue}{\left(\left(\left(n \cdot U\right) \cdot t\right) \cdot 2\right)} \cdot \left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      9. *-commutative 20.1%

         \[\leadsto {\left(\left(\left(\left(n \cdot U\right) \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(\left(\left(n \cdot U\right) \cdot t\right) \cdot 2\right)}\right)}^{\left(\frac{0.5}{2}\right)} \]

      10. swap-sqr 20.1%

          \[\leadsto {\color{blue}{\left(\left(\left(\left(n \cdot U\right) \cdot t\right) \cdot \left(\left(n \cdot U\right) \cdot t\right)\right) \cdot \left(2 \cdot 2\right)\right)}}^{\left(\frac{0.5}{2}\right)} \]

      11. pow2 20.1%

          \[\leadsto {\left(\color{blue}{{\left(\left(n \cdot U\right) \cdot t\right)}^{2}} \cdot \left(2 \cdot 2\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      12. associate-*l* 20.1%

          \[\leadsto {\left({\color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}}^{2} \cdot \left(2 \cdot 2\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

      13. metadata-eval 20.1%

          \[\leadsto {\left({\left(n \cdot \left(U \cdot t\right)\right)}^{2} \cdot \color{blue}{4}\right)}^{\left(\frac{0.5}{2}\right)} \]

      14. metadata-eval 20.1%

          \[\leadsto {\left({\left(n \cdot \left(U \cdot t\right)\right)}^{2} \cdot 4\right)}^{\color{blue}{0.25}} \]

   9. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{{\left({\left(n \cdot \left(U \cdot t\right)\right)}^{2} \cdot 4\right)}^{0.25}} \]
   10. Step-by-step derivation

       1. unpow2 20.1%

          \[\leadsto {\left(\color{blue}{\left(\left(n \cdot \left(U \cdot t\right)\right) \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)} \cdot 4\right)}^{0.25} \]

   11. Applied egg-rr 20.1%

       \[\leadsto {\left(\color{blue}{\left(\left(n \cdot \left(U \cdot t\right)\right) \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)} \cdot 4\right)}^{0.25} \]

   if 1.02e127 < l

   1. Initial program 21.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 39.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 39.7%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      2. fma-define 42.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]

      3. associate-*r* 44.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]

   6. Applied egg-rr 44.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]

   7. Taylor expanded in U around 0 24.5%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-1 \cdot \frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 24.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-\frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]

      2. associate-/l* 24.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-\color{blue}{U* \cdot \frac{{\ell}^{2}}{{Om}^{2}}}\right)\right)\right)} \]

      3. unpow2 24.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right)\right)\right)} \]

      4. unpow2 24.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

      5. times-frac 44.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]

      6. unpow2 44.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right)\right)\right)} \]

   9. Simplified 44.2%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-U* \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   10. Taylor expanded in n around inf 26.9%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 26.9%

          \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}{{Om}^{2}}}} \]

       2. associate-*r* 26.9%

          \[\leadsto \sqrt{\frac{2 \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{{Om}^{2}}} \]

   12. Simplified 26.9%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 26.9%

          \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]

       2. sqrt-div 26.9%

          \[\leadsto 1 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{\sqrt{{Om}^{2}}}} \]

       3. associate-*r* 26.9%

          \[\leadsto 1 \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot \left(U \cdot U*\right)\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}}}{\sqrt{{Om}^{2}}} \]

       4. sqrt-prod 26.9%

          \[\leadsto 1 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{{\ell}^{2} \cdot {n}^{2}}}}{\sqrt{{Om}^{2}}} \]

       5. *-commutative 26.9%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{\color{blue}{{n}^{2} \cdot {\ell}^{2}}}}{\sqrt{{Om}^{2}}} \]

       6. sqrt-prod 27.1%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\sqrt{{n}^{2}} \cdot \sqrt{{\ell}^{2}}\right)}}{\sqrt{{Om}^{2}}} \]

       7. sqrt-pow1 15.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{{n}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       8. metadata-eval 15.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left({n}^{\color{blue}{1}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       9. pow1 15.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{n} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       10. sqrt-pow1 15.8%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}\right)}{\sqrt{{Om}^{2}}} \]

       11. metadata-eval 15.8%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot {\ell}^{\color{blue}{1}}\right)}{\sqrt{{Om}^{2}}} \]

       12. pow1 15.8%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{\ell}\right)}{\sqrt{{Om}^{2}}} \]

       13. sqrt-pow1 8.7%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{{Om}^{\left(\frac{2}{2}\right)}}} \]

       14. metadata-eval 8.7%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{{Om}^{\color{blue}{1}}} \]

       15. pow1 8.7%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{Om}} \]

   14. Applied egg-rr 8.7%

       \[\leadsto \color{blue}{1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]
   15. Step-by-step derivation

       1. *-lft-identity 8.7%

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]

       2. associate-/l* 8.7%

          \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{n \cdot \ell}{Om}} \]

       3. *-commutative 8.7%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\color{blue}{\ell \cdot n}}{Om} \]

       4. associate-/l* 15.5%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\ell \cdot \frac{n}{Om}\right)} \]

   16. Simplified 15.5%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)} \]
   17. Step-by-step derivation

       1. associate-*r* 37.2%

          \[\leadsto \left|\color{blue}{\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell\right) \cdot \frac{n}{Om}}\right| \]

       2. clear-num 37.2%

          \[\leadsto \left|\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{Om}{n}}}\right| \]

       3. un-div-inv 37.3%

          \[\leadsto \left|\color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell}{\frac{Om}{n}}}\right| \]

       4. associate-*r* 37.3%

          \[\leadsto \left|\frac{\sqrt{\color{blue}{\left(2 \cdot U\right) \cdot U*}} \cdot \ell}{\frac{Om}{n}}\right| \]

   18. Applied egg-rr 15.4%

       \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot U\right) \cdot U*} \cdot \ell}{\frac{Om}{n}}} \]
   19. Step-by-step derivation

       1. *-commutative 15.4%

          \[\leadsto \frac{\color{blue}{\ell \cdot \sqrt{\left(2 \cdot U\right) \cdot U*}}}{\frac{Om}{n}} \]

       2. associate-/l* 15.5%

          \[\leadsto \color{blue}{\ell \cdot \frac{\sqrt{\left(2 \cdot U\right) \cdot U*}}{\frac{Om}{n}}} \]

       3. *-commutative 15.5%

          \[\leadsto \ell \cdot \frac{\sqrt{\color{blue}{U* \cdot \left(2 \cdot U\right)}}}{\frac{Om}{n}} \]

       4. *-commutative 15.5%

          \[\leadsto \ell \cdot \frac{\sqrt{U* \cdot \color{blue}{\left(U \cdot 2\right)}}}{\frac{Om}{n}} \]

   20. Simplified 15.5%

       \[\leadsto \color{blue}{\ell \cdot \frac{\sqrt{U* \cdot \left(U \cdot 2\right)}}{\frac{Om}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.45 \cdot 10^{+92}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right)\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{+127}:\\ \;\;\;\;{\left(\left(\left(n \cdot \left(U \cdot t\right)\right) \cdot \left(n \cdot \left(U \cdot t\right)\right)\right) \cdot 4\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+90}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+108} \lor \neg \left(\ell \leq 9.2 \cdot 10^{+126}\right):\\ \;\;\;\;\ell \cdot \frac{\sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.8e+90)
   (pow (* t (* 2.0 (* n U))) 0.5)
   (if (or (<= l 1.15e+108) (not (<= l 9.2e+126)))
     (* l (/ (sqrt (* U* (* 2.0 U))) (/ Om n)))
     (sqrt (* 2.0 (* n (* U t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.8e+90) {
		tmp = pow((t * (2.0 * (n * U))), 0.5);
	} else if ((l <= 1.15e+108) || !(l <= 9.2e+126)) {
		tmp = l * (sqrt((U_42_ * (2.0 * U))) / (Om / n));
	} else {
		tmp = sqrt((2.0 * (n * (U * t))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 3.8d+90) then
        tmp = (t * (2.0d0 * (n * u))) ** 0.5d0
    else if ((l <= 1.15d+108) .or. (.not. (l <= 9.2d+126))) then
        tmp = l * (sqrt((u_42 * (2.0d0 * u))) / (om / n))
    else
        tmp = sqrt((2.0d0 * (n * (u * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.8e+90) {
		tmp = Math.pow((t * (2.0 * (n * U))), 0.5);
	} else if ((l <= 1.15e+108) || !(l <= 9.2e+126)) {
		tmp = l * (Math.sqrt((U_42_ * (2.0 * U))) / (Om / n));
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.8e+90:
		tmp = math.pow((t * (2.0 * (n * U))), 0.5)
	elif (l <= 1.15e+108) or not (l <= 9.2e+126):
		tmp = l * (math.sqrt((U_42_ * (2.0 * U))) / (Om / n))
	else:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.8e+90)
		tmp = Float64(t * Float64(2.0 * Float64(n * U))) ^ 0.5;
	elseif ((l <= 1.15e+108) || !(l <= 9.2e+126))
		tmp = Float64(l * Float64(sqrt(Float64(U_42_ * Float64(2.0 * U))) / Float64(Om / n)));
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.8e+90)
		tmp = (t * (2.0 * (n * U))) ^ 0.5;
	elseif ((l <= 1.15e+108) || ~((l <= 9.2e+126)))
		tmp = l * (sqrt((U_42_ * (2.0 * U))) / (Om / n));
	else
		tmp = sqrt((2.0 * (n * (U * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.8e+90], N[Power[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[Or[LessEqual[l, 1.15e+108], N[Not[LessEqual[l, 9.2e+126]], $MachinePrecision]], N[(l * N[(N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.8000000000000001e90

   1. Initial program 54.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0 47.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* 46.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. cancel-sign-sub-inv 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      3. metadata-eval 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      4. *-commutative 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{\frac{{\ell}^{2}}{Om} \cdot -2}\right)\right)} \]

      5. associate-*l/ 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{\frac{{\ell}^{2} \cdot -2}{Om}}\right)\right)} \]

   7. Simplified 46.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 50.2%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 50.2%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 50.2%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 50.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 50.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 50.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}\right)}^{0.5} \]

   9. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)\right)}^{0.5}} \]

   10. Taylor expanded in l around 0 43.3%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{t}\right)}^{0.5} \]

   if 3.8000000000000001e90 < l < 1.1499999999999999e108 or 9.2000000000000002e126 < l

   1. Initial program 21.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 37.9%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 37.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      2. fma-define 42.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]

      3. associate-*r* 44.0%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]

   6. Applied egg-rr 44.0%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]

   7. Taylor expanded in U around 0 26.4%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-1 \cdot \frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 26.4%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-\frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]

      2. associate-/l* 26.4%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-\color{blue}{U* \cdot \frac{{\ell}^{2}}{{Om}^{2}}}\right)\right)\right)} \]

      3. unpow2 26.4%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right)\right)\right)} \]

      4. unpow2 26.4%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

      5. times-frac 44.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]

      6. unpow2 44.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right)\right)\right)} \]

   9. Simplified 44.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-U* \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   10. Taylor expanded in n around inf 28.6%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 28.6%

          \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}{{Om}^{2}}}} \]

       2. associate-*r* 28.5%

          \[\leadsto \sqrt{\frac{2 \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{{Om}^{2}}} \]

   12. Simplified 28.5%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 28.5%

          \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]

       2. sqrt-div 28.5%

          \[\leadsto 1 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{\sqrt{{Om}^{2}}}} \]

       3. associate-*r* 28.5%

          \[\leadsto 1 \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot \left(U \cdot U*\right)\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}}}{\sqrt{{Om}^{2}}} \]

       4. sqrt-prod 28.5%

          \[\leadsto 1 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{{\ell}^{2} \cdot {n}^{2}}}}{\sqrt{{Om}^{2}}} \]

       5. *-commutative 28.5%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{\color{blue}{{n}^{2} \cdot {\ell}^{2}}}}{\sqrt{{Om}^{2}}} \]

       6. sqrt-prod 28.7%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\sqrt{{n}^{2}} \cdot \sqrt{{\ell}^{2}}\right)}}{\sqrt{{Om}^{2}}} \]

       7. sqrt-pow1 17.8%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{{n}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       8. metadata-eval 17.8%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left({n}^{\color{blue}{1}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       9. pow1 17.8%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{n} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       10. sqrt-pow1 18.5%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}\right)}{\sqrt{{Om}^{2}}} \]

       11. metadata-eval 18.5%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot {\ell}^{\color{blue}{1}}\right)}{\sqrt{{Om}^{2}}} \]

       12. pow1 18.5%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{\ell}\right)}{\sqrt{{Om}^{2}}} \]

       13. sqrt-pow1 12.2%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{{Om}^{\left(\frac{2}{2}\right)}}} \]

       14. metadata-eval 12.2%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{{Om}^{\color{blue}{1}}} \]

       15. pow1 12.2%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{Om}} \]

   14. Applied egg-rr 12.2%

       \[\leadsto \color{blue}{1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]
   15. Step-by-step derivation

       1. *-lft-identity 12.2%

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]

       2. associate-/l* 12.2%

          \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{n \cdot \ell}{Om}} \]

       3. *-commutative 12.2%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\color{blue}{\ell \cdot n}}{Om} \]

       4. associate-/l* 18.1%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\ell \cdot \frac{n}{Om}\right)} \]

   16. Simplified 18.1%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)} \]
   17. Step-by-step derivation

       1. associate-*r* 39.7%

          \[\leadsto \left|\color{blue}{\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell\right) \cdot \frac{n}{Om}}\right| \]

       2. clear-num 39.7%

          \[\leadsto \left|\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{Om}{n}}}\right| \]

       3. un-div-inv 39.7%

          \[\leadsto \left|\color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell}{\frac{Om}{n}}}\right| \]

       4. associate-*r* 39.7%

          \[\leadsto \left|\frac{\sqrt{\color{blue}{\left(2 \cdot U\right) \cdot U*}} \cdot \ell}{\frac{Om}{n}}\right| \]

   18. Applied egg-rr 18.1%

       \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot U\right) \cdot U*} \cdot \ell}{\frac{Om}{n}}} \]
   19. Step-by-step derivation

       1. *-commutative 18.1%

          \[\leadsto \frac{\color{blue}{\ell \cdot \sqrt{\left(2 \cdot U\right) \cdot U*}}}{\frac{Om}{n}} \]

       2. associate-/l* 18.2%

          \[\leadsto \color{blue}{\ell \cdot \frac{\sqrt{\left(2 \cdot U\right) \cdot U*}}{\frac{Om}{n}}} \]

       3. *-commutative 18.2%

          \[\leadsto \ell \cdot \frac{\sqrt{\color{blue}{U* \cdot \left(2 \cdot U\right)}}}{\frac{Om}{n}} \]

       4. *-commutative 18.2%

          \[\leadsto \ell \cdot \frac{\sqrt{U* \cdot \color{blue}{\left(U \cdot 2\right)}}}{\frac{Om}{n}} \]

   20. Simplified 18.2%

       \[\leadsto \color{blue}{\ell \cdot \frac{\sqrt{U* \cdot \left(U \cdot 2\right)}}{\frac{Om}{n}}} \]

   if 1.1499999999999999e108 < l < 9.2000000000000002e126

   1. Initial program 35.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 35.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 3.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. pow1 3.1%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(U \cdot \left(n \cdot t\right)\right)}^{1}}} \]

      2. associate-*r* 3.5%

         \[\leadsto \sqrt{2 \cdot {\color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}}^{1}} \]

      3. *-commutative 3.5%

         \[\leadsto \sqrt{2 \cdot {\left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)}^{1}} \]

   7. Applied egg-rr 3.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot t\right)}^{1}}} \]
   8. Step-by-step derivation

      1. unpow1 3.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}} \]

      2. associate-*l* 3.9%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]

   9. Simplified 3.9%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+90}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+108} \lor \neg \left(\ell \leq 9.2 \cdot 10^{+126}\right):\\ \;\;\;\;\ell \cdot \frac{\sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+91}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+108} \lor \neg \left(\ell \leq 6 \cdot 10^{+127}\right):\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.2e+91)
   (pow (* t (* 2.0 (* n U))) 0.5)
   (if (or (<= l 1.9e+108) (not (<= l 6e+127)))
     (* l (* (/ n Om) (sqrt (* U* (* 2.0 U)))))
     (sqrt (* 2.0 (* n (* U t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.2e+91) {
		tmp = pow((t * (2.0 * (n * U))), 0.5);
	} else if ((l <= 1.9e+108) || !(l <= 6e+127)) {
		tmp = l * ((n / Om) * sqrt((U_42_ * (2.0 * U))));
	} else {
		tmp = sqrt((2.0 * (n * (U * t))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 5.2d+91) then
        tmp = (t * (2.0d0 * (n * u))) ** 0.5d0
    else if ((l <= 1.9d+108) .or. (.not. (l <= 6d+127))) then
        tmp = l * ((n / om) * sqrt((u_42 * (2.0d0 * u))))
    else
        tmp = sqrt((2.0d0 * (n * (u * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.2e+91) {
		tmp = Math.pow((t * (2.0 * (n * U))), 0.5);
	} else if ((l <= 1.9e+108) || !(l <= 6e+127)) {
		tmp = l * ((n / Om) * Math.sqrt((U_42_ * (2.0 * U))));
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5.2e+91:
		tmp = math.pow((t * (2.0 * (n * U))), 0.5)
	elif (l <= 1.9e+108) or not (l <= 6e+127):
		tmp = l * ((n / Om) * math.sqrt((U_42_ * (2.0 * U))))
	else:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.2e+91)
		tmp = Float64(t * Float64(2.0 * Float64(n * U))) ^ 0.5;
	elseif ((l <= 1.9e+108) || !(l <= 6e+127))
		tmp = Float64(l * Float64(Float64(n / Om) * sqrt(Float64(U_42_ * Float64(2.0 * U)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 5.2e+91)
		tmp = (t * (2.0 * (n * U))) ^ 0.5;
	elseif ((l <= 1.9e+108) || ~((l <= 6e+127)))
		tmp = l * ((n / Om) * sqrt((U_42_ * (2.0 * U))));
	else
		tmp = sqrt((2.0 * (n * (U * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 5.2e+91], N[Power[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[Or[LessEqual[l, 1.9e+108], N[Not[LessEqual[l, 6e+127]], $MachinePrecision]], N[(l * N[(N[(n / Om), $MachinePrecision] * N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.2000000000000001e91

   1. Initial program 54.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0 47.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* 46.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. cancel-sign-sub-inv 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      3. metadata-eval 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      4. *-commutative 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{\frac{{\ell}^{2}}{Om} \cdot -2}\right)\right)} \]

      5. associate-*l/ 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{\frac{{\ell}^{2} \cdot -2}{Om}}\right)\right)} \]

   7. Simplified 46.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 50.2%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 50.2%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 50.2%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 50.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 50.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 50.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}\right)}^{0.5} \]

   9. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)\right)}^{0.5}} \]

   10. Taylor expanded in l around 0 43.3%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{t}\right)}^{0.5} \]

   if 5.2000000000000001e91 < l < 1.90000000000000004e108 or 6.0000000000000005e127 < l

   1. Initial program 21.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 37.9%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 37.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      2. fma-define 42.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]

      3. associate-*r* 44.0%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]

   6. Applied egg-rr 44.0%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]

   7. Taylor expanded in U around 0 26.4%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-1 \cdot \frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 26.4%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-\frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]

      2. associate-/l* 26.4%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-\color{blue}{U* \cdot \frac{{\ell}^{2}}{{Om}^{2}}}\right)\right)\right)} \]

      3. unpow2 26.4%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right)\right)\right)} \]

      4. unpow2 26.4%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

      5. times-frac 44.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]

      6. unpow2 44.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right)\right)\right)} \]

   9. Simplified 44.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-U* \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   10. Taylor expanded in n around inf 28.6%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 28.6%

          \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}{{Om}^{2}}}} \]

       2. associate-*r* 28.5%

          \[\leadsto \sqrt{\frac{2 \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{{Om}^{2}}} \]

   12. Simplified 28.5%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 28.5%

          \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]

       2. sqrt-div 28.5%

          \[\leadsto 1 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{\sqrt{{Om}^{2}}}} \]

       3. associate-*r* 28.5%

          \[\leadsto 1 \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot \left(U \cdot U*\right)\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}}}{\sqrt{{Om}^{2}}} \]

       4. sqrt-prod 28.5%

          \[\leadsto 1 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{{\ell}^{2} \cdot {n}^{2}}}}{\sqrt{{Om}^{2}}} \]

       5. *-commutative 28.5%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{\color{blue}{{n}^{2} \cdot {\ell}^{2}}}}{\sqrt{{Om}^{2}}} \]

       6. sqrt-prod 28.7%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\sqrt{{n}^{2}} \cdot \sqrt{{\ell}^{2}}\right)}}{\sqrt{{Om}^{2}}} \]

       7. sqrt-pow1 17.8%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{{n}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       8. metadata-eval 17.8%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left({n}^{\color{blue}{1}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       9. pow1 17.8%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{n} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       10. sqrt-pow1 18.5%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}\right)}{\sqrt{{Om}^{2}}} \]

       11. metadata-eval 18.5%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot {\ell}^{\color{blue}{1}}\right)}{\sqrt{{Om}^{2}}} \]

       12. pow1 18.5%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{\ell}\right)}{\sqrt{{Om}^{2}}} \]

       13. sqrt-pow1 12.2%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{{Om}^{\left(\frac{2}{2}\right)}}} \]

       14. metadata-eval 12.2%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{{Om}^{\color{blue}{1}}} \]

       15. pow1 12.2%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{Om}} \]

   14. Applied egg-rr 12.2%

       \[\leadsto \color{blue}{1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]
   15. Step-by-step derivation

       1. *-lft-identity 12.2%

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]

       2. associate-/l* 12.2%

          \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{n \cdot \ell}{Om}} \]

       3. *-commutative 12.2%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\color{blue}{\ell \cdot n}}{Om} \]

       4. associate-/l* 18.1%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\ell \cdot \frac{n}{Om}\right)} \]

   16. Simplified 18.1%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)} \]
   17. Step-by-step derivation

       1. associate-*r* 39.7%

          \[\leadsto \left|\color{blue}{\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell\right) \cdot \frac{n}{Om}}\right| \]

       2. clear-num 39.7%

          \[\leadsto \left|\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{Om}{n}}}\right| \]

       3. un-div-inv 39.7%

          \[\leadsto \left|\color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell}{\frac{Om}{n}}}\right| \]

       4. associate-*r* 39.7%

          \[\leadsto \left|\frac{\sqrt{\color{blue}{\left(2 \cdot U\right) \cdot U*}} \cdot \ell}{\frac{Om}{n}}\right| \]

   18. Applied egg-rr 18.1%

       \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot U\right) \cdot U*} \cdot \ell}{\frac{Om}{n}}} \]
   19. Step-by-step derivation

       1. *-commutative 18.1%

          \[\leadsto \frac{\color{blue}{\ell \cdot \sqrt{\left(2 \cdot U\right) \cdot U*}}}{\frac{Om}{n}} \]

       2. associate-/l* 18.2%

          \[\leadsto \color{blue}{\ell \cdot \frac{\sqrt{\left(2 \cdot U\right) \cdot U*}}{\frac{Om}{n}}} \]

       3. *-commutative 18.2%

          \[\leadsto \ell \cdot \frac{\sqrt{\color{blue}{U* \cdot \left(2 \cdot U\right)}}}{\frac{Om}{n}} \]

       4. *-commutative 18.2%

          \[\leadsto \ell \cdot \frac{\sqrt{U* \cdot \color{blue}{\left(U \cdot 2\right)}}}{\frac{Om}{n}} \]

   20. Simplified 18.2%

       \[\leadsto \color{blue}{\ell \cdot \frac{\sqrt{U* \cdot \left(U \cdot 2\right)}}{\frac{Om}{n}}} \]
   21. Step-by-step derivation

       1. associate-*r/ 18.1%

          \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{U* \cdot \left(U \cdot 2\right)}}{\frac{Om}{n}}} \]

       2. *-commutative 18.1%

          \[\leadsto \frac{\ell \cdot \sqrt{\color{blue}{\left(U \cdot 2\right) \cdot U*}}}{\frac{Om}{n}} \]

       3. *-commutative 18.1%

          \[\leadsto \frac{\ell \cdot \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot U*}}{\frac{Om}{n}} \]

       4. associate-*l* 18.1%

          \[\leadsto \frac{\ell \cdot \sqrt{\color{blue}{2 \cdot \left(U \cdot U*\right)}}}{\frac{Om}{n}} \]

   22. Applied egg-rr 18.1%

       \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{2 \cdot \left(U \cdot U*\right)}}{\frac{Om}{n}}} \]
   23. Step-by-step derivation

       1. associate-/l* 18.2%

          \[\leadsto \color{blue}{\ell \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)}}{\frac{Om}{n}}} \]

       2. associate-/r/ 18.1%

          \[\leadsto \ell \cdot \color{blue}{\left(\frac{\sqrt{2 \cdot \left(U \cdot U*\right)}}{Om} \cdot n\right)} \]

       3. associate-*l/ 18.2%

          \[\leadsto \ell \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot n}{Om}} \]

       4. associate-*r/ 18.2%

          \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{n}{Om}\right)} \]

       5. associate-*r* 18.2%

          \[\leadsto \ell \cdot \left(\sqrt{\color{blue}{\left(2 \cdot U\right) \cdot U*}} \cdot \frac{n}{Om}\right) \]

       6. *-commutative 18.2%

          \[\leadsto \ell \cdot \left(\sqrt{\color{blue}{\left(U \cdot 2\right)} \cdot U*} \cdot \frac{n}{Om}\right) \]

   24. Simplified 18.2%

       \[\leadsto \color{blue}{\ell \cdot \left(\sqrt{\left(U \cdot 2\right) \cdot U*} \cdot \frac{n}{Om}\right)} \]

   if 1.90000000000000004e108 < l < 6.0000000000000005e127

   1. Initial program 35.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 35.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 3.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. pow1 3.1%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(U \cdot \left(n \cdot t\right)\right)}^{1}}} \]

      2. associate-*r* 3.5%

         \[\leadsto \sqrt{2 \cdot {\color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}}^{1}} \]

      3. *-commutative 3.5%

         \[\leadsto \sqrt{2 \cdot {\left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)}^{1}} \]

   7. Applied egg-rr 3.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot t\right)}^{1}}} \]
   8. Step-by-step derivation

      1. unpow1 3.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}} \]

      2. associate-*l* 3.9%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]

   9. Simplified 3.9%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{+91}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+108} \lor \neg \left(\ell \leq 6 \cdot 10^{+127}\right):\\ \;\;\;\;\ell \cdot \left(\frac{n}{Om} \cdot \sqrt{U* \cdot \left(2 \cdot U\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{+91}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.6e+91)
   (pow (* t (* 2.0 (* n U))) 0.5)
   (if (<= l 1.06e+108)
     (* (sqrt (* 2.0 (* U U*))) (* n (/ l Om)))
     (if (<= l 1.75e+127)
       (sqrt (* 2.0 (* n (* U t))))
       (* l (/ (sqrt (* U* (* 2.0 U))) (/ Om n)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.6e+91) {
		tmp = pow((t * (2.0 * (n * U))), 0.5);
	} else if (l <= 1.06e+108) {
		tmp = sqrt((2.0 * (U * U_42_))) * (n * (l / Om));
	} else if (l <= 1.75e+127) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = l * (sqrt((U_42_ * (2.0 * U))) / (Om / n));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 3.6d+91) then
        tmp = (t * (2.0d0 * (n * u))) ** 0.5d0
    else if (l <= 1.06d+108) then
        tmp = sqrt((2.0d0 * (u * u_42))) * (n * (l / om))
    else if (l <= 1.75d+127) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = l * (sqrt((u_42 * (2.0d0 * u))) / (om / n))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.6e+91) {
		tmp = Math.pow((t * (2.0 * (n * U))), 0.5);
	} else if (l <= 1.06e+108) {
		tmp = Math.sqrt((2.0 * (U * U_42_))) * (n * (l / Om));
	} else if (l <= 1.75e+127) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = l * (Math.sqrt((U_42_ * (2.0 * U))) / (Om / n));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.6e+91:
		tmp = math.pow((t * (2.0 * (n * U))), 0.5)
	elif l <= 1.06e+108:
		tmp = math.sqrt((2.0 * (U * U_42_))) * (n * (l / Om))
	elif l <= 1.75e+127:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = l * (math.sqrt((U_42_ * (2.0 * U))) / (Om / n))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.6e+91)
		tmp = Float64(t * Float64(2.0 * Float64(n * U))) ^ 0.5;
	elseif (l <= 1.06e+108)
		tmp = Float64(sqrt(Float64(2.0 * Float64(U * U_42_))) * Float64(n * Float64(l / Om)));
	elseif (l <= 1.75e+127)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = Float64(l * Float64(sqrt(Float64(U_42_ * Float64(2.0 * U))) / Float64(Om / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.6e+91)
		tmp = (t * (2.0 * (n * U))) ^ 0.5;
	elseif (l <= 1.06e+108)
		tmp = sqrt((2.0 * (U * U_42_))) * (n * (l / Om));
	elseif (l <= 1.75e+127)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = l * (sqrt((U_42_ * (2.0 * U))) / (Om / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.6e+91], N[Power[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[l, 1.06e+108], N[(N[Sqrt[N[(2.0 * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.75e+127], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l * N[(N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(Om / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < 3.6e91

   1. Initial program 54.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0 47.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* 46.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. cancel-sign-sub-inv 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      3. metadata-eval 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      4. *-commutative 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{\frac{{\ell}^{2}}{Om} \cdot -2}\right)\right)} \]

      5. associate-*l/ 46.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{\frac{{\ell}^{2} \cdot -2}{Om}}\right)\right)} \]

   7. Simplified 46.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 50.2%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 50.2%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 50.2%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 50.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 50.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 50.2%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}\right)}^{0.5} \]

   9. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)\right)}^{0.5}} \]

   10. Taylor expanded in l around 0 43.3%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{t}\right)}^{0.5} \]

   if 3.6e91 < l < 1.06e108

   1. Initial program 23.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 23.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 23.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      2. fma-define 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]

      3. associate-*r* 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]

   6. Applied egg-rr 43.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]

   7. Taylor expanded in U around 0 43.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-1 \cdot \frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-\frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]

      2. associate-/l* 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-\color{blue}{U* \cdot \frac{{\ell}^{2}}{{Om}^{2}}}\right)\right)\right)} \]

      3. unpow2 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right)\right)\right)} \]

      4. unpow2 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

      5. times-frac 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]

      6. unpow2 43.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right)\right)\right)} \]

   9. Simplified 43.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-U* \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   10. Taylor expanded in n around inf 42.1%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 42.1%

          \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}{{Om}^{2}}}} \]

       2. associate-*r* 42.1%

          \[\leadsto \sqrt{\frac{2 \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{{Om}^{2}}} \]

   12. Simplified 42.1%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 42.1%

          \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]

       2. sqrt-div 42.1%

          \[\leadsto 1 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{\sqrt{{Om}^{2}}}} \]

       3. associate-*r* 42.1%

          \[\leadsto 1 \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot \left(U \cdot U*\right)\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}}}{\sqrt{{Om}^{2}}} \]

       4. sqrt-prod 42.1%

          \[\leadsto 1 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{{\ell}^{2} \cdot {n}^{2}}}}{\sqrt{{Om}^{2}}} \]

       5. *-commutative 42.1%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{\color{blue}{{n}^{2} \cdot {\ell}^{2}}}}{\sqrt{{Om}^{2}}} \]

       6. sqrt-prod 42.1%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\sqrt{{n}^{2}} \cdot \sqrt{{\ell}^{2}}\right)}}{\sqrt{{Om}^{2}}} \]

       7. sqrt-pow1 41.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{{n}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       8. metadata-eval 41.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left({n}^{\color{blue}{1}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       9. pow1 41.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{n} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       10. sqrt-pow1 41.0%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}\right)}{\sqrt{{Om}^{2}}} \]

       11. metadata-eval 41.0%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot {\ell}^{\color{blue}{1}}\right)}{\sqrt{{Om}^{2}}} \]

       12. pow1 41.0%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{\ell}\right)}{\sqrt{{Om}^{2}}} \]

       13. sqrt-pow1 40.9%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{{Om}^{\left(\frac{2}{2}\right)}}} \]

       14. metadata-eval 40.9%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{{Om}^{\color{blue}{1}}} \]

       15. pow1 40.9%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{Om}} \]

   14. Applied egg-rr 40.9%

       \[\leadsto \color{blue}{1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]
   15. Step-by-step derivation

       1. *-lft-identity 40.9%

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]

       2. associate-/l* 40.9%

          \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{n \cdot \ell}{Om}} \]

       3. *-commutative 40.9%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\color{blue}{\ell \cdot n}}{Om} \]

       4. associate-/l* 40.6%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\ell \cdot \frac{n}{Om}\right)} \]

   16. Simplified 40.6%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)} \]
   17. Step-by-step derivation

       1. associate-*r* 60.4%

          \[\leadsto \left|\color{blue}{\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell\right) \cdot \frac{n}{Om}}\right| \]

       2. clear-num 60.4%

          \[\leadsto \left|\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{Om}{n}}}\right| \]

       3. un-div-inv 60.6%

          \[\leadsto \left|\color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell}{\frac{Om}{n}}}\right| \]

       4. associate-*r* 60.6%

          \[\leadsto \left|\frac{\sqrt{\color{blue}{\left(2 \cdot U\right) \cdot U*}} \cdot \ell}{\frac{Om}{n}}\right| \]

   18. Applied egg-rr 40.6%

       \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot U\right) \cdot U*} \cdot \ell}{\frac{Om}{n}}} \]
   19. Step-by-step derivation

       1. associate-/l* 40.6%

          \[\leadsto \color{blue}{\sqrt{\left(2 \cdot U\right) \cdot U*} \cdot \frac{\ell}{\frac{Om}{n}}} \]

       2. associate-*l* 40.6%

          \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot U*\right)}} \cdot \frac{\ell}{\frac{Om}{n}} \]

       3. associate-/r/ 40.9%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot n\right)} \]

   20. Simplified 40.9%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\frac{\ell}{Om} \cdot n\right)} \]

   if 1.06e108 < l < 1.74999999999999989e127

   1. Initial program 35.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 35.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 3.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. pow1 3.1%

         \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(U \cdot \left(n \cdot t\right)\right)}^{1}}} \]

      2. associate-*r* 3.5%

         \[\leadsto \sqrt{2 \cdot {\color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}}^{1}} \]

      3. *-commutative 3.5%

         \[\leadsto \sqrt{2 \cdot {\left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)}^{1}} \]

   7. Applied egg-rr 3.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(\left(n \cdot U\right) \cdot t\right)}^{1}}} \]
   8. Step-by-step derivation

      1. unpow1 3.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}} \]

      2. associate-*l* 3.9%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]

   9. Simplified 3.9%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]

   if 1.74999999999999989e127 < l

   1. Initial program 21.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 39.7%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 39.7%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      2. fma-define 42.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]

      3. associate-*r* 44.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]

   6. Applied egg-rr 44.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]

   7. Taylor expanded in U around 0 24.5%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-1 \cdot \frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 24.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-\frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]

      2. associate-/l* 24.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-\color{blue}{U* \cdot \frac{{\ell}^{2}}{{Om}^{2}}}\right)\right)\right)} \]

      3. unpow2 24.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right)\right)\right)} \]

      4. unpow2 24.5%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

      5. times-frac 44.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]

      6. unpow2 44.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right)\right)\right)} \]

   9. Simplified 44.2%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-U* \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   10. Taylor expanded in n around inf 26.9%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 26.9%

          \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}{{Om}^{2}}}} \]

       2. associate-*r* 26.9%

          \[\leadsto \sqrt{\frac{2 \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{{Om}^{2}}} \]

   12. Simplified 26.9%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 26.9%

          \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]

       2. sqrt-div 26.9%

          \[\leadsto 1 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{\sqrt{{Om}^{2}}}} \]

       3. associate-*r* 26.9%

          \[\leadsto 1 \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot \left(U \cdot U*\right)\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}}}{\sqrt{{Om}^{2}}} \]

       4. sqrt-prod 26.9%

          \[\leadsto 1 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{{\ell}^{2} \cdot {n}^{2}}}}{\sqrt{{Om}^{2}}} \]

       5. *-commutative 26.9%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{\color{blue}{{n}^{2} \cdot {\ell}^{2}}}}{\sqrt{{Om}^{2}}} \]

       6. sqrt-prod 27.1%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\sqrt{{n}^{2}} \cdot \sqrt{{\ell}^{2}}\right)}}{\sqrt{{Om}^{2}}} \]

       7. sqrt-pow1 15.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{{n}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       8. metadata-eval 15.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left({n}^{\color{blue}{1}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       9. pow1 15.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{n} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       10. sqrt-pow1 15.8%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}\right)}{\sqrt{{Om}^{2}}} \]

       11. metadata-eval 15.8%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot {\ell}^{\color{blue}{1}}\right)}{\sqrt{{Om}^{2}}} \]

       12. pow1 15.8%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{\ell}\right)}{\sqrt{{Om}^{2}}} \]

       13. sqrt-pow1 8.7%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{{Om}^{\left(\frac{2}{2}\right)}}} \]

       14. metadata-eval 8.7%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{{Om}^{\color{blue}{1}}} \]

       15. pow1 8.7%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{Om}} \]

   14. Applied egg-rr 8.7%

       \[\leadsto \color{blue}{1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]
   15. Step-by-step derivation

       1. *-lft-identity 8.7%

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]

       2. associate-/l* 8.7%

          \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{n \cdot \ell}{Om}} \]

       3. *-commutative 8.7%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\color{blue}{\ell \cdot n}}{Om} \]

       4. associate-/l* 15.5%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\ell \cdot \frac{n}{Om}\right)} \]

   16. Simplified 15.5%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)} \]
   17. Step-by-step derivation

       1. associate-*r* 37.2%

          \[\leadsto \left|\color{blue}{\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell\right) \cdot \frac{n}{Om}}\right| \]

       2. clear-num 37.2%

          \[\leadsto \left|\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{Om}{n}}}\right| \]

       3. un-div-inv 37.3%

          \[\leadsto \left|\color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \ell}{\frac{Om}{n}}}\right| \]

       4. associate-*r* 37.3%

          \[\leadsto \left|\frac{\sqrt{\color{blue}{\left(2 \cdot U\right) \cdot U*}} \cdot \ell}{\frac{Om}{n}}\right| \]

   18. Applied egg-rr 15.4%

       \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot U\right) \cdot U*} \cdot \ell}{\frac{Om}{n}}} \]
   19. Step-by-step derivation

       1. *-commutative 15.4%

          \[\leadsto \frac{\color{blue}{\ell \cdot \sqrt{\left(2 \cdot U\right) \cdot U*}}}{\frac{Om}{n}} \]

       2. associate-/l* 15.5%

          \[\leadsto \color{blue}{\ell \cdot \frac{\sqrt{\left(2 \cdot U\right) \cdot U*}}{\frac{Om}{n}}} \]

       3. *-commutative 15.5%

          \[\leadsto \ell \cdot \frac{\sqrt{\color{blue}{U* \cdot \left(2 \cdot U\right)}}}{\frac{Om}{n}} \]

       4. *-commutative 15.5%

          \[\leadsto \ell \cdot \frac{\sqrt{U* \cdot \color{blue}{\left(U \cdot 2\right)}}}{\frac{Om}{n}} \]

   20. Simplified 15.5%

       \[\leadsto \color{blue}{\ell \cdot \frac{\sqrt{U* \cdot \left(U \cdot 2\right)}}{\frac{Om}{n}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{+91}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\sqrt{U* \cdot \left(2 \cdot U\right)}}{\frac{Om}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{n}{Om}\\ \mathbf{if}\;\ell \leq 4.3 \cdot 10^{+26}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U* \cdot \left(2 \cdot U\right)\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ n Om))))
   (if (<= l 4.3e+26)
     (pow (* (* n t) (* 2.0 U)) 0.5)
     (sqrt (* (* U* (* 2.0 U)) (* t_1 t_1))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (n / Om);
	double tmp;
	if (l <= 4.3e+26) {
		tmp = pow(((n * t) * (2.0 * U)), 0.5);
	} else {
		tmp = sqrt(((U_42_ * (2.0 * U)) * (t_1 * t_1)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l * (n / om)
    if (l <= 4.3d+26) then
        tmp = ((n * t) * (2.0d0 * u)) ** 0.5d0
    else
        tmp = sqrt(((u_42 * (2.0d0 * u)) * (t_1 * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (n / Om);
	double tmp;
	if (l <= 4.3e+26) {
		tmp = Math.pow(((n * t) * (2.0 * U)), 0.5);
	} else {
		tmp = Math.sqrt(((U_42_ * (2.0 * U)) * (t_1 * t_1)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (n / Om)
	tmp = 0
	if l <= 4.3e+26:
		tmp = math.pow(((n * t) * (2.0 * U)), 0.5)
	else:
		tmp = math.sqrt(((U_42_ * (2.0 * U)) * (t_1 * t_1)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(n / Om))
	tmp = 0.0
	if (l <= 4.3e+26)
		tmp = Float64(Float64(n * t) * Float64(2.0 * U)) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(U_42_ * Float64(2.0 * U)) * Float64(t_1 * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (n / Om);
	tmp = 0.0;
	if (l <= 4.3e+26)
		tmp = ((n * t) * (2.0 * U)) ^ 0.5;
	else
		tmp = sqrt(((U_42_ * (2.0 * U)) * (t_1 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(n / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 4.3e+26], N[Power[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 4.2999999999999998e26

   1. Initial program 52.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 52.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0 46.4%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* 44.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. cancel-sign-sub-inv 44.4%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      3. metadata-eval 44.4%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      4. *-commutative 44.4%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{\frac{{\ell}^{2}}{Om} \cdot -2}\right)\right)} \]

      5. associate-*l/ 44.4%

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{\frac{{\ell}^{2} \cdot -2}{Om}}\right)\right)} \]

   7. Simplified 44.4%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 48.3%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 48.3%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

      3. *-commutative 48.3%

         \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

      4. +-commutative 48.3%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

      5. associate-/l* 48.3%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

      6. fma-define 48.3%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}\right)}^{0.5} \]

   9. Applied egg-rr 48.3%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)\right)}^{0.5}} \]

   10. Taylor expanded in l around 0 43.2%

       \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{t}\right)}^{0.5} \]

   11. Taylor expanded in n around 0 44.2%

       \[\leadsto {\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}^{0.5} \]
   12. Step-by-step derivation

       1. associate-*r* 44.2%

          \[\leadsto {\color{blue}{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}}^{0.5} \]

       2. *-commutative 44.2%

          \[\leadsto {\left(\color{blue}{\left(U \cdot 2\right)} \cdot \left(n \cdot t\right)\right)}^{0.5} \]

   13. Simplified 44.2%

       \[\leadsto {\color{blue}{\left(\left(U \cdot 2\right) \cdot \left(n \cdot t\right)\right)}}^{0.5} \]

   if 4.2999999999999998e26 < l

   1. Initial program 33.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 45.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 45.4%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}} + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      2. fma-define 49.9%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)} \]

      3. associate-*r* 51.1%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \color{blue}{n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)}\right)\right)} \]

   6. Applied egg-rr 51.1%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \color{blue}{\mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]

   7. Taylor expanded in U around 0 31.4%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-1 \cdot \frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 31.4%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-\frac{U* \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)} \]

      2. associate-/l* 38.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-\color{blue}{U* \cdot \frac{{\ell}^{2}}{{Om}^{2}}}\right)\right)\right)} \]

      3. unpow2 38.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right)\right)\right)} \]

      4. unpow2 38.8%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}}\right)\right)\right)} \]

      5. times-frac 51.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right)\right)\right)} \]

      6. unpow2 51.2%

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \left(-U* \cdot \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}}\right)\right)\right)} \]

   9. Simplified 51.2%

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, n \cdot \color{blue}{\left(-U* \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right)} \]

   10. Taylor expanded in n around inf 23.8%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 23.8%

          \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)\right)}{{Om}^{2}}}} \]

       2. associate-*r* 23.6%

          \[\leadsto \sqrt{\frac{2 \cdot \color{blue}{\left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{{Om}^{2}}} \]

   12. Simplified 23.6%

       \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 23.6%

          \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}{{Om}^{2}}}} \]

       2. sqrt-div 23.6%

          \[\leadsto 1 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\left(U \cdot U*\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)\right)}}{\sqrt{{Om}^{2}}}} \]

       3. associate-*r* 23.6%

          \[\leadsto 1 \cdot \frac{\sqrt{\color{blue}{\left(2 \cdot \left(U \cdot U*\right)\right) \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}}}{\sqrt{{Om}^{2}}} \]

       4. sqrt-prod 23.6%

          \[\leadsto 1 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{{\ell}^{2} \cdot {n}^{2}}}}{\sqrt{{Om}^{2}}} \]

       5. *-commutative 23.6%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{\color{blue}{{n}^{2} \cdot {\ell}^{2}}}}{\sqrt{{Om}^{2}}} \]

       6. sqrt-prod 23.7%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\sqrt{{n}^{2}} \cdot \sqrt{{\ell}^{2}}\right)}}{\sqrt{{Om}^{2}}} \]

       7. sqrt-pow1 13.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{{n}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       8. metadata-eval 13.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left({n}^{\color{blue}{1}} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       9. pow1 13.0%

          \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\color{blue}{n} \cdot \sqrt{{\ell}^{2}}\right)}{\sqrt{{Om}^{2}}} \]

       10. sqrt-pow1 13.4%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}\right)}{\sqrt{{Om}^{2}}} \]

       11. metadata-eval 13.4%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot {\ell}^{\color{blue}{1}}\right)}{\sqrt{{Om}^{2}}} \]

       12. pow1 13.4%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \color{blue}{\ell}\right)}{\sqrt{{Om}^{2}}} \]

       13. sqrt-pow1 13.6%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{{Om}^{\left(\frac{2}{2}\right)}}} \]

       14. metadata-eval 13.6%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{{Om}^{\color{blue}{1}}} \]

       15. pow1 13.6%

           \[\leadsto 1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{\color{blue}{Om}} \]

   14. Applied egg-rr 13.6%

       \[\leadsto \color{blue}{1 \cdot \frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]
   15. Step-by-step derivation

       1. *-lft-identity 13.6%

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(n \cdot \ell\right)}{Om}} \]

       2. associate-/l* 13.6%

          \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{n \cdot \ell}{Om}} \]

       3. *-commutative 13.6%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \frac{\color{blue}{\ell \cdot n}}{Om} \]

       4. associate-/l* 17.8%

          \[\leadsto \sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \color{blue}{\left(\ell \cdot \frac{n}{Om}\right)} \]

   16. Simplified 17.8%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)} \]
   17. Step-by-step derivation

       1. add-sqr-sqrt 17.6%

          \[\leadsto \color{blue}{\sqrt{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)} \cdot \sqrt{\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)}} \]

       2. sqrt-unprod 30.4%

          \[\leadsto \color{blue}{\sqrt{\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)\right) \cdot \left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)}} \]

       3. swap-sqr 27.4%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(U \cdot U*\right)} \cdot \sqrt{2 \cdot \left(U \cdot U*\right)}\right) \cdot \left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)}} \]

       4. add-sqr-sqrt 27.6%

          \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(U \cdot U*\right)\right)} \cdot \left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)} \]

       5. associate-*r* 27.6%

          \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot U\right) \cdot U*\right)} \cdot \left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)} \]

       6. pow2 27.6%

          \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot U*\right) \cdot \color{blue}{{\left(\ell \cdot \frac{n}{Om}\right)}^{2}}} \]

   18. Applied egg-rr 27.6%

       \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot U\right) \cdot U*\right) \cdot {\left(\ell \cdot \frac{n}{Om}\right)}^{2}}} \]
   19. Step-by-step derivation

       1. unpow2 27.6%

          \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot U*\right) \cdot \color{blue}{\left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)}} \]

   20. Applied egg-rr 27.6%

       \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot U*\right) \cdot \color{blue}{\left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.3 \cdot 10^{+26}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U* \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(\ell \cdot \frac{n}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 37.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 8.5 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 8.5e-141)
   (sqrt (* 2.0 (* U (* n t))))
   (pow (* 2.0 (* t (* n U))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 8.5e-141) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (n <= 8.5d-141) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 8.5e-141) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 8.5e-141:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 8.5e-141)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 8.5e-141)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 8.5e-141], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 8.50000000000000021e-141

   1. Initial program 47.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 53.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 40.5%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

   if 8.50000000000000021e-141 < n

   1. Initial program 48.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 47.3%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 26.5%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. pow1/2 30.3%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 32.9%

         \[\leadsto {\left(2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}\right)}^{0.5} \]

      3. *-commutative 32.9%

         \[\leadsto {\left(2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)\right)}^{0.5} \]

   7. Applied egg-rr 32.9%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 8.5 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 38.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (pow (* (* n t) (* 2.0 U)) 0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow(((n * t) * (2.0 * U)), 0.5);
}

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 = ((n * t) * (2.0d0 * u)) ** 0.5d0
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.pow(((n * t) * (2.0 * U)), 0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow(((n * t) * (2.0 * U)), 0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(Float64(n * t) * Float64(2.0 * U)) ^ 0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = ((n * t) * (2.0 * U)) ^ 0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 47.9%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 51.5%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in n around 0 42.9%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* 41.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

   2. cancel-sign-sub-inv 41.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

   3. metadata-eval 41.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

   4. *-commutative 41.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{\frac{{\ell}^{2}}{Om} \cdot -2}\right)\right)} \]

   5. associate-*l/ 41.7%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \color{blue}{\frac{{\ell}^{2} \cdot -2}{Om}}\right)\right)} \]

7. Simplified 41.7%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}} \]
8. Step-by-step derivation

   1. pow1/2 47.3%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(U \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)\right)}^{0.5}} \]

   2. associate-*r* 47.3%

      \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}}^{0.5} \]

   3. *-commutative 47.3%

      \[\leadsto {\left(\left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right) \cdot \left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right)\right)}^{0.5} \]

   4. +-commutative 47.3%

      \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(\frac{{\ell}^{2} \cdot -2}{Om} + t\right)}\right)}^{0.5} \]

   5. associate-/l* 47.3%

      \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)\right)}^{0.5} \]

   6. fma-define 47.3%

      \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}\right)}^{0.5} \]

9. Applied egg-rr 47.3%

   \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)\right)}^{0.5}} \]

10. Taylor expanded in l around 0 36.1%

    \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{t}\right)}^{0.5} \]

11. Taylor expanded in n around 0 37.5%

    \[\leadsto {\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}^{0.5} \]
12. Step-by-step derivation

    1. associate-*r* 37.5%

       \[\leadsto {\color{blue}{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}}^{0.5} \]

    2. *-commutative 37.5%

       \[\leadsto {\left(\color{blue}{\left(U \cdot 2\right)} \cdot \left(n \cdot t\right)\right)}^{0.5} \]

13. Simplified 37.5%

    \[\leadsto {\color{blue}{\left(\left(U \cdot 2\right) \cdot \left(n \cdot t\right)\right)}}^{0.5} \]

14. Final simplification 37.5%

    \[\leadsto {\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5} \]
15. Add Preprocessing

Alternative 19: 35.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* U (* n t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (U * (n * t))));
}

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 * (u * (n * t))))
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 * (U * (n * t))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (U * (n * t))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(U * Float64(n * t))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (U * (n * t))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 47.9%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 51.5%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 35.9%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024105 
(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*))))))