Toniolo and Linder, Equation (13)

Percentage Accurate: 49.7% → 65.9%

Time: 22.2s

Alternatives: 26

Speedup: 1.9×

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: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 65.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-320}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{\ell}}\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{n}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{Om \cdot 0.5}{U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}\right)\right)}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (if (<= t_2 2e-320)
     (*
      (sqrt
       (* (* 2.0 n) (+ t (/ (* l (+ -2.0 (/ (* n (- U* U)) Om))) (/ Om l)))))
      (sqrt U))
     (if (<= t_2 INFINITY)
       (sqrt
        (*
         t_1
         (-
          (- t (* 2.0 (/ l (/ Om l))))
          (* (- U U*) (* (/ l Om) (/ n (/ Om l)))))))
       (pow
        (/
         (* Om 0.5)
         (* U (* (* n l) (+ (* l -2.0) (/ (* (* n l) (- U* U)) Om)))))
        -0.5)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 2e-320) {
		tmp = sqrt(((2.0 * n) * (t + ((l * (-2.0 + ((n * (U_42_ - U)) / Om))) / (Om / l))))) * sqrt(U);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * ((t - (2.0 * (l / (Om / l)))) - ((U - U_42_) * ((l / Om) * (n / (Om / l)))))));
	} else {
		tmp = pow(((Om * 0.5) / (U * ((n * l) * ((l * -2.0) + (((n * l) * (U_42_ - U)) / Om))))), -0.5);
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 2e-320) {
		tmp = Math.sqrt(((2.0 * n) * (t + ((l * (-2.0 + ((n * (U_42_ - U)) / Om))) / (Om / l))))) * Math.sqrt(U);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_1 * ((t - (2.0 * (l / (Om / l)))) - ((U - U_42_) * ((l / Om) * (n / (Om / l)))))));
	} else {
		tmp = Math.pow(((Om * 0.5) / (U * ((n * l) * ((l * -2.0) + (((n * l) * (U_42_ - U)) / Om))))), -0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) + ((n * math.pow((l / Om), 2.0)) * (U_42_ - U)))
	tmp = 0
	if t_2 <= 2e-320:
		tmp = math.sqrt(((2.0 * n) * (t + ((l * (-2.0 + ((n * (U_42_ - U)) / Om))) / (Om / l))))) * math.sqrt(U)
	elif t_2 <= math.inf:
		tmp = math.sqrt((t_1 * ((t - (2.0 * (l / (Om / l)))) - ((U - U_42_) * ((l / Om) * (n / (Om / l)))))))
	else:
		tmp = math.pow(((Om * 0.5) / (U * ((n * l) * ((l * -2.0) + (((n * l) * (U_42_ - U)) / Om))))), -0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 2e-320)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * Float64(t + Float64(Float64(l * Float64(-2.0 + Float64(Float64(n * Float64(U_42_ - U)) / Om))) / Float64(Om / l))))) * sqrt(U));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(l / Float64(Om / l)))) - Float64(Float64(U - U_42_) * Float64(Float64(l / Om) * Float64(n / Float64(Om / l)))))));
	else
		tmp = Float64(Float64(Om * 0.5) / Float64(U * Float64(Float64(n * l) * Float64(Float64(l * -2.0) + Float64(Float64(Float64(n * l) * Float64(U_42_ - U)) / Om))))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) + ((n * ((l / Om) ^ 2.0)) * (U_42_ - U)));
	tmp = 0.0;
	if (t_2 <= 2e-320)
		tmp = sqrt(((2.0 * n) * (t + ((l * (-2.0 + ((n * (U_42_ - U)) / Om))) / (Om / l))))) * sqrt(U);
	elseif (t_2 <= Inf)
		tmp = sqrt((t_1 * ((t - (2.0 * (l / (Om / l)))) - ((U - U_42_) * ((l / Om) * (n / (Om / l)))))));
	else
		tmp = ((Om * 0.5) / (U * ((n * l) * ((l * -2.0) + (((n * l) * (U_42_ - U)) / Om))))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-320], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(t + N[(N[(l * N[(-2.0 + N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(n / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(Om * 0.5), $MachinePrecision] / N[(U * N[(N[(n * l), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(N[(n * l), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $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*)))) < 1.99998e-320

   1. Initial program 13.6%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 22.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 22.5%

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

   6. Applied egg-rr 24.6%

      \[\leadsto \color{blue}{{\left(\frac{1}{\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. inv-pow N/A

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

      2. pow-pow N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(2 \cdot n\right)}\right), \color{blue}{\left({U}^{\frac{1}{2}}\right)}\right) \]

   8. Applied egg-rr 49.9%

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

   if 1.99998e-320 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 71.1%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 74.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 74.5%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\left(n \cdot \left(\frac{1}{\frac{Om}{\ell}} \cdot \frac{\ell}{Om}\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\left(n \cdot \frac{1}{\frac{\frac{Om}{\ell}}{\frac{\ell}{Om}}}\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{n}{\frac{\frac{Om}{\ell}}{\frac{\ell}{Om}}}\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. associate-/r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{n}{\frac{Om}{\ell}} \cdot \frac{\ell}{Om}\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{n}{\frac{Om}{\ell}}\right), \left(\frac{\ell}{Om}\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \left(\frac{Om}{\ell}\right)\right), \left(\frac{\ell}{Om}\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \ell\right)\right), \left(\frac{\ell}{Om}\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 75.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{/.f64}\left(\ell, Om\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 75.1%

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

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

   1. Initial program 0.0%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 0.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 0.6%

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

   6. Applied egg-rr 36.5%

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

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{Om}{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)}\right)}, \frac{-1}{2}\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\frac{1}{2} \cdot Om}{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)}\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot Om\right), \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \left(\left(\ell \cdot n\right) \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\left(\ell \cdot \left(n \cdot \left(U - U*\right)\right)\right), Om\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

      14. --lowering--.f64 66.8%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(U, U*\right)\right), Om\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   9. Simplified 66.8%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{-320}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{\ell}}\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{n}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{Om \cdot 0.5}{U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}\right)\right)}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{-1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\frac{n \cdot \frac{U - U*}{\frac{Om}{\ell}} - \ell \cdot -2}{\frac{Om}{\ell}} - t\right)}\right)}^{-0.5}\\ \mathbf{if}\;n \leq -2.8 \cdot 10^{-131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.16 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\frac{\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)}{\frac{Om}{-4}} + U \cdot \left(n \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (pow
          (/
           -1.0
           (*
            (* n (* 2.0 U))
            (- (/ (- (* n (/ (- U U*) (/ Om l))) (* l -2.0)) (/ Om l)) t)))
          -0.5)))
   (if (<= n -2.8e-131)
     t_1
     (if (<= n 1.16e-171)
       (sqrt (+ (/ (* l (* U (* n l))) (/ Om -4.0)) (* U (* n (* 2.0 t)))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((-1.0 / ((n * (2.0 * U)) * ((((n * ((U - U_42_) / (Om / l))) - (l * -2.0)) / (Om / l)) - t))), -0.5);
	double tmp;
	if (n <= -2.8e-131) {
		tmp = t_1;
	} else if (n <= 1.16e-171) {
		tmp = sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	} else {
		tmp = 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 = ((-1.0d0) / ((n * (2.0d0 * u)) * ((((n * ((u - u_42) / (om / l))) - (l * (-2.0d0))) / (om / l)) - t))) ** (-0.5d0)
    if (n <= (-2.8d-131)) then
        tmp = t_1
    else if (n <= 1.16d-171) then
        tmp = sqrt((((l * (u * (n * l))) / (om / (-4.0d0))) + (u * (n * (2.0d0 * t)))))
    else
        tmp = 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 = Math.pow((-1.0 / ((n * (2.0 * U)) * ((((n * ((U - U_42_) / (Om / l))) - (l * -2.0)) / (Om / l)) - t))), -0.5);
	double tmp;
	if (n <= -2.8e-131) {
		tmp = t_1;
	} else if (n <= 1.16e-171) {
		tmp = Math.sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.pow((-1.0 / ((n * (2.0 * U)) * ((((n * ((U - U_42_) / (Om / l))) - (l * -2.0)) / (Om / l)) - t))), -0.5)
	tmp = 0
	if n <= -2.8e-131:
		tmp = t_1
	elif n <= 1.16e-171:
		tmp = math.sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(-1.0 / Float64(Float64(n * Float64(2.0 * U)) * Float64(Float64(Float64(Float64(n * Float64(Float64(U - U_42_) / Float64(Om / l))) - Float64(l * -2.0)) / Float64(Om / l)) - t))) ^ -0.5
	tmp = 0.0
	if (n <= -2.8e-131)
		tmp = t_1;
	elseif (n <= 1.16e-171)
		tmp = sqrt(Float64(Float64(Float64(l * Float64(U * Float64(n * l))) / Float64(Om / -4.0)) + Float64(U * Float64(n * Float64(2.0 * t)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (-1.0 / ((n * (2.0 * U)) * ((((n * ((U - U_42_) / (Om / l))) - (l * -2.0)) / (Om / l)) - t))) ^ -0.5;
	tmp = 0.0;
	if (n <= -2.8e-131)
		tmp = t_1;
	elseif (n <= 1.16e-171)
		tmp = sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(-1.0 / N[(N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[n, -2.8e-131], t$95$1, If[LessEqual[n, 1.16e-171], N[Sqrt[N[(N[(N[(l * N[(U * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / -4.0), $MachinePrecision]), $MachinePrecision] + N[(U * N[(n * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.8e-131 or 1.15999999999999999e-171 < n

   1. Initial program 50.4%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 55.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 55.2%

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

   6. Applied egg-rr 61.6%

      \[\leadsto \color{blue}{{\left(\frac{1}{\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(n \cdot \frac{U - U*}{\frac{Om}{\ell}}\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(n \cdot \left(\left(U - U*\right) \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right), n\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right), n\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\left(\left(U - U*\right) \cdot \frac{1}{\frac{Om}{\ell}}\right), n\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\left(\frac{U - U*}{\frac{Om}{\ell}}\right), n\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(U - U*\right), \left(\frac{Om}{\ell}\right)\right), n\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{Om}{\ell}\right)\right), n\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      12. /-lowering-/.f64 65.3%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{/.f64}\left(Om, \ell\right)\right), n\right)\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   8. Applied egg-rr 65.3%

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

   if -2.8e-131 < n < 1.15999999999999999e-171

   1. Initial program 45.2%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}{Om}\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot \frac{-4}{Om}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 50.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \mathsf{/.f64}\left(-4, Om\right)\right)\right)\right) \]

   5. Simplified 50.3%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om} + \left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{1}{\frac{Om}{-4}}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{\frac{Om}{-4}}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot U\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\ell \cdot n\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(n \cdot \ell\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(n \cdot \ell\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\ell \cdot n\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(2 \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(\left(U \cdot \left(t \cdot n\right)\right) \cdot 2\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(U \cdot \left(\left(t \cdot n\right) \cdot 2\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(\left(t \cdot n\right) \cdot 2\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(\left(n \cdot t\right) \cdot 2\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(n \cdot \left(t \cdot 2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(t \cdot 2\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 63.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(t, 2\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 63.7%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{-131}:\\ \;\;\;\;{\left(\frac{-1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\frac{n \cdot \frac{U - U*}{\frac{Om}{\ell}} - \ell \cdot -2}{\frac{Om}{\ell}} - t\right)}\right)}^{-0.5}\\ \mathbf{elif}\;n \leq 1.16 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\frac{\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)}{\frac{Om}{-4}} + U \cdot \left(n \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\frac{n \cdot \frac{U - U*}{\frac{Om}{\ell}} - \ell \cdot -2}{\frac{Om}{\ell}} - t\right)}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{-1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(\left(U - U*\right) \cdot \frac{n}{Om} - -2\right)\right) - t\right)}\right)}^{-0.5}\\ \mathbf{if}\;U \leq -5 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;U \leq 1.4 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{U}{\frac{1}{t + \frac{\ell \cdot -2 - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (pow
          (/
           -1.0
           (*
            (* n (* 2.0 U))
            (- (* (/ l Om) (* l (- (* (- U U*) (/ n Om)) -2.0))) t)))
          -0.5)))
   (if (<= U -5e+16)
     t_1
     (if (<= U 1.4e-211)
       (sqrt
        (*
         (* 2.0 n)
         (/
          U
          (/
           1.0
           (+ t (/ (- (* l -2.0) (/ (* n (- U U*)) (/ Om l))) (/ Om l)))))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((-1.0 / ((n * (2.0 * U)) * (((l / Om) * (l * (((U - U_42_) * (n / Om)) - -2.0))) - t))), -0.5);
	double tmp;
	if (U <= -5e+16) {
		tmp = t_1;
	} else if (U <= 1.4e-211) {
		tmp = sqrt(((2.0 * n) * (U / (1.0 / (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l)))))));
	} else {
		tmp = 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 = ((-1.0d0) / ((n * (2.0d0 * u)) * (((l / om) * (l * (((u - u_42) * (n / om)) - (-2.0d0)))) - t))) ** (-0.5d0)
    if (u <= (-5d+16)) then
        tmp = t_1
    else if (u <= 1.4d-211) then
        tmp = sqrt(((2.0d0 * n) * (u / (1.0d0 / (t + (((l * (-2.0d0)) - ((n * (u - u_42)) / (om / l))) / (om / l)))))))
    else
        tmp = 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 = Math.pow((-1.0 / ((n * (2.0 * U)) * (((l / Om) * (l * (((U - U_42_) * (n / Om)) - -2.0))) - t))), -0.5);
	double tmp;
	if (U <= -5e+16) {
		tmp = t_1;
	} else if (U <= 1.4e-211) {
		tmp = Math.sqrt(((2.0 * n) * (U / (1.0 / (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.pow((-1.0 / ((n * (2.0 * U)) * (((l / Om) * (l * (((U - U_42_) * (n / Om)) - -2.0))) - t))), -0.5)
	tmp = 0
	if U <= -5e+16:
		tmp = t_1
	elif U <= 1.4e-211:
		tmp = math.sqrt(((2.0 * n) * (U / (1.0 / (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l)))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(-1.0 / Float64(Float64(n * Float64(2.0 * U)) * Float64(Float64(Float64(l / Om) * Float64(l * Float64(Float64(Float64(U - U_42_) * Float64(n / Om)) - -2.0))) - t))) ^ -0.5
	tmp = 0.0
	if (U <= -5e+16)
		tmp = t_1;
	elseif (U <= 1.4e-211)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U / Float64(1.0 / Float64(t + Float64(Float64(Float64(l * -2.0) - Float64(Float64(n * Float64(U - U_42_)) / Float64(Om / l))) / Float64(Om / l)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (-1.0 / ((n * (2.0 * U)) * (((l / Om) * (l * (((U - U_42_) * (n / Om)) - -2.0))) - t))) ^ -0.5;
	tmp = 0.0;
	if (U <= -5e+16)
		tmp = t_1;
	elseif (U <= 1.4e-211)
		tmp = sqrt(((2.0 * n) * (U / (1.0 / (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l)))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(-1.0 / N[(N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * N[(l * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[U, -5e+16], t$95$1, If[LessEqual[U, 1.4e-211], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U / N[(1.0 / N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] - N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if U < -5e16 or 1.3999999999999999e-211 < U

   1. Initial program 52.6%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 56.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 56.0%

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

   6. Applied egg-rr 62.9%

      \[\leadsto \color{blue}{{\left(\frac{1}{\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{1}{\frac{\frac{Om}{\ell}}{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}}\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{Om} \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\ell \cdot \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\left(n \cdot \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      12. --lowering--.f64 62.9%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U, U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   8. Applied egg-rr 62.9%

      \[\leadsto {\left(\frac{1}{\left(t + \color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{\left(U - U*\right) \cdot n}{Om}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\left(U - U*\right) \cdot \frac{n}{Om}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(\left(U - U*\right), \left(\frac{n}{Om}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{n}{Om}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      5. /-lowering-/.f64 65.1%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{/.f64}\left(n, Om\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   10. Applied egg-rr 65.1%

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

   if -5e16 < U < 1.3999999999999999e-211

   1. Initial program 44.2%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 48.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 48.1%

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

   6. Applied egg-rr 59.0%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \frac{U}{\frac{1}{t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -5 \cdot 10^{+16}:\\ \;\;\;\;{\left(\frac{-1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(\left(U - U*\right) \cdot \frac{n}{Om} - -2\right)\right) - t\right)}\right)}^{-0.5}\\ \mathbf{elif}\;U \leq 1.4 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{U}{\frac{1}{t + \frac{\ell \cdot -2 - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(\left(U - U*\right) \cdot \frac{n}{Om} - -2\right)\right) - t\right)}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{-1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(\left(U - U*\right) \cdot \frac{n}{Om} - -2\right)\right) - t\right)}\right)}^{-0.5}\\ \mathbf{if}\;U \leq -3 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;U \leq 1.85 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2 - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (pow
          (/
           -1.0
           (*
            (* n (* 2.0 U))
            (- (* (/ l Om) (* l (- (* (- U U*) (/ n Om)) -2.0))) t)))
          -0.5)))
   (if (<= U -3e+16)
     t_1
     (if (<= U 1.85e-211)
       (sqrt
        (*
         (* 2.0 n)
         (* U (+ t (/ (- (* l -2.0) (/ (* n (- U U*)) (/ Om l))) (/ Om l))))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((-1.0 / ((n * (2.0 * U)) * (((l / Om) * (l * (((U - U_42_) * (n / Om)) - -2.0))) - t))), -0.5);
	double tmp;
	if (U <= -3e+16) {
		tmp = t_1;
	} else if (U <= 1.85e-211) {
		tmp = sqrt(((2.0 * n) * (U * (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l))))));
	} else {
		tmp = 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 = ((-1.0d0) / ((n * (2.0d0 * u)) * (((l / om) * (l * (((u - u_42) * (n / om)) - (-2.0d0)))) - t))) ** (-0.5d0)
    if (u <= (-3d+16)) then
        tmp = t_1
    else if (u <= 1.85d-211) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + (((l * (-2.0d0)) - ((n * (u - u_42)) / (om / l))) / (om / l))))))
    else
        tmp = 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 = Math.pow((-1.0 / ((n * (2.0 * U)) * (((l / Om) * (l * (((U - U_42_) * (n / Om)) - -2.0))) - t))), -0.5);
	double tmp;
	if (U <= -3e+16) {
		tmp = t_1;
	} else if (U <= 1.85e-211) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.pow((-1.0 / ((n * (2.0 * U)) * (((l / Om) * (l * (((U - U_42_) * (n / Om)) - -2.0))) - t))), -0.5)
	tmp = 0
	if U <= -3e+16:
		tmp = t_1
	elif U <= 1.85e-211:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(-1.0 / Float64(Float64(n * Float64(2.0 * U)) * Float64(Float64(Float64(l / Om) * Float64(l * Float64(Float64(Float64(U - U_42_) * Float64(n / Om)) - -2.0))) - t))) ^ -0.5
	tmp = 0.0
	if (U <= -3e+16)
		tmp = t_1;
	elseif (U <= 1.85e-211)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(l * -2.0) - Float64(Float64(n * Float64(U - U_42_)) / Float64(Om / l))) / Float64(Om / l))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (-1.0 / ((n * (2.0 * U)) * (((l / Om) * (l * (((U - U_42_) * (n / Om)) - -2.0))) - t))) ^ -0.5;
	tmp = 0.0;
	if (U <= -3e+16)
		tmp = t_1;
	elseif (U <= 1.85e-211)
		tmp = sqrt(((2.0 * n) * (U * (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(-1.0 / N[(N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * N[(l * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[U, -3e+16], t$95$1, If[LessEqual[U, 1.85e-211], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] - N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if U < -3e16 or 1.8499999999999999e-211 < U

   1. Initial program 52.6%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 56.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 56.0%

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

   6. Applied egg-rr 62.9%

      \[\leadsto \color{blue}{{\left(\frac{1}{\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{1}{\frac{\frac{Om}{\ell}}{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}}\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{Om} \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\ell \cdot \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\left(n \cdot \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      12. --lowering--.f64 62.9%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U, U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   8. Applied egg-rr 62.9%

      \[\leadsto {\left(\frac{1}{\left(t + \color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{\left(U - U*\right) \cdot n}{Om}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\left(U - U*\right) \cdot \frac{n}{Om}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(\left(U - U*\right), \left(\frac{n}{Om}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(\frac{n}{Om}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      5. /-lowering-/.f64 65.1%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{/.f64}\left(n, Om\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   10. Applied egg-rr 65.1%

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

   if -3e16 < U < 1.8499999999999999e-211

   1. Initial program 44.2%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 48.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 48.1%

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

   6. Applied egg-rr 59.0%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -3 \cdot 10^{+16}:\\ \;\;\;\;{\left(\frac{-1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(\left(U - U*\right) \cdot \frac{n}{Om} - -2\right)\right) - t\right)}\right)}^{-0.5}\\ \mathbf{elif}\;U \leq 1.85 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2 - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{-1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot \left(\left(U - U*\right) \cdot \frac{n}{Om} - -2\right)\right) - t\right)}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(2 \cdot U\right)\\ \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-67}:\\ \;\;\;\;{\left(\frac{1}{t\_1 \cdot \left(t + \frac{\ell}{Om} \cdot \frac{U* \cdot \left(n \cdot \ell\right)}{Om}\right)}\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(t + \frac{\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{\ell}}\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{Om \cdot 0.5}{U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}\right)\right)}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* 2.0 U))))
   (if (<= l 1.5e-67)
     (pow (/ 1.0 (* t_1 (+ t (* (/ l Om) (/ (* U* (* n l)) Om))))) -0.5)
     (if (<= l 6e+151)
       (sqrt (* (+ t (/ (* l (+ -2.0 (/ (* n (- U* U)) Om))) (/ Om l))) t_1))
       (pow
        (/
         (* Om 0.5)
         (* U (* (* n l) (+ (* l -2.0) (/ (* (* n l) (- U* U)) Om)))))
        -0.5)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (2.0 * U);
	double tmp;
	if (l <= 1.5e-67) {
		tmp = pow((1.0 / (t_1 * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))), -0.5);
	} else if (l <= 6e+151) {
		tmp = sqrt(((t + ((l * (-2.0 + ((n * (U_42_ - U)) / Om))) / (Om / l))) * t_1));
	} else {
		tmp = pow(((Om * 0.5) / (U * ((n * l) * ((l * -2.0) + (((n * l) * (U_42_ - U)) / Om))))), -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) :: t_1
    real(8) :: tmp
    t_1 = n * (2.0d0 * u)
    if (l <= 1.5d-67) then
        tmp = (1.0d0 / (t_1 * (t + ((l / om) * ((u_42 * (n * l)) / om))))) ** (-0.5d0)
    else if (l <= 6d+151) then
        tmp = sqrt(((t + ((l * ((-2.0d0) + ((n * (u_42 - u)) / om))) / (om / l))) * t_1))
    else
        tmp = ((om * 0.5d0) / (u * ((n * l) * ((l * (-2.0d0)) + (((n * l) * (u_42 - u)) / om))))) ** (-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 t_1 = n * (2.0 * U);
	double tmp;
	if (l <= 1.5e-67) {
		tmp = Math.pow((1.0 / (t_1 * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))), -0.5);
	} else if (l <= 6e+151) {
		tmp = Math.sqrt(((t + ((l * (-2.0 + ((n * (U_42_ - U)) / Om))) / (Om / l))) * t_1));
	} else {
		tmp = Math.pow(((Om * 0.5) / (U * ((n * l) * ((l * -2.0) + (((n * l) * (U_42_ - U)) / Om))))), -0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * (2.0 * U)
	tmp = 0
	if l <= 1.5e-67:
		tmp = math.pow((1.0 / (t_1 * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))), -0.5)
	elif l <= 6e+151:
		tmp = math.sqrt(((t + ((l * (-2.0 + ((n * (U_42_ - U)) / Om))) / (Om / l))) * t_1))
	else:
		tmp = math.pow(((Om * 0.5) / (U * ((n * l) * ((l * -2.0) + (((n * l) * (U_42_ - U)) / Om))))), -0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(2.0 * U))
	tmp = 0.0
	if (l <= 1.5e-67)
		tmp = Float64(1.0 / Float64(t_1 * Float64(t + Float64(Float64(l / Om) * Float64(Float64(U_42_ * Float64(n * l)) / Om))))) ^ -0.5;
	elseif (l <= 6e+151)
		tmp = sqrt(Float64(Float64(t + Float64(Float64(l * Float64(-2.0 + Float64(Float64(n * Float64(U_42_ - U)) / Om))) / Float64(Om / l))) * t_1));
	else
		tmp = Float64(Float64(Om * 0.5) / Float64(U * Float64(Float64(n * l) * Float64(Float64(l * -2.0) + Float64(Float64(Float64(n * l) * Float64(U_42_ - U)) / Om))))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * (2.0 * U);
	tmp = 0.0;
	if (l <= 1.5e-67)
		tmp = (1.0 / (t_1 * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))) ^ -0.5;
	elseif (l <= 6e+151)
		tmp = sqrt(((t + ((l * (-2.0 + ((n * (U_42_ - U)) / Om))) / (Om / l))) * t_1));
	else
		tmp = ((Om * 0.5) / (U * ((n * l) * ((l * -2.0) + (((n * l) * (U_42_ - U)) / Om))))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.5e-67], N[Power[N[(1.0 / N[(t$95$1 * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(U$42$ * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], If[LessEqual[l, 6e+151], N[Sqrt[N[(N[(t + N[(N[(l * N[(-2.0 + N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(Om * 0.5), $MachinePrecision] / N[(U * N[(N[(n * l), $MachinePrecision] * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(N[(n * l), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.50000000000000016e-67

   1. Initial program 51.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 55.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 55.6%

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

   6. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{{\left(\frac{1}{\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{1}{\frac{\frac{Om}{\ell}}{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}}\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{Om} \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\ell \cdot \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\left(n \cdot \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      12. --lowering--.f64 55.1%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U, U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   8. Applied egg-rr 55.1%

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

   9. Taylor expanded in U* around inf

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

       3. *-lowering-*.f64 51.2%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   11. Simplified 51.2%

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

   if 1.50000000000000016e-67 < l < 5.9999999999999998e151

   1. Initial program 63.6%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 63.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 63.6%

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

   6. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. inv-pow N/A

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

      2. pow-pow N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right), \left(n \cdot \left(2 \cdot U\right)\right)\right)\right) \]

   8. Applied egg-rr 68.9%

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

   if 5.9999999999999998e151 < l

   1. Initial program 16.3%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 23.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 23.0%

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

   6. Applied egg-rr 51.1%

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

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{Om}{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)}\right)}, \frac{-1}{2}\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\frac{1}{2} \cdot Om}{U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)}\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot Om\right), \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \left(U \cdot \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \left(\ell \cdot \left(n \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \left(\left(\ell \cdot n\right) \cdot \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(-2 \cdot \ell - \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\left(-2 \cdot \ell\right), \left(\frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \left(\frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\left(\ell \cdot \left(n \cdot \left(U - U*\right)\right)\right), Om\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\left(\left(\ell \cdot n\right) \cdot \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot n\right), \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

      14. --lowering--.f64 56.8%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, Om\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \mathsf{\_.f64}\left(U, U*\right)\right), Om\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   9. Simplified 56.8%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-67}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \frac{U* \cdot \left(n \cdot \ell\right)}{Om}\right)}\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(t + \frac{\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{Om \cdot 0.5}{U \cdot \left(\left(n \cdot \ell\right) \cdot \left(\ell \cdot -2 + \frac{\left(n \cdot \ell\right) \cdot \left(U* - U\right)}{Om}\right)\right)}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(2 \cdot U\right)\\ \mathbf{if}\;n \leq -1.8 \cdot 10^{-127}:\\ \;\;\;\;{\left(\frac{1}{t\_1 \cdot \left(t + \frac{\frac{U* \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \mathbf{elif}\;n \leq 2.25 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\frac{\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)}{\frac{Om}{-4}} + U \cdot \left(n \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t + \frac{\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{\ell}}\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* 2.0 U))))
   (if (<= n -1.8e-127)
     (pow (/ 1.0 (* t_1 (+ t (/ (/ (* U* (* n l)) Om) (/ Om l))))) -0.5)
     (if (<= n 2.25e-171)
       (sqrt (+ (/ (* l (* U (* n l))) (/ Om -4.0)) (* U (* n (* 2.0 t)))))
       (sqrt
        (* (+ t (/ (* l (+ -2.0 (/ (* n (- U* U)) Om))) (/ Om l))) t_1))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (2.0 * U);
	double tmp;
	if (n <= -1.8e-127) {
		tmp = pow((1.0 / (t_1 * (t + (((U_42_ * (n * l)) / Om) / (Om / l))))), -0.5);
	} else if (n <= 2.25e-171) {
		tmp = sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	} else {
		tmp = sqrt(((t + ((l * (-2.0 + ((n * (U_42_ - U)) / Om))) / (Om / l))) * 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 = n * (2.0d0 * u)
    if (n <= (-1.8d-127)) then
        tmp = (1.0d0 / (t_1 * (t + (((u_42 * (n * l)) / om) / (om / l))))) ** (-0.5d0)
    else if (n <= 2.25d-171) then
        tmp = sqrt((((l * (u * (n * l))) / (om / (-4.0d0))) + (u * (n * (2.0d0 * t)))))
    else
        tmp = sqrt(((t + ((l * ((-2.0d0) + ((n * (u_42 - u)) / om))) / (om / l))) * 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 = n * (2.0 * U);
	double tmp;
	if (n <= -1.8e-127) {
		tmp = Math.pow((1.0 / (t_1 * (t + (((U_42_ * (n * l)) / Om) / (Om / l))))), -0.5);
	} else if (n <= 2.25e-171) {
		tmp = Math.sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	} else {
		tmp = Math.sqrt(((t + ((l * (-2.0 + ((n * (U_42_ - U)) / Om))) / (Om / l))) * t_1));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * (2.0 * U)
	tmp = 0
	if n <= -1.8e-127:
		tmp = math.pow((1.0 / (t_1 * (t + (((U_42_ * (n * l)) / Om) / (Om / l))))), -0.5)
	elif n <= 2.25e-171:
		tmp = math.sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))))
	else:
		tmp = math.sqrt(((t + ((l * (-2.0 + ((n * (U_42_ - U)) / Om))) / (Om / l))) * t_1))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(2.0 * U))
	tmp = 0.0
	if (n <= -1.8e-127)
		tmp = Float64(1.0 / Float64(t_1 * Float64(t + Float64(Float64(Float64(U_42_ * Float64(n * l)) / Om) / Float64(Om / l))))) ^ -0.5;
	elseif (n <= 2.25e-171)
		tmp = sqrt(Float64(Float64(Float64(l * Float64(U * Float64(n * l))) / Float64(Om / -4.0)) + Float64(U * Float64(n * Float64(2.0 * t)))));
	else
		tmp = sqrt(Float64(Float64(t + Float64(Float64(l * Float64(-2.0 + Float64(Float64(n * Float64(U_42_ - U)) / Om))) / Float64(Om / l))) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * (2.0 * U);
	tmp = 0.0;
	if (n <= -1.8e-127)
		tmp = (1.0 / (t_1 * (t + (((U_42_ * (n * l)) / Om) / (Om / l))))) ^ -0.5;
	elseif (n <= 2.25e-171)
		tmp = sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	else
		tmp = sqrt(((t + ((l * (-2.0 + ((n * (U_42_ - U)) / Om))) / (Om / l))) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.8e-127], N[Power[N[(1.0 / N[(t$95$1 * N[(t + N[(N[(N[(U$42$ * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], If[LessEqual[n, 2.25e-171], N[Sqrt[N[(N[(N[(l * N[(U * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / -4.0), $MachinePrecision]), $MachinePrecision] + N[(U * N[(n * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(t + N[(N[(l * N[(-2.0 + N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.8e-127

   1. Initial program 49.1%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 51.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 51.7%

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

   6. Applied egg-rr 58.1%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      3. *-lowering-*.f64 57.3%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   9. Simplified 57.3%

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

   if -1.8e-127 < n < 2.2500000000000002e-171

   1. Initial program 45.2%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}{Om}\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot \frac{-4}{Om}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 50.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \mathsf{/.f64}\left(-4, Om\right)\right)\right)\right) \]

   5. Simplified 50.3%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om} + \left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{1}{\frac{Om}{-4}}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{\frac{Om}{-4}}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot U\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\ell \cdot n\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(n \cdot \ell\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(n \cdot \ell\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\ell \cdot n\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(2 \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(\left(U \cdot \left(t \cdot n\right)\right) \cdot 2\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(U \cdot \left(\left(t \cdot n\right) \cdot 2\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(\left(t \cdot n\right) \cdot 2\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(\left(n \cdot t\right) \cdot 2\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(n \cdot \left(t \cdot 2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(t \cdot 2\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 63.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(t, 2\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 63.7%

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

   if 2.2500000000000002e-171 < n

   1. Initial program 51.4%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 57.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 57.8%

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

   6. Applied egg-rr 64.1%

      \[\leadsto \color{blue}{{\left(\frac{1}{\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. inv-pow N/A

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

      2. pow-pow N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right), \left(n \cdot \left(2 \cdot U\right)\right)\right)\right) \]

   8. Applied egg-rr 60.7%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.8 \cdot 10^{-127}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\frac{U* \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \mathbf{elif}\;n \leq 2.25 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\frac{\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)}{\frac{Om}{-4}} + U \cdot \left(n \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t + \frac{\ell \cdot \left(-2 + \frac{n \cdot \left(U* - U\right)}{Om}\right)}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{-91}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \frac{U* \cdot \left(n \cdot \ell\right)}{Om}\right)}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot -2 - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.5e-91)
   (pow
    (/ 1.0 (* (* n (* 2.0 U)) (+ t (* (/ l Om) (/ (* U* (* n l)) Om)))))
    -0.5)
   (sqrt
    (*
     U
     (*
      (* 2.0 n)
      (+ t (/ (- (* l -2.0) (/ (* n (- U U*)) (/ Om l))) (/ Om l))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.5e-91) {
		tmp = pow((1.0 / ((n * (2.0 * U)) * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))), -0.5);
	} else {
		tmp = sqrt((U * ((2.0 * n) * (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l))))));
	}
	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 <= 6.5d-91) then
        tmp = (1.0d0 / ((n * (2.0d0 * u)) * (t + ((l / om) * ((u_42 * (n * l)) / om))))) ** (-0.5d0)
    else
        tmp = sqrt((u * ((2.0d0 * n) * (t + (((l * (-2.0d0)) - ((n * (u - u_42)) / (om / l))) / (om / l))))))
    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 <= 6.5e-91) {
		tmp = Math.pow((1.0 / ((n * (2.0 * U)) * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))), -0.5);
	} else {
		tmp = Math.sqrt((U * ((2.0 * n) * (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.5e-91:
		tmp = math.pow((1.0 / ((n * (2.0 * U)) * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))), -0.5)
	else:
		tmp = math.sqrt((U * ((2.0 * n) * (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.5e-91)
		tmp = Float64(1.0 / Float64(Float64(n * Float64(2.0 * U)) * Float64(t + Float64(Float64(l / Om) * Float64(Float64(U_42_ * Float64(n * l)) / Om))))) ^ -0.5;
	else
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * Float64(t + Float64(Float64(Float64(l * -2.0) - Float64(Float64(n * Float64(U - U_42_)) / Float64(Om / l))) / Float64(Om / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6.5e-91)
		tmp = (1.0 / ((n * (2.0 * U)) * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))) ^ -0.5;
	else
		tmp = sqrt((U * ((2.0 * n) * (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.5e-91], N[Power[N[(1.0 / N[(N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(U$42$ * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] - N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.5000000000000001e-91

   1. Initial program 51.4%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 55.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 55.3%

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

   6. Applied egg-rr 57.4%

      \[\leadsto \color{blue}{{\left(\frac{1}{\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{1}{\frac{\frac{Om}{\ell}}{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}}\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{Om} \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\ell \cdot \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\left(n \cdot \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      12. --lowering--.f64 55.3%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U, U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   8. Applied egg-rr 55.3%

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

   9. Taylor expanded in U* around inf

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

       3. *-lowering-*.f64 50.7%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   11. Simplified 50.7%

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

   if 6.5000000000000001e-91 < l

   1. Initial program 43.3%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 46.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 46.3%

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

   6. Applied egg-rr 63.8%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{-91}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \frac{U* \cdot \left(n \cdot \ell\right)}{Om}\right)}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{\ell \cdot -2 - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.1 \cdot 10^{-193}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \frac{U* \cdot \left(n \cdot \ell\right)}{Om}\right)}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2 - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.1e-193)
   (pow
    (/ 1.0 (* (* n (* 2.0 U)) (+ t (* (/ l Om) (/ (* U* (* n l)) Om)))))
    -0.5)
   (sqrt
    (*
     (* 2.0 n)
     (* U (+ t (/ (- (* l -2.0) (/ (* n (- U U*)) (/ Om l))) (/ Om l))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.1e-193) {
		tmp = pow((1.0 / ((n * (2.0 * U)) * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))), -0.5);
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l))))));
	}
	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 <= 4.1d-193) then
        tmp = (1.0d0 / ((n * (2.0d0 * u)) * (t + ((l / om) * ((u_42 * (n * l)) / om))))) ** (-0.5d0)
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t + (((l * (-2.0d0)) - ((n * (u - u_42)) / (om / l))) / (om / l))))))
    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 <= 4.1e-193) {
		tmp = Math.pow((1.0 / ((n * (2.0 * U)) * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))), -0.5);
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.1e-193:
		tmp = math.pow((1.0 / ((n * (2.0 * U)) * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))), -0.5)
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.1e-193)
		tmp = Float64(1.0 / Float64(Float64(n * Float64(2.0 * U)) * Float64(t + Float64(Float64(l / Om) * Float64(Float64(U_42_ * Float64(n * l)) / Om))))) ^ -0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(l * -2.0) - Float64(Float64(n * Float64(U - U_42_)) / Float64(Om / l))) / Float64(Om / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4.1e-193)
		tmp = (1.0 / ((n * (2.0 * U)) * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))) ^ -0.5;
	else
		tmp = sqrt(((2.0 * n) * (U * (t + (((l * -2.0) - ((n * (U - U_42_)) / (Om / l))) / (Om / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.1e-193], N[Power[N[(1.0 / N[(N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(U$42$ * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(l * -2.0), $MachinePrecision] - N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.10000000000000003e-193

   1. Initial program 50.4%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 54.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 54.7%

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

   6. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{{\left(\frac{1}{\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{1}{\frac{\frac{Om}{\ell}}{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}}\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{Om} \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\ell \cdot \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\left(n \cdot \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      12. --lowering--.f64 55.4%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U, U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   8. Applied egg-rr 55.4%

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

   9. Taylor expanded in U* around inf

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

       3. *-lowering-*.f64 50.2%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   11. Simplified 50.2%

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

   if 4.10000000000000003e-193 < l

   1. Initial program 46.6%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 49.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 49.0%

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

   6. Applied egg-rr 59.9%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.1 \cdot 10^{-193}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \frac{U* \cdot \left(n \cdot \ell\right)}{Om}\right)}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot -2 - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(2 \cdot U\right)\\ \mathbf{if}\;n \leq -4 \cdot 10^{-133}:\\ \;\;\;\;{\left(\frac{1}{t\_1 \cdot \left(t + \frac{\frac{U* \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\frac{\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)}{\frac{Om}{-4}} + U \cdot \left(n \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{t\_1 \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* 2.0 U))))
   (if (<= n -4e-133)
     (pow (/ 1.0 (* t_1 (+ t (/ (/ (* U* (* n l)) Om) (/ Om l))))) -0.5)
     (if (<= n 2.6e-171)
       (sqrt (+ (/ (* l (* U (* n l))) (/ Om -4.0)) (* U (* n (* 2.0 t)))))
       (pow (/ 1.0 (* t_1 (+ t (/ (* l -2.0) (/ Om l))))) -0.5)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (2.0 * U);
	double tmp;
	if (n <= -4e-133) {
		tmp = pow((1.0 / (t_1 * (t + (((U_42_ * (n * l)) / Om) / (Om / l))))), -0.5);
	} else if (n <= 2.6e-171) {
		tmp = sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	} else {
		tmp = pow((1.0 / (t_1 * (t + ((l * -2.0) / (Om / l))))), -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) :: t_1
    real(8) :: tmp
    t_1 = n * (2.0d0 * u)
    if (n <= (-4d-133)) then
        tmp = (1.0d0 / (t_1 * (t + (((u_42 * (n * l)) / om) / (om / l))))) ** (-0.5d0)
    else if (n <= 2.6d-171) then
        tmp = sqrt((((l * (u * (n * l))) / (om / (-4.0d0))) + (u * (n * (2.0d0 * t)))))
    else
        tmp = (1.0d0 / (t_1 * (t + ((l * (-2.0d0)) / (om / l))))) ** (-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 t_1 = n * (2.0 * U);
	double tmp;
	if (n <= -4e-133) {
		tmp = Math.pow((1.0 / (t_1 * (t + (((U_42_ * (n * l)) / Om) / (Om / l))))), -0.5);
	} else if (n <= 2.6e-171) {
		tmp = Math.sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	} else {
		tmp = Math.pow((1.0 / (t_1 * (t + ((l * -2.0) / (Om / l))))), -0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * (2.0 * U)
	tmp = 0
	if n <= -4e-133:
		tmp = math.pow((1.0 / (t_1 * (t + (((U_42_ * (n * l)) / Om) / (Om / l))))), -0.5)
	elif n <= 2.6e-171:
		tmp = math.sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))))
	else:
		tmp = math.pow((1.0 / (t_1 * (t + ((l * -2.0) / (Om / l))))), -0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(2.0 * U))
	tmp = 0.0
	if (n <= -4e-133)
		tmp = Float64(1.0 / Float64(t_1 * Float64(t + Float64(Float64(Float64(U_42_ * Float64(n * l)) / Om) / Float64(Om / l))))) ^ -0.5;
	elseif (n <= 2.6e-171)
		tmp = sqrt(Float64(Float64(Float64(l * Float64(U * Float64(n * l))) / Float64(Om / -4.0)) + Float64(U * Float64(n * Float64(2.0 * t)))));
	else
		tmp = Float64(1.0 / Float64(t_1 * Float64(t + Float64(Float64(l * -2.0) / Float64(Om / l))))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * (2.0 * U);
	tmp = 0.0;
	if (n <= -4e-133)
		tmp = (1.0 / (t_1 * (t + (((U_42_ * (n * l)) / Om) / (Om / l))))) ^ -0.5;
	elseif (n <= 2.6e-171)
		tmp = sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	else
		tmp = (1.0 / (t_1 * (t + ((l * -2.0) / (Om / l))))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4e-133], N[Power[N[(1.0 / N[(t$95$1 * N[(t + N[(N[(N[(U$42$ * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], If[LessEqual[n, 2.6e-171], N[Sqrt[N[(N[(N[(l * N[(U * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / -4.0), $MachinePrecision]), $MachinePrecision] + N[(U * N[(n * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(1.0 / N[(t$95$1 * N[(t + N[(N[(l * -2.0), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.0000000000000003e-133

   1. Initial program 49.1%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 51.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 51.7%

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

   6. Applied egg-rr 58.1%

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

   7. Taylor expanded in U* around inf

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}, \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      3. *-lowering-*.f64 57.3%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   9. Simplified 57.3%

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

   if -4.0000000000000003e-133 < n < 2.60000000000000005e-171

   1. Initial program 45.2%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}{Om}\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot \frac{-4}{Om}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 50.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \mathsf{/.f64}\left(-4, Om\right)\right)\right)\right) \]

   5. Simplified 50.3%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om} + \left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{1}{\frac{Om}{-4}}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{\frac{Om}{-4}}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot U\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\ell \cdot n\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(n \cdot \ell\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(n \cdot \ell\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\ell \cdot n\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(2 \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(\left(U \cdot \left(t \cdot n\right)\right) \cdot 2\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(U \cdot \left(\left(t \cdot n\right) \cdot 2\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(\left(t \cdot n\right) \cdot 2\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(\left(n \cdot t\right) \cdot 2\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(n \cdot \left(t \cdot 2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(t \cdot 2\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 63.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(t, 2\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 63.7%

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

   if 2.60000000000000005e-171 < n

   1. Initial program 51.4%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 57.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 57.8%

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

   6. Applied egg-rr 64.1%

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

   7. Taylor expanded in n around 0

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot \ell\right)}, \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 58.2%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   9. Simplified 58.2%

      \[\leadsto {\left(\frac{1}{\left(t + \frac{\color{blue}{-2 \cdot \ell}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{-133}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\frac{U* \cdot \left(n \cdot \ell\right)}{Om}}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\frac{\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)}{\frac{Om}{-4}} + U \cdot \left(n \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(2 \cdot U\right)\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{-130}:\\ \;\;\;\;{\left(\frac{1}{t\_1 \cdot \left(t + \frac{\ell}{Om} \cdot \frac{U* \cdot \left(n \cdot \ell\right)}{Om}\right)}\right)}^{-0.5}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\frac{\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)}{\frac{Om}{-4}} + U \cdot \left(n \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{t\_1 \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* 2.0 U))))
   (if (<= n -6.2e-130)
     (pow (/ 1.0 (* t_1 (+ t (* (/ l Om) (/ (* U* (* n l)) Om))))) -0.5)
     (if (<= n 1.8e-171)
       (sqrt (+ (/ (* l (* U (* n l))) (/ Om -4.0)) (* U (* n (* 2.0 t)))))
       (pow (/ 1.0 (* t_1 (+ t (/ (* l -2.0) (/ Om l))))) -0.5)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (2.0 * U);
	double tmp;
	if (n <= -6.2e-130) {
		tmp = pow((1.0 / (t_1 * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))), -0.5);
	} else if (n <= 1.8e-171) {
		tmp = sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	} else {
		tmp = pow((1.0 / (t_1 * (t + ((l * -2.0) / (Om / l))))), -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) :: t_1
    real(8) :: tmp
    t_1 = n * (2.0d0 * u)
    if (n <= (-6.2d-130)) then
        tmp = (1.0d0 / (t_1 * (t + ((l / om) * ((u_42 * (n * l)) / om))))) ** (-0.5d0)
    else if (n <= 1.8d-171) then
        tmp = sqrt((((l * (u * (n * l))) / (om / (-4.0d0))) + (u * (n * (2.0d0 * t)))))
    else
        tmp = (1.0d0 / (t_1 * (t + ((l * (-2.0d0)) / (om / l))))) ** (-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 t_1 = n * (2.0 * U);
	double tmp;
	if (n <= -6.2e-130) {
		tmp = Math.pow((1.0 / (t_1 * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))), -0.5);
	} else if (n <= 1.8e-171) {
		tmp = Math.sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	} else {
		tmp = Math.pow((1.0 / (t_1 * (t + ((l * -2.0) / (Om / l))))), -0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * (2.0 * U)
	tmp = 0
	if n <= -6.2e-130:
		tmp = math.pow((1.0 / (t_1 * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))), -0.5)
	elif n <= 1.8e-171:
		tmp = math.sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))))
	else:
		tmp = math.pow((1.0 / (t_1 * (t + ((l * -2.0) / (Om / l))))), -0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(2.0 * U))
	tmp = 0.0
	if (n <= -6.2e-130)
		tmp = Float64(1.0 / Float64(t_1 * Float64(t + Float64(Float64(l / Om) * Float64(Float64(U_42_ * Float64(n * l)) / Om))))) ^ -0.5;
	elseif (n <= 1.8e-171)
		tmp = sqrt(Float64(Float64(Float64(l * Float64(U * Float64(n * l))) / Float64(Om / -4.0)) + Float64(U * Float64(n * Float64(2.0 * t)))));
	else
		tmp = Float64(1.0 / Float64(t_1 * Float64(t + Float64(Float64(l * -2.0) / Float64(Om / l))))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * (2.0 * U);
	tmp = 0.0;
	if (n <= -6.2e-130)
		tmp = (1.0 / (t_1 * (t + ((l / Om) * ((U_42_ * (n * l)) / Om))))) ^ -0.5;
	elseif (n <= 1.8e-171)
		tmp = sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	else
		tmp = (1.0 / (t_1 * (t + ((l * -2.0) / (Om / l))))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.2e-130], N[Power[N[(1.0 / N[(t$95$1 * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(N[(U$42$ * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], If[LessEqual[n, 1.8e-171], N[Sqrt[N[(N[(N[(l * N[(U * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / -4.0), $MachinePrecision]), $MachinePrecision] + N[(U * N[(n * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(1.0 / N[(t$95$1 * N[(t + N[(N[(l * -2.0), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.20000000000000021e-130

   1. Initial program 49.1%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 51.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 51.7%

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

   6. Applied egg-rr 58.1%

      \[\leadsto \color{blue}{{\left(\frac{1}{\left(t + \frac{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{1}{\frac{\frac{Om}{\ell}}{-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}}}\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{1}{\frac{Om}{\ell}} \cdot \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \left(\frac{\ell}{Om} \cdot \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{\ell}{Om}\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(-2 \cdot \ell - \frac{n \cdot \left(U - U*\right)}{Om} \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \left(\ell \cdot \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \left(-2 - \frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \left(\frac{n \cdot \left(U - U*\right)}{Om}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\left(n \cdot \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(U - U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

      12. --lowering--.f64 55.7%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{\_.f64}\left(U, U*\right)\right), Om\right)\right)\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   8. Applied egg-rr 55.7%

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

   9. Taylor expanded in U* around inf

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \color{blue}{\left(\frac{U* \cdot \left(\ell \cdot n\right)}{Om}\right)}\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(\left(U* \cdot \left(\ell \cdot n\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \left(\ell \cdot n\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

       3. *-lowering-*.f64 57.3%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(U*, \mathsf{*.f64}\left(\ell, n\right)\right), Om\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   11. Simplified 57.3%

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

   if -6.20000000000000021e-130 < n < 1.80000000000000002e-171

   1. Initial program 45.2%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}{Om}\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot \frac{-4}{Om}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 50.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \mathsf{/.f64}\left(-4, Om\right)\right)\right)\right) \]

   5. Simplified 50.3%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om} + \left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{1}{\frac{Om}{-4}}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{\frac{Om}{-4}}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot U\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\ell \cdot n\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(n \cdot \ell\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(n \cdot \ell\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\ell \cdot n\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(2 \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(\left(U \cdot \left(t \cdot n\right)\right) \cdot 2\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(U \cdot \left(\left(t \cdot n\right) \cdot 2\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(\left(t \cdot n\right) \cdot 2\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(\left(n \cdot t\right) \cdot 2\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(n \cdot \left(t \cdot 2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(t \cdot 2\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 63.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(t, 2\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 63.7%

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

   if 1.80000000000000002e-171 < n

   1. Initial program 51.4%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 57.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 57.8%

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

   6. Applied egg-rr 64.1%

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

   7. Taylor expanded in n around 0

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot \ell\right)}, \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 58.2%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   9. Simplified 58.2%

      \[\leadsto {\left(\frac{1}{\left(t + \frac{\color{blue}{-2 \cdot \ell}}{\frac{Om}{\ell}}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\right)}^{-0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-130}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell}{Om} \cdot \frac{U* \cdot \left(n \cdot \ell\right)}{Om}\right)}\right)}^{-0.5}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\frac{\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)}{\frac{Om}{-4}} + U \cdot \left(n \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \mathbf{if}\;n \leq -5.5 \cdot 10^{-132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.72 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\frac{\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)}{\frac{Om}{-4}} + U \cdot \left(n \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (pow (/ 1.0 (* (* n (* 2.0 U)) (+ t (/ (* l -2.0) (/ Om l))))) -0.5)))
   (if (<= n -5.5e-132)
     t_1
     (if (<= n 1.72e-171)
       (sqrt (+ (/ (* l (* U (* n l))) (/ Om -4.0)) (* U (* n (* 2.0 t)))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((1.0 / ((n * (2.0 * U)) * (t + ((l * -2.0) / (Om / l))))), -0.5);
	double tmp;
	if (n <= -5.5e-132) {
		tmp = t_1;
	} else if (n <= 1.72e-171) {
		tmp = sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	} else {
		tmp = 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 = (1.0d0 / ((n * (2.0d0 * u)) * (t + ((l * (-2.0d0)) / (om / l))))) ** (-0.5d0)
    if (n <= (-5.5d-132)) then
        tmp = t_1
    else if (n <= 1.72d-171) then
        tmp = sqrt((((l * (u * (n * l))) / (om / (-4.0d0))) + (u * (n * (2.0d0 * t)))))
    else
        tmp = 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 = Math.pow((1.0 / ((n * (2.0 * U)) * (t + ((l * -2.0) / (Om / l))))), -0.5);
	double tmp;
	if (n <= -5.5e-132) {
		tmp = t_1;
	} else if (n <= 1.72e-171) {
		tmp = Math.sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.pow((1.0 / ((n * (2.0 * U)) * (t + ((l * -2.0) / (Om / l))))), -0.5)
	tmp = 0
	if n <= -5.5e-132:
		tmp = t_1
	elif n <= 1.72e-171:
		tmp = math.sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(1.0 / Float64(Float64(n * Float64(2.0 * U)) * Float64(t + Float64(Float64(l * -2.0) / Float64(Om / l))))) ^ -0.5
	tmp = 0.0
	if (n <= -5.5e-132)
		tmp = t_1;
	elseif (n <= 1.72e-171)
		tmp = sqrt(Float64(Float64(Float64(l * Float64(U * Float64(n * l))) / Float64(Om / -4.0)) + Float64(U * Float64(n * Float64(2.0 * t)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (1.0 / ((n * (2.0 * U)) * (t + ((l * -2.0) / (Om / l))))) ^ -0.5;
	tmp = 0.0;
	if (n <= -5.5e-132)
		tmp = t_1;
	elseif (n <= 1.72e-171)
		tmp = sqrt((((l * (U * (n * l))) / (Om / -4.0)) + (U * (n * (2.0 * t)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(1.0 / N[(N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l * -2.0), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[n, -5.5e-132], t$95$1, If[LessEqual[n, 1.72e-171], N[Sqrt[N[(N[(N[(l * N[(U * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / -4.0), $MachinePrecision]), $MachinePrecision] + N[(U * N[(n * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -5.4999999999999999e-132 or 1.7199999999999999e-171 < n

   1. Initial program 50.4%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 55.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 55.2%

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

   6. Applied egg-rr 61.6%

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

   7. Taylor expanded in n around 0

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot \ell\right)}, \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 56.6%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   9. Simplified 56.6%

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

   if -5.4999999999999999e-132 < n < 1.7199999999999999e-171

   1. Initial program 45.2%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}{Om}\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot \frac{-4}{Om}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 50.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \mathsf{/.f64}\left(-4, Om\right)\right)\right)\right) \]

   5. Simplified 50.3%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om} + \left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{1}{\frac{Om}{-4}}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{\frac{Om}{-4}}\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot U\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\left(\ell \cdot n\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\ell \cdot n\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(n \cdot \ell\right) \cdot U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(n \cdot \ell\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\ell \cdot n\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \left(\frac{Om}{-4}\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(2 \cdot \left(U \cdot \left(t \cdot n\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(\left(U \cdot \left(t \cdot n\right)\right) \cdot 2\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \left(U \cdot \left(\left(t \cdot n\right) \cdot 2\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(\left(t \cdot n\right) \cdot 2\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(\left(n \cdot t\right) \cdot 2\right)\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \left(n \cdot \left(t \cdot 2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \left(t \cdot 2\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 63.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), U\right)\right), \mathsf{/.f64}\left(Om, -4\right)\right), \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(t, 2\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 63.7%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.5 \cdot 10^{-132}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \mathbf{elif}\;n \leq 1.72 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\frac{\ell \cdot \left(U \cdot \left(n \cdot \ell\right)\right)}{\frac{Om}{-4}} + U \cdot \left(n \cdot \left(2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \mathbf{if}\;n \leq -9.6 \cdot 10^{-129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot \frac{-4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (pow (/ 1.0 (* (* n (* 2.0 U)) (+ t (/ (* l -2.0) (/ Om l))))) -0.5)))
   (if (<= n -9.6e-129)
     t_1
     (if (<= n 5.6e-173)
       (sqrt (+ (* (* 2.0 U) (* n t)) (* (* U (* l (* n l))) (/ -4.0 Om))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((1.0 / ((n * (2.0 * U)) * (t + ((l * -2.0) / (Om / l))))), -0.5);
	double tmp;
	if (n <= -9.6e-129) {
		tmp = t_1;
	} else if (n <= 5.6e-173) {
		tmp = sqrt((((2.0 * U) * (n * t)) + ((U * (l * (n * l))) * (-4.0 / Om))));
	} else {
		tmp = 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 = (1.0d0 / ((n * (2.0d0 * u)) * (t + ((l * (-2.0d0)) / (om / l))))) ** (-0.5d0)
    if (n <= (-9.6d-129)) then
        tmp = t_1
    else if (n <= 5.6d-173) then
        tmp = sqrt((((2.0d0 * u) * (n * t)) + ((u * (l * (n * l))) * ((-4.0d0) / om))))
    else
        tmp = 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 = Math.pow((1.0 / ((n * (2.0 * U)) * (t + ((l * -2.0) / (Om / l))))), -0.5);
	double tmp;
	if (n <= -9.6e-129) {
		tmp = t_1;
	} else if (n <= 5.6e-173) {
		tmp = Math.sqrt((((2.0 * U) * (n * t)) + ((U * (l * (n * l))) * (-4.0 / Om))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.pow((1.0 / ((n * (2.0 * U)) * (t + ((l * -2.0) / (Om / l))))), -0.5)
	tmp = 0
	if n <= -9.6e-129:
		tmp = t_1
	elif n <= 5.6e-173:
		tmp = math.sqrt((((2.0 * U) * (n * t)) + ((U * (l * (n * l))) * (-4.0 / Om))))
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(1.0 / Float64(Float64(n * Float64(2.0 * U)) * Float64(t + Float64(Float64(l * -2.0) / Float64(Om / l))))) ^ -0.5
	tmp = 0.0
	if (n <= -9.6e-129)
		tmp = t_1;
	elseif (n <= 5.6e-173)
		tmp = sqrt(Float64(Float64(Float64(2.0 * U) * Float64(n * t)) + Float64(Float64(U * Float64(l * Float64(n * l))) * Float64(-4.0 / Om))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (1.0 / ((n * (2.0 * U)) * (t + ((l * -2.0) / (Om / l))))) ^ -0.5;
	tmp = 0.0;
	if (n <= -9.6e-129)
		tmp = t_1;
	elseif (n <= 5.6e-173)
		tmp = sqrt((((2.0 * U) * (n * t)) + ((U * (l * (n * l))) * (-4.0 / Om))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(1.0 / N[(N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l * -2.0), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[n, -9.6e-129], t$95$1, If[LessEqual[n, 5.6e-173], N[Sqrt[N[(N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(U * N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -9.59999999999999954e-129 or 5.5999999999999998e-173 < n

   1. Initial program 50.4%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 55.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 55.2%

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

   6. Applied egg-rr 61.6%

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

   7. Taylor expanded in n around 0

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot \ell\right)}, \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 56.6%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

   9. Simplified 56.6%

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

   if -9.59999999999999954e-129 < n < 5.5999999999999998e-173

   1. Initial program 45.2%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}{Om}\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot \frac{-4}{Om}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 50.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \mathsf{/.f64}\left(-4, Om\right)\right)\right)\right) \]

   5. Simplified 50.3%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot \left(\ell \cdot \ell\right)\right)\right), \mathsf{/.f64}\left(-4, Om\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(\left(n \cdot \ell\right) \cdot \ell\right)\right), \mathsf{/.f64}\left(-4, Om\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(n \cdot \ell\right), \ell\right)\right), \mathsf{/.f64}\left(-4, Om\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot n\right), \ell\right)\right), \mathsf{/.f64}\left(-4, Om\right)\right)\right)\right) \]

      5. *-lowering-*.f64 61.5%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, n\right), \ell\right)\right), \mathsf{/.f64}\left(-4, Om\right)\right)\right)\right) \]

   7. Applied egg-rr 61.5%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.6 \cdot 10^{-129}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right) + \left(U \cdot \left(\ell \cdot \left(n \cdot \ell\right)\right)\right) \cdot \frac{-4}{Om}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{if}\;Om \leq -5.7 \cdot 10^{-279}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 1.26 \cdot 10^{-169}:\\ \;\;\;\;\frac{\ell \cdot \sqrt{\left(n \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \left(U \cdot -2\right)\right)}}{Om}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (* (* 2.0 n) U) (+ t (* (/ (* l l) Om) -2.0))))))
   (if (<= Om -5.7e-279)
     t_1
     (if (<= Om 1.26e-169)
       (/ (* l (sqrt (* (* n (- U U*)) (* n (* U -2.0))))) Om)
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * (t + (((l * l) / Om) * -2.0))));
	double tmp;
	if (Om <= -5.7e-279) {
		tmp = t_1;
	} else if (Om <= 1.26e-169) {
		tmp = (l * sqrt(((n * (U - U_42_)) * (n * (U * -2.0))))) / Om;
	} else {
		tmp = 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 = sqrt((((2.0d0 * n) * u) * (t + (((l * l) / om) * (-2.0d0)))))
    if (om <= (-5.7d-279)) then
        tmp = t_1
    else if (om <= 1.26d-169) then
        tmp = (l * sqrt(((n * (u - u_42)) * (n * (u * (-2.0d0)))))) / om
    else
        tmp = 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 = Math.sqrt((((2.0 * n) * U) * (t + (((l * l) / Om) * -2.0))));
	double tmp;
	if (Om <= -5.7e-279) {
		tmp = t_1;
	} else if (Om <= 1.26e-169) {
		tmp = (l * Math.sqrt(((n * (U - U_42_)) * (n * (U * -2.0))))) / Om;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((2.0 * n) * U) * (t + (((l * l) / Om) * -2.0))))
	tmp = 0
	if Om <= -5.7e-279:
		tmp = t_1
	elif Om <= 1.26e-169:
		tmp = (l * math.sqrt(((n * (U - U_42_)) * (n * (U * -2.0))))) / Om
	else:
		tmp = t_1
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0))))
	tmp = 0.0
	if (Om <= -5.7e-279)
		tmp = t_1;
	elseif (Om <= 1.26e-169)
		tmp = Float64(Float64(l * sqrt(Float64(Float64(n * Float64(U - U_42_)) * Float64(n * Float64(U * -2.0))))) / Om);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * (t + (((l * l) / Om) * -2.0))));
	tmp = 0.0;
	if (Om <= -5.7e-279)
		tmp = t_1;
	elseif (Om <= 1.26e-169)
		tmp = (l * sqrt(((n * (U - U_42_)) * (n * (U * -2.0))))) / Om;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -5.7e-279], t$95$1, If[LessEqual[Om, 1.26e-169], N[(N[(l * N[Sqrt[N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(n * N[(U * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if Om < -5.7000000000000001e-279 or 1.26e-169 < Om

   1. Initial program 50.2%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \left(\frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{Om}\right), -2\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right), -2\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right), -2\right)\right)\right)\right) \]

      6. *-lowering-*.f64 46.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right), -2\right)\right)\right)\right) \]

   5. Simplified 46.2%

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

   if -5.7000000000000001e-279 < Om < 1.26e-169

   1. Initial program 39.6%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}{{Om}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot U\right), \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(U - U*\right) \cdot {n}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(U - U*\right), \left({n}^{2}\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left({n}^{2}\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(n, n\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(n, n\right)\right)\right)\right), \left(Om \cdot Om\right)\right)\right) \]

      15. *-lowering-*.f64 39.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(n, n\right)\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right) \]

   5. Simplified 39.3%

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

      1. sqrt-div N/A

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

      2. sqrt-prod N/A

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

      3. rem-square-sqrt N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right)\right)}\right), \color{blue}{Om}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right)\right)\right)\right), Om\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right)\right) \cdot \left(-2 \cdot U\right)\right)\right), Om\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot \left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right), \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(U - U*\right), \left(n \cdot n\right)\right), \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot n\right)\right), \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(n, n\right)\right), \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(n, n\right)\right), \left(U \cdot -2\right)\right)\right)\right), Om\right) \]

      15. *-lowering-*.f64 48.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(n, n\right)\right), \mathsf{*.f64}\left(U, -2\right)\right)\right)\right), Om\right) \]

   7. Applied egg-rr 48.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U \cdot -2\right)\right) \cdot \left(\ell \cdot \ell\right)}\right), Om\right) \]

      2. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U \cdot -2\right)} \cdot \sqrt{\ell \cdot \ell}\right), Om\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U \cdot -2\right)\right)}^{\frac{1}{2}} \cdot \sqrt{\ell \cdot \ell}\right), Om\right) \]

      4. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U \cdot -2\right)\right)}^{\frac{1}{2}} \cdot \left(\sqrt{\ell} \cdot \sqrt{\ell}\right)\right), Om\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U \cdot -2\right)\right)}^{\frac{1}{2}} \cdot \ell\right), Om\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U \cdot -2\right)\right)}^{\frac{1}{2}}\right), \ell\right), Om\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U \cdot -2\right)}\right), \ell\right), Om\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U \cdot -2\right)\right)\right), \ell\right), Om\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(U - U*\right) \cdot n\right) \cdot n\right) \cdot \left(U \cdot -2\right)\right)\right), \ell\right), Om\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(n \cdot \left(U - U*\right)\right) \cdot n\right) \cdot \left(U \cdot -2\right)\right)\right), \ell\right), Om\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(n \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \left(U \cdot -2\right)\right)\right)\right), \ell\right), Om\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(n \cdot \left(U - U*\right)\right), \left(n \cdot \left(U \cdot -2\right)\right)\right)\right), \ell\right), Om\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(U - U*\right) \cdot n\right), \left(n \cdot \left(U \cdot -2\right)\right)\right)\right), \ell\right), Om\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(U - U*\right), n\right), \left(n \cdot \left(U \cdot -2\right)\right)\right)\right), \ell\right), Om\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), n\right), \left(n \cdot \left(U \cdot -2\right)\right)\right)\right), \ell\right), Om\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), n\right), \mathsf{*.f64}\left(n, \left(U \cdot -2\right)\right)\right)\right), \ell\right), Om\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), n\right), \mathsf{*.f64}\left(n, \left(-2 \cdot U\right)\right)\right)\right), \ell\right), Om\right) \]

      18. *-lowering-*.f64 29.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), n\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(-2, U\right)\right)\right)\right), \ell\right), Om\right) \]

   9. Applied egg-rr 29.9%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -5.7 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{elif}\;Om \leq 1.26 \cdot 10^{-169}:\\ \;\;\;\;\frac{\ell \cdot \sqrt{\left(n \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \left(U \cdot -2\right)\right)}}{Om}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\sqrt{\left(n \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \left(U \cdot -2\right)\right)}}{Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.8e+162)
   (sqrt (* (* (* 2.0 n) U) (+ t (* (/ (* l l) Om) -2.0))))
   (* l (/ (sqrt (* (* n (- U U*)) (* n (* U -2.0)))) Om))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.8e+162) {
		tmp = sqrt((((2.0 * n) * U) * (t + (((l * l) / Om) * -2.0))));
	} else {
		tmp = l * (sqrt(((n * (U - U_42_)) * (n * (U * -2.0)))) / Om);
	}
	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 <= 2.8d+162) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + (((l * l) / om) * (-2.0d0)))))
    else
        tmp = l * (sqrt(((n * (u - u_42)) * (n * (u * (-2.0d0))))) / om)
    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 <= 2.8e+162) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + (((l * l) / Om) * -2.0))));
	} else {
		tmp = l * (Math.sqrt(((n * (U - U_42_)) * (n * (U * -2.0)))) / Om);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.8e+162:
		tmp = math.sqrt((((2.0 * n) * U) * (t + (((l * l) / Om) * -2.0))))
	else:
		tmp = l * (math.sqrt(((n * (U - U_42_)) * (n * (U * -2.0)))) / Om)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.8e+162)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0))));
	else
		tmp = Float64(l * Float64(sqrt(Float64(Float64(n * Float64(U - U_42_)) * Float64(n * Float64(U * -2.0)))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.8e+162)
		tmp = sqrt((((2.0 * n) * U) * (t + (((l * l) / Om) * -2.0))));
	else
		tmp = l * (sqrt(((n * (U - U_42_)) * (n * (U * -2.0)))) / Om);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.8e+162], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l * N[(N[Sqrt[N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(n * N[(U * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.79999999999999991e162

   1. Initial program 53.9%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \left(\frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{Om}\right), -2\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right), -2\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right), -2\right)\right)\right)\right) \]

      6. *-lowering-*.f64 48.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right), -2\right)\right)\right)\right) \]

   5. Simplified 48.4%

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

   if 2.79999999999999991e162 < l

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{{Om}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)}{{Om}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(-2 \cdot U\right), \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({n}^{2} \cdot \left(U - U*\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(U - U*\right) \cdot {n}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(U - U*\right), \left({n}^{2}\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left({n}^{2}\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot n\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(n, n\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(n, n\right)\right)\right)\right), \left(Om \cdot Om\right)\right)\right) \]

      15. *-lowering-*.f64 40.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, U\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(n, n\right)\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right) \]

   5. Simplified 40.1%

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

      1. sqrt-div N/A

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

      2. sqrt-prod N/A

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

      3. rem-square-sqrt N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right)\right)}\right), \color{blue}{Om}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right)\right)\right)\right), Om\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right)\right) \cdot \left(-2 \cdot U\right)\right)\right), Om\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot \left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right), \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(U - U*\right), \left(n \cdot n\right)\right), \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \left(n \cdot n\right)\right), \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(n, n\right)\right), \left(-2 \cdot U\right)\right)\right)\right), Om\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(n, n\right)\right), \left(U \cdot -2\right)\right)\right)\right), Om\right) \]

      15. *-lowering-*.f64 28.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(U, U*\right), \mathsf{*.f64}\left(n, n\right)\right), \mathsf{*.f64}\left(U, -2\right)\right)\right)\right), Om\right) \]

   7. Applied egg-rr 28.0%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-/l* N/A

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

      4. pow1/2 N/A

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

      5. sqrt-prod N/A

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

      6. rem-square-sqrt N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{{\left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U \cdot -2\right)\right)}^{\frac{1}{2}}}{Om}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left({\left(\left(\left(U - U*\right) \cdot \left(n \cdot n\right)\right) \cdot \left(U \cdot -2\right)\right)}^{\frac{1}{2}}\right), \color{blue}{Om}\right)\right) \]

   9. Applied egg-rr 35.9%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\sqrt{\left(n \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \left(U \cdot -2\right)\right)}}{Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.6e-91)
   (sqrt (* (* (* 2.0 n) U) t))
   (sqrt (* (* 2.0 U) (* n (+ t (* (/ (* l l) Om) -2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.6e-91) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt(((2.0 * U) * (n * (t + (((l * l) / Om) * -2.0)))));
	}
	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 <= 2.6d-91) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt(((2.0d0 * u) * (n * (t + (((l * l) / om) * (-2.0d0))))))
    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 <= 2.6e-91) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * (t + (((l * l) / Om) * -2.0)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.6e-91:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt(((2.0 * U) * (n * (t + (((l * l) / Om) * -2.0)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.6e-91)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.6e-91)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt(((2.0 * U) * (n * (t + (((l * l) / Om) * -2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.6e-91], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.60000000000000014e-91

   1. Initial program 51.4%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 39.6%

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

   if 2.60000000000000014e-91 < l

   1. Initial program 43.3%

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

   3. Taylor expanded in n around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \left(\frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{Om}\right), -2\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right), -2\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right), -2\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 38.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right), -2\right)\right)\right)\right)\right) \]

   5. Simplified 38.9%

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

Alternative 16: 51.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (pow (/ 1.0 (* (* n (* 2.0 U)) (+ t (/ (* l -2.0) (/ Om l))))) -0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow((1.0 / ((n * (2.0 * U)) * (t + ((l * -2.0) / (Om / l))))), -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 = (1.0d0 / ((n * (2.0d0 * u)) * (t + ((l * (-2.0d0)) / (om / l))))) ** (-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((1.0 / ((n * (2.0 * U)) * (t + ((l * -2.0) / (Om / l))))), -0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow((1.0 / ((n * (2.0 * U)) * (t + ((l * -2.0) / (Om / l))))), -0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(1.0 / Float64(Float64(n * Float64(2.0 * U)) * Float64(t + Float64(Float64(l * -2.0) / Float64(Om / l))))) ^ -0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = (1.0 / ((n * (2.0 * U)) * (t + ((l * -2.0) / (Om / l))))) ^ -0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(1.0 / N[(N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(l * -2.0), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 49.0%

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

   1. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f64 52.6%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

4. Applied egg-rr 52.6%

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

6. Applied egg-rr 58.5%

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

7. Taylor expanded in n around 0

   \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot \ell\right)}, \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 54.2%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \ell\right), \mathsf{/.f64}\left(Om, \ell\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right)\right), \frac{-1}{2}\right) \]

9. Simplified 54.2%

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

10. Final simplification 54.2%

    \[\leadsto {\left(\frac{1}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(t + \frac{\ell \cdot -2}{\frac{Om}{\ell}}\right)}\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 17: 38.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{+67}:\\ \;\;\;\;{\left(t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.3e+67)
   (pow (* t (* n (* 2.0 U))) 0.5)
   (sqrt (/ (* -4.0 (* U (* n (* l l)))) Om))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.3e+67) {
		tmp = pow((t * (n * (2.0 * U))), 0.5);
	} else {
		tmp = sqrt(((-4.0 * (U * (n * (l * l)))) / Om));
	}
	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 <= 2.3d+67) then
        tmp = (t * (n * (2.0d0 * u))) ** 0.5d0
    else
        tmp = sqrt((((-4.0d0) * (u * (n * (l * l)))) / om))
    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 <= 2.3e+67) {
		tmp = Math.pow((t * (n * (2.0 * U))), 0.5);
	} else {
		tmp = Math.sqrt(((-4.0 * (U * (n * (l * l)))) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.3e+67:
		tmp = math.pow((t * (n * (2.0 * U))), 0.5)
	else:
		tmp = math.sqrt(((-4.0 * (U * (n * (l * l)))) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.3e+67)
		tmp = Float64(t * Float64(n * Float64(2.0 * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(U * Float64(n * Float64(l * l)))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.3e+67)
		tmp = (t * (n * (2.0 * U))) ^ 0.5;
	else
		tmp = sqrt(((-4.0 * (U * (n * (l * l)))) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.3e+67], N[Power[N[(t * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(-4.0 * N[(U * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.2999999999999999e67

   1. Initial program 52.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 38.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   5. Simplified 38.7%

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

      1. pow1/2 N/A

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

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

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right), \frac{1}{2}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot t\right), \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right), \frac{1}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right), \frac{1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(n \cdot U\right)\right)\right), \frac{1}{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(\left(2 \cdot n\right) \cdot U\right)\right), \frac{1}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(\left(n \cdot 2\right) \cdot U\right)\right), \frac{1}{2}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(n \cdot \left(2 \cdot U\right)\right)\right), \frac{1}{2}\right) \]

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

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(n, \left(2 \cdot U\right)\right)\right), \frac{1}{2}\right) \]

      12. *-lowering-*.f64 41.7%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 41.7%

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

   if 2.2999999999999999e67 < l

   1. Initial program 33.1%

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

   3. Taylor expanded in Om around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\frac{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}{Om}\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot \frac{-4}{Om}\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\left(U \cdot \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \left(\frac{-4}{Om}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 24.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right), \mathsf{/.f64}\left(-4, Om\right)\right)\right)\right) \]

   5. Simplified 24.1%

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

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \left({\ell}^{2} \cdot n\right)\right)\right), Om\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left({\ell}^{2}\right), n\right)\right)\right), Om\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), n\right)\right)\right), Om\right)\right) \]

      7. *-lowering-*.f64 20.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), n\right)\right)\right), Om\right)\right) \]

   8. Simplified 20.1%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{+67}:\\ \;\;\;\;{\left(t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-4 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 47.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{0.5}{\left(n \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (pow (/ 0.5 (* (* n U) (+ t (* (/ (* l l) Om) -2.0)))) -0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow((0.5 / ((n * U) * (t + (((l * l) / Om) * -2.0)))), -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 = (0.5d0 / ((n * u) * (t + (((l * l) / om) * (-2.0d0))))) ** (-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((0.5 / ((n * U) * (t + (((l * l) / Om) * -2.0)))), -0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow((0.5 / ((n * U) * (t + (((l * l) / Om) * -2.0)))), -0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(0.5 / Float64(Float64(n * U) * Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)))) ^ -0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = (0.5 / ((n * U) * (t + (((l * l) / Om) * -2.0)))) ^ -0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(0.5 / N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 49.0%

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

   1. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f64 52.6%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

4. Applied egg-rr 52.6%

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

6. Applied egg-rr 58.5%

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

7. Taylor expanded in n around 0

   \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left(\frac{\frac{1}{2}}{U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right)}, \frac{-1}{2}\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right), \frac{-1}{2}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(\left(U \cdot n\right) \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right), \frac{-1}{2}\right) \]

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

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(U \cdot n\right), \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right), \frac{-1}{2}\right) \]

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

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right), \frac{-1}{2}\right) \]

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

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(-2, \left(\frac{{\ell}^{2}}{Om}\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

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

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   9. *-lowering-*.f64 51.8%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(U, n\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

9. Simplified 51.8%

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

10. Final simplification 51.8%

    \[\leadsto {\left(\frac{0.5}{\left(n \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 19: 43.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 + (((l * l) / om) * (-2.0d0)))))
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 + (((l * l) / Om) * -2.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.0%

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

3. Taylor expanded in Om around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \left(-2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \left(\frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{Om}\right), -2\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right), -2\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right), -2\right)\right)\right)\right) \]

   6. *-lowering-*.f64 44.3%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right), -2\right)\right)\right)\right) \]

5. Simplified 44.3%

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

Alternative 20: 36.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.9 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{0.5}{U \cdot \left(n \cdot t\right)}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.9e-28)
   (sqrt (* (* (* 2.0 n) U) t))
   (pow (/ 0.5 (* U (* n t))) -0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.9e-28) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = pow((0.5 / (U * (n * t))), -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 (l <= 3.9d-28) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = (0.5d0 / (u * (n * t))) ** (-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 (l <= 3.9e-28) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.pow((0.5 / (U * (n * t))), -0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.9e-28:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.pow((0.5 / (U * (n * t))), -0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.9e-28)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = Float64(0.5 / Float64(U * Float64(n * t))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.9e-28)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = (0.5 / (U * (n * t))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.9e-28], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Power[N[(0.5 / N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.89999999999999999e-28

   1. Initial program 52.4%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 39.5%

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

   if 3.89999999999999999e-28 < l

   1. Initial program 38.3%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\ell \cdot \frac{1}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \left(\frac{\ell}{\frac{Om}{\ell}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \left(\frac{Om}{\ell}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 41.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(Om, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 2\right)\right), \mathsf{\_.f64}\left(U, U*\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 41.9%

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

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in t around inf

      \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left(\frac{\frac{1}{2}}{U \cdot \left(n \cdot t\right)}\right)}, \frac{-1}{2}\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(U \cdot \left(n \cdot t\right)\right)\right), \frac{-1}{2}\right) \]

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

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(U, \left(n \cdot t\right)\right)\right), \frac{-1}{2}\right) \]

      3. *-lowering-*.f64 23.1%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(n, t\right)\right)\right), \frac{-1}{2}\right) \]

   9. Simplified 23.1%

      \[\leadsto {\color{blue}{\left(\frac{0.5}{U \cdot \left(n \cdot t\right)}\right)}}^{-0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 35.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2e-190) (sqrt (* (* (* 2.0 n) U) t)) (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 <= 2e-190) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} 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 <= 2d-190) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    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 <= 2e-190) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} 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 <= 2e-190:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	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 <= 2e-190)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	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 <= 2e-190)
		tmp = sqrt((((2.0 * n) * U) * t));
	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, 2e-190], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2e-190

   1. Initial program 49.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 38.0%

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

   if 2e-190 < l

   1. Initial program 47.5%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 28.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   5. Simplified 28.2%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(n \cdot t\right)\right) \cdot 2\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(n \cdot t\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot t\right)\right), 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(t \cdot n\right)\right), 2\right)\right) \]

      6. *-lowering-*.f64 28.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, n\right)\right), 2\right)\right) \]

   7. Applied egg-rr 28.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(U \cdot t\right) \cdot n\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(U \cdot t\right), n\right), 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot U\right), n\right), 2\right)\right) \]

      4. *-lowering-*.f64 31.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, U\right), n\right), 2\right)\right) \]

   9. Applied egg-rr 31.2%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 35.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 8e-190) (sqrt (* (* (* 2.0 n) U) t)) (sqrt (* n (* t (* 2.0 U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 8e-190) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt((n * (t * (2.0 * 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) :: tmp
    if (l <= 8d-190) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt((n * (t * (2.0d0 * 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 tmp;
	if (l <= 8e-190) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt((n * (t * (2.0 * U))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 8e-190:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt((n * (t * (2.0 * U))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 8e-190)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(n * Float64(t * Float64(2.0 * U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 8e-190)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt((n * (t * (2.0 * U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 8e-190], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(n * N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.0000000000000002e-190

   1. Initial program 49.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, n\right), U\right), \color{blue}{t}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 38.0%

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

   if 8.0000000000000002e-190 < l

   1. Initial program 47.5%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 28.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

   5. Simplified 28.2%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(t \cdot n\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot t\right) \cdot n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot U\right) \cdot t\right), n\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), t\right), n\right)\right) \]

      5. *-lowering-*.f64 31.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), t\right), n\right)\right) \]

   7. Applied egg-rr 31.2%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 37.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (pow (* t (* n (* 2.0 U))) 0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow((t * (n * (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 = (t * (n * (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((t * (n * (2.0 * U))), 0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow((t * (n * (2.0 * U))), 0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(t * Float64(n * Float64(2.0 * U))) ^ 0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = (t * (n * (2.0 * U))) ^ 0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(t * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 49.0%

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

3. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

   4. *-lowering-*.f64 34.0%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

5. Simplified 34.0%

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

   1. pow1/2 N/A

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

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

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right), \color{blue}{\frac{1}{2}}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right), \frac{1}{2}\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot t\right), \frac{1}{2}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right), \frac{1}{2}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right), \frac{1}{2}\right) \]

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

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(2 \cdot \left(n \cdot U\right)\right)\right), \frac{1}{2}\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(\left(2 \cdot n\right) \cdot U\right)\right), \frac{1}{2}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(\left(n \cdot 2\right) \cdot U\right)\right), \frac{1}{2}\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \left(n \cdot \left(2 \cdot U\right)\right)\right), \frac{1}{2}\right) \]

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

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(n, \left(2 \cdot U\right)\right)\right), \frac{1}{2}\right) \]

   12. *-lowering-*.f64 36.0%

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(2, U\right)\right)\right), \frac{1}{2}\right) \]

7. Applied egg-rr 36.0%

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

Alternative 24: 35.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{n \cdot \left(t \cdot \left(2 \cdot U\right)\right)} \end{array} \]

FPCore

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

C

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

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((n * (t * (2.0d0 * u))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.0%

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

3. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

   4. *-lowering-*.f64 34.0%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

5. Simplified 34.0%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(t \cdot n\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot U\right) \cdot t\right) \cdot n\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot U\right) \cdot t\right), n\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), t\right), n\right)\right) \]

   5. *-lowering-*.f64 35.2%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), t\right), n\right)\right) \]

7. Applied egg-rr 35.2%

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

8. Final simplification 35.2%

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

Alternative 25: 35.5% 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 49.0%

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

3. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

   4. *-lowering-*.f64 34.0%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

5. Simplified 34.0%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(U \cdot \left(n \cdot t\right)\right) \cdot 2\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(U \cdot \left(n \cdot t\right)\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(n \cdot t\right)\right), 2\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \left(t \cdot n\right)\right), 2\right)\right) \]

   6. *-lowering-*.f64 34.0%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(U, \mathsf{*.f64}\left(t, n\right)\right), 2\right)\right) \]

7. Applied egg-rr 34.0%

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

8. Final simplification 34.0%

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

Alternative 26: 35.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)} \end{array} \]

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(Float64(2.0 * 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[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.0%

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

3. Taylor expanded in t around inf

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot U\right), \left(n \cdot t\right)\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \left(n \cdot t\right)\right)\right) \]

   4. *-lowering-*.f64 34.0%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, U\right), \mathsf{*.f64}\left(n, t\right)\right)\right) \]

5. Simplified 34.0%

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

Reproduce

herbie shell --seed 2024192 
(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*))))))