Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 64.9%

Time: 43.9s

Alternatives: 15

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U* U)))))))
   (if (<= t_1 0.0)
     (*
      (sqrt (* 2.0 n))
      (sqrt (* U (+ t (* (/ l Om) (fma l -2.0 (* (/ l Om) (* n (- U* U)))))))))
     (if (<= t_1 5e+135)
       t_1
       (if (<= t_1 INFINITY)
         (pow
          (*
           2.0
           (+
            (* n (* U t))
            (/ (+ (/ n (/ Om (* l U*))) (* l -2.0)) (/ Om (* n (* U l))))))
          0.5)
         (sqrt
          (* -2.0 (/ (* (* l (* n l)) (* U (- 2.0 (/ (* n U*) Om)))) Om))))))))

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 - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t + ((l / Om) * fma(l, -2.0, ((l / Om) * (n * (U_42_ - U))))))));
	} else if (t_1 <= 5e+135) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = pow((2.0 * ((n * (U * t)) + (((n / (Om / (l * U_42_))) + (l * -2.0)) / (Om / (n * (U * l)))))), 0.5);
	} else {
		tmp = sqrt((-2.0 * (((l * (n * l)) * (U * (2.0 - ((n * U_42_) / Om)))) / Om)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t + Float64(Float64(l / Om) * fma(l, -2.0, Float64(Float64(l / Om) * Float64(n * Float64(U_42_ - U)))))))));
	elseif (t_1 <= 5e+135)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(U * t)) + Float64(Float64(Float64(n / Float64(Om / Float64(l * U_42_))) + Float64(l * -2.0)) / Float64(Om / Float64(n * Float64(U * l)))))) ^ 0.5;
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(Float64(l * Float64(n * l)) * Float64(U * Float64(2.0 - Float64(Float64(n * U_42_) / Om)))) / Om)));
	end
	return 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[(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]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(N[(l / Om), $MachinePrecision] * N[(l * -2.0 + N[(N[(l / Om), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+135], t$95$1, If[LessEqual[t$95$1, Infinity], N[Power[N[(2.0 * N[(N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision] * N[(U * N[(2.0 - N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 13.2%

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

      1. associate-*l* 28.3%

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

      2. sub-neg 28.3%

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

      3. associate--l+ 28.3%

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

      4. *-commutative 28.3%

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

      5. distribute-rgt-neg-in 28.3%

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

      6. associate-*l/ 28.4%

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

      7. associate-*l* 28.4%

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

      8. *-commutative 28.4%

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

      9. *-commutative 28.4%

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

      10. associate-*l* 28.4%

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

      11. unpow2 28.4%

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

      12. associate-*l* 28.4%

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

   3. Simplified 28.4%

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

      1. sqrt-prod 32.8%

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

   5. Applied egg-rr 32.8%

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

      1. *-commutative 32.8%

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

      2. *-commutative 32.8%

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

   7. Simplified 32.8%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 5.00000000000000029e135

   1. Initial program 98.0%

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

   if 5.00000000000000029e135 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 30.3%

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

      1. associate-*l* 34.1%

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

      2. sub-neg 34.1%

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

      3. associate--l+ 34.1%

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

      4. *-commutative 34.1%

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

      5. distribute-rgt-neg-in 34.1%

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

      6. associate-*l/ 37.9%

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

      7. associate-*l* 37.9%

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

      8. *-commutative 37.9%

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

      9. *-commutative 37.9%

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

      10. associate-*l* 32.2%

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

      11. unpow2 32.2%

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

      12. associate-*l* 33.5%

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

   3. Simplified 33.5%

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

   4. Taylor expanded in t around inf 37.9%

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

   5. Taylor expanded in U* around inf 41.0%

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

      1. pow1/2 41.1%

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

      2. distribute-lft-out 41.1%

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

      3. associate-/l* 41.1%

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

      4. associate-/l* 42.4%

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

      5. *-commutative 42.4%

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

      6. *-commutative 42.4%

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

   7. Applied egg-rr 42.4%

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.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. Step-by-step derivation

      1. associate-*l* 0.9%

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

      2. sub-neg 0.9%

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

      3. associate--l+ 0.9%

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

      4. *-commutative 0.9%

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

      5. distribute-rgt-neg-in 0.9%

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

      6. associate-*l/ 10.0%

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

      7. associate-*l* 10.0%

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

      8. *-commutative 10.0%

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

      9. *-commutative 10.0%

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

      10. associate-*l* 9.9%

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

      11. unpow2 9.9%

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

      12. associate-*l* 12.9%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in l around -inf 49.3%

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

      1. associate-/l* 45.5%

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

      2. unpow2 45.5%

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

      3. *-commutative 45.5%

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

      4. mul-1-neg 45.5%

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

      5. associate-/l* 45.1%

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

   6. Simplified 45.1%

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

   7. Taylor expanded in U around 0 49.3%

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

      1. associate-*r* 53.7%

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

      2. unpow2 53.7%

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

      3. *-commutative 53.7%

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

   9. Simplified 53.7%

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

   10. Taylor expanded in n around 0 53.7%

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

       1. unpow2 53.7%

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

       2. associate-*r* 71.3%

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

   12. Simplified 71.3%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 2: 61.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U* U)))))))
   (if (<= t_1 5e+135)
     t_1
     (if (<= t_1 INFINITY)
       (pow
        (*
         2.0
         (+
          (* n (* U t))
          (/ (+ (/ n (/ Om (* l U*))) (* l -2.0)) (/ Om (* n (* U l))))))
        0.5)
       (sqrt
        (* -2.0 (/ (* (* l (* n l)) (* U (- 2.0 (/ (* n U*) Om)))) Om)))))))

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 - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 5e+135) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = pow((2.0 * ((n * (U * t)) + (((n / (Om / (l * U_42_))) + (l * -2.0)) / (Om / (n * (U * l)))))), 0.5);
	} else {
		tmp = sqrt((-2.0 * (((l * (n * l)) * (U * (2.0 - ((n * U_42_) / Om)))) / Om)));
	}
	return tmp;
}

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 - (2.0 * ((l * l) / Om))) + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 5e+135) {
		tmp = t_1;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow((2.0 * ((n * (U * t)) + (((n / (Om / (l * U_42_))) + (l * -2.0)) / (Om / (n * (U * l)))))), 0.5);
	} else {
		tmp = Math.sqrt((-2.0 * (((l * (n * l)) * (U * (2.0 - ((n * U_42_) / Om)))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * math.pow((l / Om), 2.0)) * (U_42_ - U)))))
	tmp = 0
	if t_1 <= 5e+135:
		tmp = t_1
	elif t_1 <= math.inf:
		tmp = math.pow((2.0 * ((n * (U * t)) + (((n / (Om / (l * U_42_))) + (l * -2.0)) / (Om / (n * (U * l)))))), 0.5)
	else:
		tmp = math.sqrt((-2.0 * (((l * (n * l)) * (U * (2.0 - ((n * U_42_) / Om)))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 5e+135)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(2.0 * Float64(Float64(n * Float64(U * t)) + Float64(Float64(Float64(n / Float64(Om / Float64(l * U_42_))) + Float64(l * -2.0)) / Float64(Om / Float64(n * Float64(U * l)))))) ^ 0.5;
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(Float64(l * Float64(n * l)) * Float64(U * Float64(2.0 - Float64(Float64(n * U_42_) / Om)))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * ((l / Om) ^ 2.0)) * (U_42_ - U)))));
	tmp = 0.0;
	if (t_1 <= 5e+135)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = (2.0 * ((n * (U * t)) + (((n / (Om / (l * U_42_))) + (l * -2.0)) / (Om / (n * (U * l)))))) ^ 0.5;
	else
		tmp = sqrt((-2.0 * (((l * (n * l)) * (U * (2.0 - ((n * U_42_) / Om)))) / Om)));
	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[(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]], $MachinePrecision]}, If[LessEqual[t$95$1, 5e+135], t$95$1, If[LessEqual[t$95$1, Infinity], N[Power[N[(2.0 * N[(N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision] * N[(U * N[(2.0 - N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 5.00000000000000029e135

   1. Initial program 82.4%

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

   if 5.00000000000000029e135 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 30.3%

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

      1. associate-*l* 34.1%

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

      2. sub-neg 34.1%

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

      3. associate--l+ 34.1%

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

      4. *-commutative 34.1%

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

      5. distribute-rgt-neg-in 34.1%

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

      6. associate-*l/ 37.9%

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

      7. associate-*l* 37.9%

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

      8. *-commutative 37.9%

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

      9. *-commutative 37.9%

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

      10. associate-*l* 32.2%

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

      11. unpow2 32.2%

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

      12. associate-*l* 33.5%

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

   3. Simplified 33.5%

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

   4. Taylor expanded in t around inf 37.9%

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

   5. Taylor expanded in U* around inf 41.0%

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

      1. pow1/2 41.1%

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

      2. distribute-lft-out 41.1%

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

      3. associate-/l* 41.1%

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

      4. associate-/l* 42.4%

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

      5. *-commutative 42.4%

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

      6. *-commutative 42.4%

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

   7. Applied egg-rr 42.4%

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.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. Step-by-step derivation

      1. associate-*l* 0.9%

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

      2. sub-neg 0.9%

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

      3. associate--l+ 0.9%

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

      4. *-commutative 0.9%

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

      5. distribute-rgt-neg-in 0.9%

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

      6. associate-*l/ 10.0%

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

      7. associate-*l* 10.0%

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

      8. *-commutative 10.0%

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

      9. *-commutative 10.0%

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

      10. associate-*l* 9.9%

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

      11. unpow2 9.9%

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

      12. associate-*l* 12.9%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in l around -inf 49.3%

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

      1. associate-/l* 45.5%

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

      2. unpow2 45.5%

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

      3. *-commutative 45.5%

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

      4. mul-1-neg 45.5%

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

      5. associate-/l* 45.1%

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

   6. Simplified 45.1%

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

   7. Taylor expanded in U around 0 49.3%

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

      1. associate-*r* 53.7%

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

      2. unpow2 53.7%

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

      3. *-commutative 53.7%

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

   9. Simplified 53.7%

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

   10. Taylor expanded in n around 0 53.7%

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

       1. unpow2 53.7%

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

       2. associate-*r* 71.3%

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

   12. Simplified 71.3%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 3: 57.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;U \leq -6.8 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 3.7 \cdot 10^{-258}:\\ \;\;\;\;\sqrt{2 \cdot t_3 + 2 \cdot \frac{\left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right) \cdot t_1}{Om}}\\ \mathbf{elif}\;U \leq 1.55 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{n \cdot -4}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)}\\ \mathbf{elif}\;U \leq 1.92 \cdot 10^{+59}:\\ \;\;\;\;{\left(2 \cdot \left(t_3 + \frac{\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2}{\frac{Om}{t_1}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* U l)))
        (t_2 (sqrt (* 2.0 (* U (* n (fma -2.0 (* l (/ l Om)) t))))))
        (t_3 (* n (* U t))))
   (if (<= U -6.8e+68)
     t_2
     (if (<= U 3.7e-258)
       (sqrt
        (+
         (* 2.0 t_3)
         (* 2.0 (/ (* (+ (* l -2.0) (/ (* n (* l U*)) Om)) t_1) Om))))
       (if (<= U 1.55e-227)
         (sqrt (fma 2.0 (* U (* n t)) (/ (* n -4.0) (/ (/ Om U) (* l l)))))
         (if (<= U 1.92e+59)
           (pow
            (* 2.0 (+ t_3 (/ (+ (/ n (/ Om (* l U*))) (* l -2.0)) (/ Om t_1))))
            0.5)
           t_2))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * l);
	double t_2 = sqrt((2.0 * (U * (n * fma(-2.0, (l * (l / Om)), t)))));
	double t_3 = n * (U * t);
	double tmp;
	if (U <= -6.8e+68) {
		tmp = t_2;
	} else if (U <= 3.7e-258) {
		tmp = sqrt(((2.0 * t_3) + (2.0 * ((((l * -2.0) + ((n * (l * U_42_)) / Om)) * t_1) / Om))));
	} else if (U <= 1.55e-227) {
		tmp = sqrt(fma(2.0, (U * (n * t)), ((n * -4.0) / ((Om / U) / (l * l)))));
	} else if (U <= 1.92e+59) {
		tmp = pow((2.0 * (t_3 + (((n / (Om / (l * U_42_))) + (l * -2.0)) / (Om / t_1)))), 0.5);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U * l))
	t_2 = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(-2.0, Float64(l * Float64(l / Om)), t)))))
	t_3 = Float64(n * Float64(U * t))
	tmp = 0.0
	if (U <= -6.8e+68)
		tmp = t_2;
	elseif (U <= 3.7e-258)
		tmp = sqrt(Float64(Float64(2.0 * t_3) + Float64(2.0 * Float64(Float64(Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om)) * t_1) / Om))));
	elseif (U <= 1.55e-227)
		tmp = sqrt(fma(2.0, Float64(U * Float64(n * t)), Float64(Float64(n * -4.0) / Float64(Float64(Om / U) / Float64(l * l)))));
	elseif (U <= 1.92e+59)
		tmp = Float64(2.0 * Float64(t_3 + Float64(Float64(Float64(n / Float64(Om / Float64(l * U_42_))) + Float64(l * -2.0)) / Float64(Om / t_1)))) ^ 0.5;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * N[(U * N[(n * N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -6.8e+68], t$95$2, If[LessEqual[U, 3.7e-258], N[Sqrt[N[(N[(2.0 * t$95$3), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 1.55e-227], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(N[(n * -4.0), $MachinePrecision] / N[(N[(Om / U), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 1.92e+59], N[Power[N[(2.0 * N[(t$95$3 + N[(N[(N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if U < -6.8000000000000003e68 or 1.92e59 < U

   1. Initial program 62.2%

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

      1. associate-*l* 47.1%

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

      2. sub-neg 47.1%

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

      3. associate--l+ 47.1%

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

      4. *-commutative 47.1%

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

      5. distribute-rgt-neg-in 47.1%

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

      6. associate-*l/ 48.2%

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

      7. associate-*l* 48.2%

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

      8. *-commutative 48.2%

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

      9. *-commutative 48.2%

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

      10. associate-*l* 42.2%

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

      11. unpow2 42.2%

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

      12. associate-*l* 43.5%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in n around 0 44.8%

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

      1. associate-*r* 63.5%

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

      2. +-commutative 63.5%

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

      3. unpow2 63.5%

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

      4. fma-def 63.5%

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

      5. associate-*r/ 67.8%

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

   6. Simplified 67.8%

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

   if -6.8000000000000003e68 < U < 3.7e-258

   1. Initial program 46.9%

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

      1. associate-*l* 48.3%

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

      2. sub-neg 48.3%

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

      3. associate--l+ 48.3%

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

      4. *-commutative 48.3%

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

      5. distribute-rgt-neg-in 48.3%

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

      6. associate-*l/ 51.2%

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

      7. associate-*l* 51.2%

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

      8. *-commutative 51.2%

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

      9. *-commutative 51.2%

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

      10. associate-*l* 46.3%

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

      11. unpow2 46.3%

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

      12. associate-*l* 47.3%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in t around inf 59.8%

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

   5. Taylor expanded in U* around inf 59.8%

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

   if 3.7e-258 < U < 1.5499999999999999e-227

   1. Initial program 71.2%

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

      1. associate-*l* 16.3%

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

      2. sub-neg 16.3%

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

      3. associate--l+ 16.3%

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

      4. *-commutative 16.3%

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

      5. distribute-rgt-neg-in 16.3%

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

      6. associate-*l/ 16.3%

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

      7. associate-*l* 16.3%

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

      8. *-commutative 16.3%

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

      9. *-commutative 16.3%

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

      10. associate-*l* 15.7%

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

      11. unpow2 15.7%

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

      12. associate-*l* 16.1%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in t around inf 15.3%

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

   5. Taylor expanded in U* around inf 15.3%

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

   6. Taylor expanded in U* around 0 15.5%

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

      1. fma-def 15.5%

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

      2. associate-*r* 70.6%

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

      3. associate-/l* 84.3%

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

      4. associate-*r/ 84.3%

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

      5. *-commutative 84.3%

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

      6. associate-/r* 84.3%

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

      7. unpow2 84.3%

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

   8. Simplified 84.3%

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

   if 1.5499999999999999e-227 < U < 1.92e59

   1. Initial program 46.4%

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

      1. associate-*l* 48.5%

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

      2. sub-neg 48.5%

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

      3. associate--l+ 48.5%

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

      4. *-commutative 48.5%

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

      5. distribute-rgt-neg-in 48.5%

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

      6. associate-*l/ 53.6%

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

      7. associate-*l* 53.6%

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

      8. *-commutative 53.6%

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

      9. *-commutative 53.6%

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

      10. associate-*l* 52.0%

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

      11. unpow2 52.0%

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

      12. associate-*l* 53.7%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in t around inf 62.7%

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

   5. Taylor expanded in U* around inf 62.7%

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

      1. pow1/2 62.8%

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

      2. distribute-lft-out 62.8%

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

      3. associate-/l* 64.4%

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

      4. associate-/l* 64.4%

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

      5. *-commutative 64.4%

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

      6. *-commutative 64.4%

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

   7. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -6.8 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ \mathbf{elif}\;U \leq 3.7 \cdot 10^{-258}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{elif}\;U \leq 1.55 \cdot 10^{-227}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{n \cdot -4}{\frac{\frac{Om}{U}}{\ell \cdot \ell}}\right)}\\ \mathbf{elif}\;U \leq 1.92 \cdot 10^{+59}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ \end{array} \]

Alternative 4: 56.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U \cdot \ell\right)\\ t_2 := \sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ t_3 := n \cdot \left(U \cdot t\right)\\ \mathbf{if}\;U \leq -1.28 \cdot 10^{+67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;U \leq 2.1 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{2 \cdot t_3 + 2 \cdot \frac{\left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right) \cdot t_1}{Om}}\\ \mathbf{elif}\;U \leq 8 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{elif}\;U \leq 1.22 \cdot 10^{+54}:\\ \;\;\;\;{\left(2 \cdot \left(t_3 + \frac{\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2}{\frac{Om}{t_1}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (* U l)))
        (t_2 (sqrt (* 2.0 (* U (* n (fma -2.0 (* l (/ l Om)) t))))))
        (t_3 (* n (* U t))))
   (if (<= U -1.28e+67)
     t_2
     (if (<= U 2.1e-302)
       (sqrt
        (+
         (* 2.0 t_3)
         (* 2.0 (/ (* (+ (* l -2.0) (/ (* n (* l U*)) Om)) t_1) Om))))
       (if (<= U 8e-228)
         (* (sqrt 2.0) (sqrt (* (* n U) (+ t (* (/ (* l l) Om) -2.0)))))
         (if (<= U 1.22e+54)
           (pow
            (* 2.0 (+ t_3 (/ (+ (/ n (/ Om (* l U*))) (* l -2.0)) (/ Om t_1))))
            0.5)
           t_2))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U * l);
	double t_2 = sqrt((2.0 * (U * (n * fma(-2.0, (l * (l / Om)), t)))));
	double t_3 = n * (U * t);
	double tmp;
	if (U <= -1.28e+67) {
		tmp = t_2;
	} else if (U <= 2.1e-302) {
		tmp = sqrt(((2.0 * t_3) + (2.0 * ((((l * -2.0) + ((n * (l * U_42_)) / Om)) * t_1) / Om))));
	} else if (U <= 8e-228) {
		tmp = sqrt(2.0) * sqrt(((n * U) * (t + (((l * l) / Om) * -2.0))));
	} else if (U <= 1.22e+54) {
		tmp = pow((2.0 * (t_3 + (((n / (Om / (l * U_42_))) + (l * -2.0)) / (Om / t_1)))), 0.5);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U * l))
	t_2 = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(-2.0, Float64(l * Float64(l / Om)), t)))))
	t_3 = Float64(n * Float64(U * t))
	tmp = 0.0
	if (U <= -1.28e+67)
		tmp = t_2;
	elseif (U <= 2.1e-302)
		tmp = sqrt(Float64(Float64(2.0 * t_3) + Float64(2.0 * Float64(Float64(Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om)) * t_1) / Om))));
	elseif (U <= 8e-228)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(Float64(n * U) * Float64(t + Float64(Float64(Float64(l * l) / Om) * -2.0)))));
	elseif (U <= 1.22e+54)
		tmp = Float64(2.0 * Float64(t_3 + Float64(Float64(Float64(n / Float64(Om / Float64(l * U_42_))) + Float64(l * -2.0)) / Float64(Om / t_1)))) ^ 0.5;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * N[(U * N[(n * N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -1.28e+67], t$95$2, If[LessEqual[U, 2.1e-302], N[Sqrt[N[(N[(2.0 * t$95$3), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 8e-228], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 1.22e+54], N[Power[N[(2.0 * N[(t$95$3 + N[(N[(N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if U < -1.28e67 or 1.22e54 < U

   1. Initial program 62.2%

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

      1. associate-*l* 47.1%

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

      2. sub-neg 47.1%

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

      3. associate--l+ 47.1%

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

      4. *-commutative 47.1%

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

      5. distribute-rgt-neg-in 47.1%

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

      6. associate-*l/ 48.2%

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

      7. associate-*l* 48.2%

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

      8. *-commutative 48.2%

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

      9. *-commutative 48.2%

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

      10. associate-*l* 42.2%

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

      11. unpow2 42.2%

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

      12. associate-*l* 43.5%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in n around 0 44.8%

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

      1. associate-*r* 63.5%

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

      2. +-commutative 63.5%

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

      3. unpow2 63.5%

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

      4. fma-def 63.5%

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

      5. associate-*r/ 67.8%

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

   6. Simplified 67.8%

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

   if -1.28e67 < U < 2.10000000000000013e-302

   1. Initial program 47.1%

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

      1. associate-*l* 49.6%

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

      2. sub-neg 49.6%

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

      3. associate--l+ 49.6%

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

      4. *-commutative 49.6%

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

      5. distribute-rgt-neg-in 49.6%

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

      6. associate-*l/ 52.7%

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

      7. associate-*l* 52.7%

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

      8. *-commutative 52.7%

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

      9. *-commutative 52.7%

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

      10. associate-*l* 47.4%

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

      11. unpow2 47.4%

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

      12. associate-*l* 48.5%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in t around inf 62.9%

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

   5. Taylor expanded in U* around inf 62.9%

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

   if 2.10000000000000013e-302 < U < 8.00000000000000026e-228

   1. Initial program 58.1%

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

      1. associate-*l* 24.2%

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

      2. sub-neg 24.2%

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

      3. associate--l+ 24.2%

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

      4. *-commutative 24.2%

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

      5. distribute-rgt-neg-in 24.2%

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

      6. associate-*l/ 24.2%

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

      7. associate-*l* 24.2%

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

      8. *-commutative 24.2%

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

      9. *-commutative 24.2%

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

      10. associate-*l* 23.9%

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

      11. unpow2 23.9%

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

      12. associate-*l* 24.1%

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

   3. Simplified 24.1%

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

   4. Taylor expanded in U* around 0 16.6%

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

      1. associate-*r* 44.0%

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

      2. associate-/l* 44.0%

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

      3. +-commutative 44.0%

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

      4. mul-1-neg 44.0%

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

      5. unsub-neg 44.0%

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

      6. *-commutative 44.0%

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

   6. Simplified 44.0%

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

   7. Taylor expanded in n around 0 50.6%

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

      1. *-commutative 50.6%

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

      2. unpow2 50.6%

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

   9. Simplified 50.6%

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

   if 8.00000000000000026e-228 < U < 1.22e54

   1. Initial program 46.4%

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

      1. associate-*l* 48.5%

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

      2. sub-neg 48.5%

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

      3. associate--l+ 48.5%

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

      4. *-commutative 48.5%

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

      5. distribute-rgt-neg-in 48.5%

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

      6. associate-*l/ 53.6%

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

      7. associate-*l* 53.6%

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

      8. *-commutative 53.6%

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

      9. *-commutative 53.6%

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

      10. associate-*l* 52.0%

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

      11. unpow2 52.0%

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

      12. associate-*l* 53.7%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in t around inf 62.7%

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

   5. Taylor expanded in U* around inf 62.7%

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

      1. pow1/2 62.8%

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

      2. distribute-lft-out 62.8%

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

      3. associate-/l* 64.4%

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

      4. associate-/l* 64.4%

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

      5. *-commutative 64.4%

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

      6. *-commutative 64.4%

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

   7. Applied egg-rr 64.4%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(t \cdot U\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.28 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ \mathbf{elif}\;U \leq 2.1 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right) + 2 \cdot \frac{\left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right) \cdot \left(n \cdot \left(U \cdot \ell\right)\right)}{Om}}\\ \mathbf{elif}\;U \leq 8 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t + \frac{\ell \cdot \ell}{Om} \cdot -2\right)}\\ \mathbf{elif}\;U \leq 1.22 \cdot 10^{+54}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ \end{array} \]

Alternative 5: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := U* \cdot \frac{n}{Om}\\ \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+21}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \left(U \cdot \left(2 - t_1\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + t_1\right)\right)}{Om}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U* (/ n Om))))
   (if (<= l -1.2e+21)
     (sqrt (* -2.0 (* (/ (* l (* n l)) Om) (* U (- 2.0 t_1)))))
     (if (<= l 9.2e+86)
       (sqrt
        (*
         (* 2.0 n)
         (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
       (* (sqrt 2.0) (* l (sqrt (/ (* n (* U (+ -2.0 t_1))) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U_42_ * (n / Om);
	double tmp;
	if (l <= -1.2e+21) {
		tmp = sqrt((-2.0 * (((l * (n * l)) / Om) * (U * (2.0 - t_1)))));
	} else if (l <= 9.2e+86) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = sqrt(2.0) * (l * sqrt(((n * (U * (-2.0 + t_1))) / 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) :: t_1
    real(8) :: tmp
    t_1 = u_42 * (n / om)
    if (l <= (-1.2d+21)) then
        tmp = sqrt(((-2.0d0) * (((l * (n * l)) / om) * (u * (2.0d0 - t_1)))))
    else if (l <= 9.2d+86) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else
        tmp = sqrt(2.0d0) * (l * sqrt(((n * (u * ((-2.0d0) + t_1))) / 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 t_1 = U_42_ * (n / Om);
	double tmp;
	if (l <= -1.2e+21) {
		tmp = Math.sqrt((-2.0 * (((l * (n * l)) / Om) * (U * (2.0 - t_1)))));
	} else if (l <= 9.2e+86) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = Math.sqrt(2.0) * (l * Math.sqrt(((n * (U * (-2.0 + t_1))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = U_42_ * (n / Om)
	tmp = 0
	if l <= -1.2e+21:
		tmp = math.sqrt((-2.0 * (((l * (n * l)) / Om) * (U * (2.0 - t_1)))))
	elif l <= 9.2e+86:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	else:
		tmp = math.sqrt(2.0) * (l * math.sqrt(((n * (U * (-2.0 + t_1))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U_42_ * Float64(n / Om))
	tmp = 0.0
	if (l <= -1.2e+21)
		tmp = sqrt(Float64(-2.0 * Float64(Float64(Float64(l * Float64(n * l)) / Om) * Float64(U * Float64(2.0 - t_1)))));
	elseif (l <= 9.2e+86)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(l * sqrt(Float64(Float64(n * Float64(U * Float64(-2.0 + t_1))) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = U_42_ * (n / Om);
	tmp = 0.0;
	if (l <= -1.2e+21)
		tmp = sqrt((-2.0 * (((l * (n * l)) / Om) * (U * (2.0 - t_1)))));
	elseif (l <= 9.2e+86)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt(2.0) * (l * sqrt(((n * (U * (-2.0 + t_1))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.2e+21], N[Sqrt[N[(-2.0 * N[(N[(N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(U * N[(2.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 9.2e+86], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(N[(n * N[(U * N[(-2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.2e21

   1. Initial program 22.4%

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

      1. associate-*l* 22.7%

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

      2. sub-neg 22.7%

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

      3. associate--l+ 22.7%

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

      4. *-commutative 22.7%

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

      5. distribute-rgt-neg-in 22.7%

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

      6. associate-*l/ 29.5%

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

      7. associate-*l* 29.5%

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

      8. *-commutative 29.5%

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

      9. *-commutative 29.5%

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

      10. associate-*l* 29.5%

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

      11. unpow2 29.5%

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

      12. associate-*l* 29.7%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in l around -inf 43.5%

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

      1. associate-/l* 41.9%

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

      2. unpow2 41.9%

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

      3. *-commutative 41.9%

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

      4. mul-1-neg 41.9%

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

      5. associate-/l* 41.6%

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

   6. Simplified 41.6%

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

   7. Taylor expanded in U around 0 43.5%

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

      1. associate-*r* 44.7%

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

      2. unpow2 44.7%

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

      3. *-commutative 44.7%

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

   9. Simplified 44.7%

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

   10. Taylor expanded in l around 0 43.5%

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

       1. associate-*r* 44.7%

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

       2. unpow2 44.7%

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

       3. *-commutative 44.7%

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

       4. associate-/l* 44.4%

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

       5. associate-*l/ 44.6%

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

       6. associate-*r* 60.2%

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

       7. associate-/r/ 60.5%

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

   12. Simplified 60.5%

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

   if -1.2e21 < l < 9.19999999999999958e86

   1. Initial program 68.4%

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

      1. associate-*l* 60.8%

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

      2. sub-neg 60.8%

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

      3. associate--l+ 60.8%

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

      4. *-commutative 60.8%

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

      5. distribute-rgt-neg-in 60.8%

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

      6. associate-*l/ 60.8%

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

      7. associate-*l* 60.8%

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

      8. *-commutative 60.8%

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

      9. *-commutative 60.8%

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

      10. associate-*l* 53.7%

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

      11. unpow2 53.7%

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

      12. associate-*l* 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in U around 0 61.3%

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

   if 9.19999999999999958e86 < l

   1. Initial program 31.1%

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

      1. associate-*l* 26.1%

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

      2. sub-neg 26.1%

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

      3. associate--l+ 26.1%

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

      4. *-commutative 26.1%

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

      5. distribute-rgt-neg-in 26.1%

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

      6. associate-*l/ 33.5%

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

      7. associate-*l* 33.5%

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

      8. *-commutative 33.5%

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

      9. *-commutative 33.5%

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

      10. associate-*l* 33.4%

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

      11. unpow2 33.4%

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

      12. associate-*l* 33.6%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in t around inf 47.7%

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

   5. Taylor expanded in U* around inf 48.0%

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

      1. pow1/2 48.2%

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

      2. distribute-lft-out 48.2%

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

      3. associate-/l* 48.3%

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

      4. associate-/l* 50.0%

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

      5. *-commutative 50.0%

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

      6. *-commutative 50.0%

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

   7. Applied egg-rr 50.0%

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

   8. Taylor expanded in l around inf 71.1%

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

      1. associate-*l* 71.2%

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

      2. *-commutative 71.2%

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

      3. sub-neg 71.2%

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

      4. associate-*l/ 71.2%

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

      5. metadata-eval 71.2%

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

   10. Simplified 71.2%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+21}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \left(U \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{n \cdot \left(U \cdot \left(-2 + U* \cdot \frac{n}{Om}\right)\right)}{Om}}\right)\\ \end{array} \]

Alternative 6: 58.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -9.5 \cdot 10^{+69} \lor \neg \left(U \leq 6.4 \cdot 10^{+56}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= U -9.5e+69) (not (<= U 6.4e+56)))
   (sqrt (* 2.0 (* U (* n (fma -2.0 (* l (/ l Om)) t)))))
   (pow
    (*
     2.0
     (+
      (* n (* U t))
      (/ (+ (/ n (/ Om (* l U*))) (* l -2.0)) (/ Om (* n (* U l))))))
    0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U <= -9.5e+69) || !(U <= 6.4e+56)) {
		tmp = sqrt((2.0 * (U * (n * fma(-2.0, (l * (l / Om)), t)))));
	} else {
		tmp = pow((2.0 * ((n * (U * t)) + (((n / (Om / (l * U_42_))) + (l * -2.0)) / (Om / (n * (U * l)))))), 0.5);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((U <= -9.5e+69) || !(U <= 6.4e+56))
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * fma(-2.0, Float64(l * Float64(l / Om)), t)))));
	else
		tmp = Float64(2.0 * Float64(Float64(n * Float64(U * t)) + Float64(Float64(Float64(n / Float64(Om / Float64(l * U_42_))) + Float64(l * -2.0)) / Float64(Om / Float64(n * Float64(U * l)))))) ^ 0.5;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[U, -9.5e+69], N[Not[LessEqual[U, 6.4e+56]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(n / N[(Om / N[(l * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * -2.0), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(n * N[(U * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -9.4999999999999995e69 or 6.40000000000000007e56 < U

   1. Initial program 62.2%

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

      1. associate-*l* 47.1%

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

      2. sub-neg 47.1%

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

      3. associate--l+ 47.1%

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

      4. *-commutative 47.1%

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

      5. distribute-rgt-neg-in 47.1%

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

      6. associate-*l/ 48.2%

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

      7. associate-*l* 48.2%

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

      8. *-commutative 48.2%

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

      9. *-commutative 48.2%

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

      10. associate-*l* 42.2%

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

      11. unpow2 42.2%

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

      12. associate-*l* 43.5%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in n around 0 44.8%

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

      1. associate-*r* 63.5%

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

      2. +-commutative 63.5%

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

      3. unpow2 63.5%

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

      4. fma-def 63.5%

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

      5. associate-*r/ 67.8%

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

   6. Simplified 67.8%

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

   if -9.4999999999999995e69 < U < 6.40000000000000007e56

   1. Initial program 47.7%

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

      1. associate-*l* 47.1%

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

      2. sub-neg 47.1%

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

      3. associate--l+ 47.1%

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

      4. *-commutative 47.1%

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

      5. distribute-rgt-neg-in 47.1%

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

      6. associate-*l/ 50.6%

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

      7. associate-*l* 50.6%

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

      8. *-commutative 50.6%

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

      9. *-commutative 50.6%

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

      10. associate-*l* 47.0%

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

      11. unpow2 47.0%

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

      12. associate-*l* 48.3%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in t around inf 58.9%

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

   5. Taylor expanded in U* around inf 58.9%

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

      1. pow1/2 59.0%

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

      2. distribute-lft-out 59.0%

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

      3. associate-/l* 58.4%

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

      4. associate-/l* 59.0%

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

      5. *-commutative 59.0%

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

      6. *-commutative 59.0%

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

   7. Applied egg-rr 59.0%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -9.5 \cdot 10^{+69} \lor \neg \left(U \leq 6.4 \cdot 10^{+56}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right) + \frac{\frac{n}{\frac{Om}{\ell \cdot U*}} + \ell \cdot -2}{\frac{Om}{n \cdot \left(U \cdot \ell\right)}}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 7: 57.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \left(n \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{t_1}{Om} \cdot \left(U \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{t_1 \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (* n l))))
   (if (<= l -6.5e+20)
     (sqrt (* -2.0 (* (/ t_1 Om) (* U (- 2.0 (* U* (/ n Om)))))))
     (if (<= l 9.8e+86)
       (sqrt
        (*
         (* 2.0 n)
         (* U (+ t (/ (* l (+ (* l -2.0) (/ (* n (* l U*)) Om))) Om)))))
       (sqrt (* -2.0 (/ (* t_1 (* U (- 2.0 (/ (* n U*) Om)))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (n * l);
	double tmp;
	if (l <= -6.5e+20) {
		tmp = sqrt((-2.0 * ((t_1 / Om) * (U * (2.0 - (U_42_ * (n / Om)))))));
	} else if (l <= 9.8e+86) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = sqrt((-2.0 * ((t_1 * (U * (2.0 - ((n * U_42_) / Om)))) / 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) :: t_1
    real(8) :: tmp
    t_1 = l * (n * l)
    if (l <= (-6.5d+20)) then
        tmp = sqrt(((-2.0d0) * ((t_1 / om) * (u * (2.0d0 - (u_42 * (n / om)))))))
    else if (l <= 9.8d+86) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((l * ((l * (-2.0d0)) + ((n * (l * u_42)) / om))) / om)))))
    else
        tmp = sqrt(((-2.0d0) * ((t_1 * (u * (2.0d0 - ((n * u_42) / om)))) / 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 t_1 = l * (n * l);
	double tmp;
	if (l <= -6.5e+20) {
		tmp = Math.sqrt((-2.0 * ((t_1 / Om) * (U * (2.0 - (U_42_ * (n / Om)))))));
	} else if (l <= 9.8e+86) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	} else {
		tmp = Math.sqrt((-2.0 * ((t_1 * (U * (2.0 - ((n * U_42_) / Om)))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (n * l)
	tmp = 0
	if l <= -6.5e+20:
		tmp = math.sqrt((-2.0 * ((t_1 / Om) * (U * (2.0 - (U_42_ * (n / Om)))))))
	elif l <= 9.8e+86:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))))
	else:
		tmp = math.sqrt((-2.0 * ((t_1 * (U * (2.0 - ((n * U_42_) / Om)))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(n * l))
	tmp = 0.0
	if (l <= -6.5e+20)
		tmp = sqrt(Float64(-2.0 * Float64(Float64(t_1 / Om) * Float64(U * Float64(2.0 - Float64(U_42_ * Float64(n / Om)))))));
	elseif (l <= 9.8e+86)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(l * Float64(Float64(l * -2.0) + Float64(Float64(n * Float64(l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(t_1 * Float64(U * Float64(2.0 - Float64(Float64(n * U_42_) / Om)))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (n * l);
	tmp = 0.0;
	if (l <= -6.5e+20)
		tmp = sqrt((-2.0 * ((t_1 / Om) * (U * (2.0 - (U_42_ * (n / Om)))))));
	elseif (l <= 9.8e+86)
		tmp = sqrt(((2.0 * n) * (U * (t + ((l * ((l * -2.0) + ((n * (l * U_42_)) / Om))) / Om)))));
	else
		tmp = sqrt((-2.0 * ((t_1 * (U * (2.0 - ((n * U_42_) / Om)))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.5e+20], N[Sqrt[N[(-2.0 * N[(N[(t$95$1 / Om), $MachinePrecision] * N[(U * N[(2.0 - N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 9.8e+86], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(l * N[(N[(l * -2.0), $MachinePrecision] + N[(N[(n * N[(l * U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(t$95$1 * N[(U * N[(2.0 - N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.5e20

   1. Initial program 22.4%

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

      1. associate-*l* 22.7%

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

      2. sub-neg 22.7%

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

      3. associate--l+ 22.7%

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

      4. *-commutative 22.7%

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

      5. distribute-rgt-neg-in 22.7%

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

      6. associate-*l/ 29.5%

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

      7. associate-*l* 29.5%

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

      8. *-commutative 29.5%

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

      9. *-commutative 29.5%

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

      10. associate-*l* 29.5%

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

      11. unpow2 29.5%

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

      12. associate-*l* 29.7%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in l around -inf 43.5%

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

      1. associate-/l* 41.9%

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

      2. unpow2 41.9%

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

      3. *-commutative 41.9%

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

      4. mul-1-neg 41.9%

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

      5. associate-/l* 41.6%

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

   6. Simplified 41.6%

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

   7. Taylor expanded in U around 0 43.5%

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

      1. associate-*r* 44.7%

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

      2. unpow2 44.7%

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

      3. *-commutative 44.7%

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

   9. Simplified 44.7%

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

   10. Taylor expanded in l around 0 43.5%

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

       1. associate-*r* 44.7%

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

       2. unpow2 44.7%

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

       3. *-commutative 44.7%

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

       4. associate-/l* 44.4%

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

       5. associate-*l/ 44.6%

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

       6. associate-*r* 60.2%

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

       7. associate-/r/ 60.5%

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

   12. Simplified 60.5%

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

   if -6.5e20 < l < 9.7999999999999999e86

   1. Initial program 68.4%

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

      1. associate-*l* 60.8%

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

      2. sub-neg 60.8%

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

      3. associate--l+ 60.8%

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

      4. *-commutative 60.8%

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

      5. distribute-rgt-neg-in 60.8%

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

      6. associate-*l/ 60.8%

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

      7. associate-*l* 60.8%

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

      8. *-commutative 60.8%

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

      9. *-commutative 60.8%

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

      10. associate-*l* 53.7%

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

      11. unpow2 53.7%

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

      12. associate-*l* 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in U around 0 61.3%

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

   if 9.7999999999999999e86 < l

   1. Initial program 31.1%

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

      1. associate-*l* 26.1%

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

      2. sub-neg 26.1%

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

      3. associate--l+ 26.1%

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

      4. *-commutative 26.1%

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

      5. distribute-rgt-neg-in 26.1%

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

      6. associate-*l/ 33.5%

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

      7. associate-*l* 33.5%

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

      8. *-commutative 33.5%

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

      9. *-commutative 33.5%

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

      10. associate-*l* 33.4%

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

      11. unpow2 33.4%

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

      12. associate-*l* 33.6%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in l around -inf 39.5%

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

      1. associate-/l* 37.7%

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

      2. unpow2 37.7%

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

      3. *-commutative 37.7%

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

      4. mul-1-neg 37.7%

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

      5. associate-/l* 35.9%

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

   6. Simplified 35.9%

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

   7. Taylor expanded in U around 0 39.5%

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

      1. associate-*r* 43.2%

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

      2. unpow2 43.2%

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

      3. *-commutative 43.2%

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

   9. Simplified 43.2%

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

   10. Taylor expanded in n around 0 43.2%

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

       1. unpow2 43.2%

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

       2. associate-*r* 54.0%

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

   12. Simplified 54.0%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \left(U \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell \cdot \left(\ell \cdot -2 + \frac{n \cdot \left(\ell \cdot U*\right)}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 8: 50.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+19} \lor \neg \left(\ell \leq 3 \cdot 10^{-20}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \left(U \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= l -6.2e+19) (not (<= l 3e-20)))
   (sqrt (* -2.0 (* (/ (* l (* n l)) Om) (* U (- 2.0 (* U* (/ n Om)))))))
   (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((l <= -6.2e+19) || !(l <= 3e-20)) {
		tmp = sqrt((-2.0 * (((l * (n * l)) / Om) * (U * (2.0 - (U_42_ * (n / Om)))))));
	} else {
		tmp = sqrt((2.0 * (U * (n * 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 <= (-6.2d+19)) .or. (.not. (l <= 3d-20))) then
        tmp = sqrt(((-2.0d0) * (((l * (n * l)) / om) * (u * (2.0d0 - (u_42 * (n / om)))))))
    else
        tmp = sqrt((2.0d0 * (u * (n * 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 <= -6.2e+19) || !(l <= 3e-20)) {
		tmp = Math.sqrt((-2.0 * (((l * (n * l)) / Om) * (U * (2.0 - (U_42_ * (n / Om)))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (l <= -6.2e+19) or not (l <= 3e-20):
		tmp = math.sqrt((-2.0 * (((l * (n * l)) / Om) * (U * (2.0 - (U_42_ * (n / Om)))))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((l <= -6.2e+19) || !(l <= 3e-20))
		tmp = sqrt(Float64(-2.0 * Float64(Float64(Float64(l * Float64(n * l)) / Om) * Float64(U * Float64(2.0 - Float64(U_42_ * Float64(n / Om)))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((l <= -6.2e+19) || ~((l <= 3e-20)))
		tmp = sqrt((-2.0 * (((l * (n * l)) / Om) * (U * (2.0 - (U_42_ * (n / Om)))))));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[l, -6.2e+19], N[Not[LessEqual[l, 3e-20]], $MachinePrecision]], N[Sqrt[N[(-2.0 * N[(N[(N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(U * N[(2.0 - N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.2e19 or 3.00000000000000029e-20 < l

   1. Initial program 32.7%

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

      1. associate-*l* 30.5%

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

      2. sub-neg 30.5%

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

      3. associate--l+ 30.5%

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

      4. *-commutative 30.5%

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

      5. distribute-rgt-neg-in 30.5%

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

      6. associate-*l/ 36.6%

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

      7. associate-*l* 36.6%

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

      8. *-commutative 36.6%

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

      9. *-commutative 36.6%

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

      10. associate-*l* 36.6%

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

      11. unpow2 36.6%

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

      12. associate-*l* 36.8%

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

   3. Simplified 48.7%

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

   4. Taylor expanded in l around -inf 42.3%

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

      1. associate-/l* 40.8%

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

      2. unpow2 40.8%

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

      3. *-commutative 40.8%

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

      4. mul-1-neg 40.8%

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

      5. associate-/l* 40.6%

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

   6. Simplified 40.6%

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

   7. Taylor expanded in U around 0 42.3%

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

      1. associate-*r* 44.6%

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

      2. unpow2 44.6%

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

      3. *-commutative 44.6%

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

   9. Simplified 44.6%

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

   10. Taylor expanded in l around 0 42.3%

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

       1. associate-*r* 44.6%

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

       2. unpow2 44.6%

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

       3. *-commutative 44.6%

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

       4. associate-/l* 43.6%

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

       5. associate-*l/ 42.8%

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

       6. associate-*r* 53.1%

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

       7. associate-/r/ 54.9%

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

   12. Simplified 54.9%

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

   if -6.2e19 < l < 3.00000000000000029e-20

   1. Initial program 68.3%

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

      1. associate-*l* 60.0%

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

      2. sub-neg 60.0%

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

      3. associate--l+ 60.0%

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

      4. *-commutative 60.0%

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

      5. distribute-rgt-neg-in 60.0%

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

      6. associate-*l/ 60.0%

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

      7. associate-*l* 60.0%

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

      8. *-commutative 60.0%

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

      9. *-commutative 60.0%

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

      10. associate-*l* 52.2%

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

      11. unpow2 52.2%

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

      12. associate-*l* 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in t around inf 48.5%

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

   5. Taylor expanded in n around 0 48.5%

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

      1. associate-*r* 55.6%

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

   7. Simplified 55.6%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+19} \lor \neg \left(\ell \leq 3 \cdot 10^{-20}\right):\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \left(U \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 9: 50.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \left(n \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{t_1}{Om} \cdot \left(U \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{t_1 \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (* n l))))
   (if (<= l -4.2e+14)
     (sqrt (* -2.0 (* (/ t_1 Om) (* U (- 2.0 (* U* (/ n Om)))))))
     (if (<= l 4.3e-20)
       (sqrt (* 2.0 (* U (* n t))))
       (sqrt (* -2.0 (/ (* t_1 (* U (- 2.0 (/ (* n U*) Om)))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (n * l);
	double tmp;
	if (l <= -4.2e+14) {
		tmp = sqrt((-2.0 * ((t_1 / Om) * (U * (2.0 - (U_42_ * (n / Om)))))));
	} else if (l <= 4.3e-20) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt((-2.0 * ((t_1 * (U * (2.0 - ((n * U_42_) / Om)))) / 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) :: t_1
    real(8) :: tmp
    t_1 = l * (n * l)
    if (l <= (-4.2d+14)) then
        tmp = sqrt(((-2.0d0) * ((t_1 / om) * (u * (2.0d0 - (u_42 * (n / om)))))))
    else if (l <= 4.3d-20) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = sqrt(((-2.0d0) * ((t_1 * (u * (2.0d0 - ((n * u_42) / om)))) / 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 t_1 = l * (n * l);
	double tmp;
	if (l <= -4.2e+14) {
		tmp = Math.sqrt((-2.0 * ((t_1 / Om) * (U * (2.0 - (U_42_ * (n / Om)))))));
	} else if (l <= 4.3e-20) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.sqrt((-2.0 * ((t_1 * (U * (2.0 - ((n * U_42_) / Om)))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (n * l)
	tmp = 0
	if l <= -4.2e+14:
		tmp = math.sqrt((-2.0 * ((t_1 / Om) * (U * (2.0 - (U_42_ * (n / Om)))))))
	elif l <= 4.3e-20:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.sqrt((-2.0 * ((t_1 * (U * (2.0 - ((n * U_42_) / Om)))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(n * l))
	tmp = 0.0
	if (l <= -4.2e+14)
		tmp = sqrt(Float64(-2.0 * Float64(Float64(t_1 / Om) * Float64(U * Float64(2.0 - Float64(U_42_ * Float64(n / Om)))))));
	elseif (l <= 4.3e-20)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(t_1 * Float64(U * Float64(2.0 - Float64(Float64(n * U_42_) / Om)))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (n * l);
	tmp = 0.0;
	if (l <= -4.2e+14)
		tmp = sqrt((-2.0 * ((t_1 / Om) * (U * (2.0 - (U_42_ * (n / Om)))))));
	elseif (l <= 4.3e-20)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = sqrt((-2.0 * ((t_1 * (U * (2.0 - ((n * U_42_) / Om)))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(n * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.2e+14], N[Sqrt[N[(-2.0 * N[(N[(t$95$1 / Om), $MachinePrecision] * N[(U * N[(2.0 - N[(U$42$ * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.3e-20], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(t$95$1 * N[(U * N[(2.0 - N[(N[(n * U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.2e14

   1. Initial program 22.4%

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

      1. associate-*l* 22.7%

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

      2. sub-neg 22.7%

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

      3. associate--l+ 22.7%

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

      4. *-commutative 22.7%

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

      5. distribute-rgt-neg-in 22.7%

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

      6. associate-*l/ 29.5%

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

      7. associate-*l* 29.5%

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

      8. *-commutative 29.5%

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

      9. *-commutative 29.5%

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

      10. associate-*l* 29.5%

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

      11. unpow2 29.5%

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

      12. associate-*l* 29.7%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in l around -inf 43.5%

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

      1. associate-/l* 41.9%

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

      2. unpow2 41.9%

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

      3. *-commutative 41.9%

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

      4. mul-1-neg 41.9%

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

      5. associate-/l* 41.6%

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

   6. Simplified 41.6%

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

   7. Taylor expanded in U around 0 43.5%

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

      1. associate-*r* 44.7%

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

      2. unpow2 44.7%

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

      3. *-commutative 44.7%

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

   9. Simplified 44.7%

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

   10. Taylor expanded in l around 0 43.5%

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

       1. associate-*r* 44.7%

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

       2. unpow2 44.7%

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

       3. *-commutative 44.7%

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

       4. associate-/l* 44.4%

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

       5. associate-*l/ 44.6%

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

       6. associate-*r* 60.2%

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

       7. associate-/r/ 60.5%

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

   12. Simplified 60.5%

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

   if -4.2e14 < l < 4.30000000000000011e-20

   1. Initial program 68.3%

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

      1. associate-*l* 60.0%

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

      2. sub-neg 60.0%

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

      3. associate--l+ 60.0%

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

      4. *-commutative 60.0%

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

      5. distribute-rgt-neg-in 60.0%

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

      6. associate-*l/ 60.0%

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

      7. associate-*l* 60.0%

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

      8. *-commutative 60.0%

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

      9. *-commutative 60.0%

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

      10. associate-*l* 52.2%

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

      11. unpow2 52.2%

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

      12. associate-*l* 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in t around inf 48.5%

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

   5. Taylor expanded in n around 0 48.5%

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

      1. associate-*r* 55.6%

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

   7. Simplified 55.6%

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

   if 4.30000000000000011e-20 < l

   1. Initial program 39.2%

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

      1. associate-*l* 35.3%

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

      2. sub-neg 35.3%

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

      3. associate--l+ 35.3%

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

      4. *-commutative 35.3%

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

      5. distribute-rgt-neg-in 35.3%

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

      6. associate-*l/ 41.1%

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

      7. associate-*l* 41.1%

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

      8. *-commutative 41.1%

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

      9. *-commutative 41.1%

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

      10. associate-*l* 41.0%

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

      11. unpow2 41.0%

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

      12. associate-*l* 41.2%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in l around -inf 41.5%

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

      1. associate-/l* 40.1%

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

      2. unpow2 40.1%

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

      3. *-commutative 40.1%

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

      4. mul-1-neg 40.1%

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

      5. associate-/l* 40.0%

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

   6. Simplified 40.0%

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

   7. Taylor expanded in U around 0 41.5%

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

      1. associate-*r* 44.5%

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

      2. unpow2 44.5%

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

      3. *-commutative 44.5%

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

   9. Simplified 44.5%

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

   10. Taylor expanded in n around 0 44.5%

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

       1. unpow2 44.5%

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

       2. associate-*r* 52.9%

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

   12. Simplified 52.9%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om} \cdot \left(U \cdot \left(2 - U* \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}}\\ \end{array} \]

Alternative 10: 46.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -2400 \lor \neg \left(Om \leq 7.2 \cdot 10^{-233}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{Om}{\frac{-n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)}}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -2400.0) (not (<= Om 7.2e-233)))
   (sqrt (* (* 2.0 n) (* U (+ t (* -2.0 (* l (/ l Om)))))))
   (sqrt (* -2.0 (/ n (/ Om (/ (- n) (/ Om (* (* l l) (* U U*))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -2400.0) || !(Om <= 7.2e-233)) {
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((-2.0 * (n / (Om / (-n / (Om / ((l * l) * (U * U_42_))))))));
	}
	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 ((om <= (-2400.0d0)) .or. (.not. (om <= 7.2d-233))) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((-2.0d0) * (l * (l / om)))))))
    else
        tmp = sqrt(((-2.0d0) * (n / (om / (-n / (om / ((l * l) * (u * u_42))))))))
    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 ((Om <= -2400.0) || !(Om <= 7.2e-233)) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt((-2.0 * (n / (Om / (-n / (Om / ((l * l) * (U * U_42_))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -2400.0) or not (Om <= 7.2e-233):
		tmp = math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt((-2.0 * (n / (Om / (-n / (Om / ((l * l) * (U * U_42_))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -2400.0) || !(Om <= 7.2e-233))
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(n / Float64(Om / Float64(Float64(-n) / Float64(Om / Float64(Float64(l * l) * Float64(U * U_42_))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -2400.0) || ~((Om <= 7.2e-233)))
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((-2.0 * (n / (Om / (-n / (Om / ((l * l) * (U * U_42_))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -2400.0], N[Not[LessEqual[Om, 7.2e-233]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(n / N[(Om / N[((-n) / N[(Om / N[(N[(l * l), $MachinePrecision] * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -2400 or 7.20000000000000014e-233 < Om

   1. Initial program 52.7%

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

      1. associate-*l* 48.7%

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

      2. sub-neg 48.7%

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

      3. associate--l+ 48.7%

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

      4. *-commutative 48.7%

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

      5. distribute-rgt-neg-in 48.7%

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

      6. associate-*l/ 52.3%

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

      7. associate-*l* 52.3%

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

      8. *-commutative 52.3%

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

      9. *-commutative 52.3%

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

      10. associate-*l* 46.8%

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

      11. unpow2 46.8%

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

      12. associate-*l* 48.0%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in n around 0 46.0%

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

      1. *-commutative 46.0%

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

      2. unpow2 46.0%

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

      3. associate-*r/ 49.7%

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

   6. Simplified 49.7%

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

   if -2400 < Om < 7.20000000000000014e-233

   1. Initial program 53.0%

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

      1. associate-*l* 42.8%

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

      2. sub-neg 42.8%

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

      3. associate--l+ 42.8%

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

      4. *-commutative 42.8%

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

      5. distribute-rgt-neg-in 42.8%

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

      6. associate-*l/ 42.8%

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

      7. associate-*l* 42.8%

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

      8. *-commutative 42.8%

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

      9. *-commutative 42.8%

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

      10. associate-*l* 41.5%

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

      11. unpow2 41.5%

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

      12. associate-*l* 43.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in l around -inf 47.5%

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

      1. associate-/l* 45.1%

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

      2. unpow2 45.1%

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

      3. *-commutative 45.1%

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

      4. mul-1-neg 45.1%

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

      5. associate-/l* 44.8%

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

   6. Simplified 44.8%

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

   7. Taylor expanded in U* around inf 42.5%

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

      1. mul-1-neg 42.5%

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

      2. associate-/l* 42.5%

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

      3. unpow2 42.5%

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

      4. *-commutative 42.5%

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

   9. Simplified 42.5%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2400 \lor \neg \left(Om \leq 7.2 \cdot 10^{-233}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{n}{\frac{Om}{\frac{-n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)}}}}}\\ \end{array} \]

Alternative 11: 43.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)\right) \cdot \left(n \cdot n\right)}{Om \cdot Om}}\\ \mathbf{elif}\;\ell \leq 6.7 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -1.9e+55)
   (sqrt (* 2.0 (/ (* (* (* l l) (* U U*)) (* n n)) (* Om Om))))
   (if (<= l 6.7e-26)
     (sqrt (* 2.0 (* U (* n t))))
     (sqrt (* (* 2.0 n) (* U (+ t (* -2.0 (* l (/ l Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -1.9e+55) {
		tmp = sqrt((2.0 * ((((l * l) * (U * U_42_)) * (n * n)) / (Om * Om))));
	} else if (l <= 6.7e-26) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (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 <= (-1.9d+55)) then
        tmp = sqrt((2.0d0 * ((((l * l) * (u * u_42)) * (n * n)) / (om * om))))
    else if (l <= 6.7d-26) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((-2.0d0) * (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 <= -1.9e+55) {
		tmp = Math.sqrt((2.0 * ((((l * l) * (U * U_42_)) * (n * n)) / (Om * Om))));
	} else if (l <= 6.7e-26) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -1.9e+55:
		tmp = math.sqrt((2.0 * ((((l * l) * (U * U_42_)) * (n * n)) / (Om * Om))))
	elif l <= 6.7e-26:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -1.9e+55)
		tmp = sqrt(Float64(2.0 * Float64(Float64(Float64(Float64(l * l) * Float64(U * U_42_)) * Float64(n * n)) / Float64(Om * Om))));
	elseif (l <= 6.7e-26)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -1.9e+55)
		tmp = sqrt((2.0 * ((((l * l) * (U * U_42_)) * (n * n)) / (Om * Om))));
	elseif (l <= 6.7e-26)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -1.9e+55], N[Sqrt[N[(2.0 * N[(N[(N[(N[(l * l), $MachinePrecision] * N[(U * U$42$), $MachinePrecision]), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 6.7e-26], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.9e55

   1. Initial program 16.6%

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

      1. associate-*l* 16.9%

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

      2. sub-neg 16.9%

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

      3. associate--l+ 16.9%

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

      4. *-commutative 16.9%

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

      5. distribute-rgt-neg-in 16.9%

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

      6. associate-*l/ 25.7%

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

      7. associate-*l* 25.7%

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

      8. *-commutative 25.7%

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

      9. *-commutative 25.7%

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

      10. associate-*l* 25.7%

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

      11. unpow2 25.7%

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

      12. associate-*l* 25.9%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in U* around inf 31.0%

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

      1. unpow2 31.0%

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

      2. unpow2 31.0%

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

      3. *-commutative 31.0%

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

      4. unpow2 31.0%

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

   6. Simplified 31.0%

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

   if -1.9e55 < l < 6.70000000000000054e-26

   1. Initial program 67.0%

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

      1. associate-*l* 59.2%

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

      2. sub-neg 59.2%

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

      3. associate--l+ 59.2%

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

      4. *-commutative 59.2%

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

      5. distribute-rgt-neg-in 59.2%

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

      6. associate-*l/ 59.2%

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

      7. associate-*l* 59.2%

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

      8. *-commutative 59.2%

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

      9. *-commutative 59.2%

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

      10. associate-*l* 51.9%

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

      11. unpow2 51.9%

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

      12. associate-*l* 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in t around inf 47.8%

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

   5. Taylor expanded in n around 0 47.7%

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

      1. associate-*r* 54.5%

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

   7. Simplified 54.5%

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

   if 6.70000000000000054e-26 < l

   1. Initial program 38.7%

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

      1. associate-*l* 34.9%

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

      2. sub-neg 34.9%

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

      3. associate--l+ 34.9%

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

      4. *-commutative 34.9%

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

      5. distribute-rgt-neg-in 34.9%

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

      6. associate-*l/ 40.6%

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

      7. associate-*l* 40.6%

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

      8. *-commutative 40.6%

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

      9. *-commutative 40.6%

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

      10. associate-*l* 40.4%

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

      11. unpow2 40.4%

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

      12. associate-*l* 40.6%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in n around 0 31.0%

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

      1. *-commutative 31.0%

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

      2. unpow2 31.0%

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

      3. associate-*r/ 36.6%

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

   6. Simplified 36.6%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\left(\left(\ell \cdot \ell\right) \cdot \left(U \cdot U*\right)\right) \cdot \left(n \cdot n\right)}{Om \cdot Om}}\\ \mathbf{elif}\;\ell \leq 6.7 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 12: 43.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\frac{\frac{Om \cdot Om}{\left(\ell \cdot \ell\right) \cdot U*}}{U}}}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -1.4e+53)
   (sqrt (* (* 2.0 n) (/ n (/ (/ (* Om Om) (* (* l l) U*)) U))))
   (if (<= l 3.5e-23)
     (sqrt (* 2.0 (* U (* n t))))
     (sqrt (* (* 2.0 n) (* U (+ t (* -2.0 (* l (/ l Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= -1.4e+53) {
		tmp = sqrt(((2.0 * n) * (n / (((Om * Om) / ((l * l) * U_42_)) / U))));
	} else if (l <= 3.5e-23) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (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 <= (-1.4d+53)) then
        tmp = sqrt(((2.0d0 * n) * (n / (((om * om) / ((l * l) * u_42)) / u))))
    else if (l <= 3.5d-23) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((-2.0d0) * (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 <= -1.4e+53) {
		tmp = Math.sqrt(((2.0 * n) * (n / (((Om * Om) / ((l * l) * U_42_)) / U))));
	} else if (l <= 3.5e-23) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= -1.4e+53:
		tmp = math.sqrt(((2.0 * n) * (n / (((Om * Om) / ((l * l) * U_42_)) / U))))
	elif l <= 3.5e-23:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= -1.4e+53)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(n / Float64(Float64(Float64(Om * Om) / Float64(Float64(l * l) * U_42_)) / U))));
	elseif (l <= 3.5e-23)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= -1.4e+53)
		tmp = sqrt(((2.0 * n) * (n / (((Om * Om) / ((l * l) * U_42_)) / U))));
	elseif (l <= 3.5e-23)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, -1.4e+53], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(n / N[(N[(N[(Om * Om), $MachinePrecision] / N[(N[(l * l), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.5e-23], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.4e53

   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. Step-by-step derivation

      1. associate-*l* 16.5%

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

      2. sub-neg 16.5%

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

      3. associate--l+ 16.5%

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

      4. *-commutative 16.5%

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

      5. distribute-rgt-neg-in 16.5%

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

      6. associate-*l/ 25.1%

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

      7. associate-*l* 25.1%

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

      8. *-commutative 25.1%

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

      9. *-commutative 25.1%

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

      10. associate-*l* 25.1%

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

      11. unpow2 25.1%

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

      12. associate-*l* 25.3%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in U* around inf 30.7%

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

      1. associate-/l* 30.7%

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

      2. associate-*r* 33.7%

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

      3. associate-/r* 33.9%

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

      4. unpow2 33.9%

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

      5. *-commutative 33.9%

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

      6. unpow2 33.9%

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

   6. Simplified 33.9%

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

   if -1.4e53 < l < 3.49999999999999993e-23

   1. Initial program 67.4%

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

      1. associate-*l* 59.5%

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

      2. sub-neg 59.5%

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

      3. associate--l+ 59.5%

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

      4. *-commutative 59.5%

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

      5. distribute-rgt-neg-in 59.5%

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

      6. associate-*l/ 59.5%

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

      7. associate-*l* 59.5%

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

      8. *-commutative 59.5%

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

      9. *-commutative 59.5%

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

      10. associate-*l* 52.2%

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

      11. unpow2 52.2%

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

      12. associate-*l* 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in t around inf 48.1%

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

   5. Taylor expanded in n around 0 48.0%

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

      1. associate-*r* 54.8%

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

   7. Simplified 54.8%

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

   if 3.49999999999999993e-23 < l

   1. Initial program 38.7%

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

      1. associate-*l* 34.9%

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

      2. sub-neg 34.9%

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

      3. associate--l+ 34.9%

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

      4. *-commutative 34.9%

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

      5. distribute-rgt-neg-in 34.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 40.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 40.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 40.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 40.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 40.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 40.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 40.6%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 48.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in n around 0 31.0%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 31.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. unpow2 31.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

      3. associate-*r/ 36.6%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

   6. Simplified 36.6%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{n}{\frac{\frac{Om \cdot Om}{\left(\ell \cdot \ell\right) \cdot U*}}{U}}}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 13: 46.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -2.6e+62)
   (sqrt (* 2.0 (* U (* n t))))
   (sqrt (* (* 2.0 n) (* U (+ t (* -2.0 (* l (/ l Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -2.6e+62) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (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 (t <= (-2.6d+62)) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((-2.0d0) * (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 (t <= -2.6e+62) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -2.6e+62:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -2.6e+62)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(-2.0 * Float64(l * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -2.6e+62)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (l * (l / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -2.6e+62], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.59999999999999984e62

   1. Initial program 53.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 40.2%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 40.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 40.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 40.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 40.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 42.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 42.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 42.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 42.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 43.5%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 43.5%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 43.7%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 52.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 43.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]

   5. Taylor expanded in n around 0 43.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* 54.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)}} \]

   7. Simplified 54.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot t\right) \cdot U\right)}} \]

   if -2.59999999999999984e62 < t

   1. Initial program 52.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 49.0%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 49.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 49.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 49.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 49.0%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 51.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 51.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 51.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 51.9%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 45.8%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 45.8%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 47.4%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 51.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in n around 0 39.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 39.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. unpow2 39.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

      3. associate-*r/ 42.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)}\right)\right)} \]

   6. Simplified 42.2%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 14: 38.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.6e+30)
   (sqrt (* 2.0 (* U (* n t))))
   (sqrt (* -2.0 (/ (* 2.0 (* 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 <= 3.6e+30) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt((-2.0 * ((2.0 * (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 <= 3.6d+30) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = sqrt(((-2.0d0) * ((2.0d0 * (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 <= 3.6e+30) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.sqrt((-2.0 * ((2.0 * (U * (n * (l * l)))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.6e+30:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.sqrt((-2.0 * ((2.0 * (U * (n * (l * l)))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.6e+30)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(2.0 * 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 <= 3.6e+30)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = sqrt((-2.0 * ((2.0 * (U * (n * (l * l)))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.6e+30], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(2.0 * N[(U * N[(n * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.6000000000000002e30

   1. Initial program 58.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 52.7%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 52.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 52.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 52.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 52.7%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 54.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 54.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 54.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 54.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 48.3%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 48.3%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 50.0%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 40.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]

   5. Taylor expanded in n around 0 40.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* 46.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)}} \]

   7. Simplified 46.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot t\right) \cdot U\right)}} \]

   if 3.6000000000000002e30 < l

   1. Initial program 34.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 30.3%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. sub-neg 30.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. associate--l+ 30.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

      4. *-commutative 30.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      5. distribute-rgt-neg-in 30.3%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      6. associate-*l/ 36.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      7. associate-*l* 36.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      8. *-commutative 36.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

      9. *-commutative 36.5%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

      10. associate-*l* 36.3%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

      11. unpow2 36.3%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

      12. associate-*l* 36.6%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 44.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in l around -inf 39.9%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \frac{n \cdot \left({\ell}^{2} \cdot \left(\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot U\right)\right)}{Om}}} \]
   5. Step-by-step derivation

      1. associate-/l* 38.4%

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\frac{n}{\frac{Om}{{\ell}^{2} \cdot \left(\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot U\right)}}}} \]

      2. unpow2 38.4%

         \[\leadsto \sqrt{-2 \cdot \frac{n}{\frac{Om}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right) \cdot U\right)}}} \]

      3. *-commutative 38.4%

         \[\leadsto \sqrt{-2 \cdot \frac{n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(U \cdot \left(2 + -1 \cdot \frac{n \cdot \left(U* - U\right)}{Om}\right)\right)}}}} \]

      4. mul-1-neg 38.4%

         \[\leadsto \sqrt{-2 \cdot \frac{n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(2 + \color{blue}{\left(-\frac{n \cdot \left(U* - U\right)}{Om}\right)}\right)\right)}}} \]

      5. associate-/l* 38.4%

         \[\leadsto \sqrt{-2 \cdot \frac{n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(2 + \left(-\color{blue}{\frac{n}{\frac{Om}{U* - U}}}\right)\right)\right)}}} \]

   6. Simplified 38.4%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \frac{n}{\frac{Om}{\left(\ell \cdot \ell\right) \cdot \left(U \cdot \left(2 + \left(-\frac{n}{\frac{Om}{U* - U}}\right)\right)\right)}}}} \]

   7. Taylor expanded in U around 0 40.0%

      \[\leadsto \sqrt{-2 \cdot \color{blue}{\frac{n \cdot \left({\ell}^{2} \cdot \left(\left(2 - \frac{n \cdot U*}{Om}\right) \cdot U\right)\right)}{Om}}} \]
   8. Step-by-step derivation

      1. associate-*r* 43.2%

         \[\leadsto \sqrt{-2 \cdot \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right) \cdot \left(\left(2 - \frac{n \cdot U*}{Om}\right) \cdot U\right)}}{Om}} \]

      2. unpow2 43.2%

         \[\leadsto \sqrt{-2 \cdot \frac{\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(\left(2 - \frac{n \cdot U*}{Om}\right) \cdot U\right)}{Om}} \]

      3. *-commutative 43.2%

         \[\leadsto \sqrt{-2 \cdot \frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}}{Om}} \]

   9. Simplified 43.2%

      \[\leadsto \sqrt{-2 \cdot \color{blue}{\frac{\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}}} \]

   10. Taylor expanded in n around 0 43.2%

       \[\leadsto \sqrt{-2 \cdot \frac{\color{blue}{\left(n \cdot {\ell}^{2}\right)} \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}} \]
   11. Step-by-step derivation

       1. unpow2 43.2%

          \[\leadsto \sqrt{-2 \cdot \frac{\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}} \]

       2. associate-*r* 52.2%

          \[\leadsto \sqrt{-2 \cdot \frac{\color{blue}{\left(\left(n \cdot \ell\right) \cdot \ell\right)} \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}} \]

   12. Simplified 52.2%

       \[\leadsto \sqrt{-2 \cdot \frac{\color{blue}{\left(\left(n \cdot \ell\right) \cdot \ell\right)} \cdot \left(U \cdot \left(2 - \frac{n \cdot U*}{Om}\right)\right)}{Om}} \]

   13. Taylor expanded in n around 0 22.9%

       \[\leadsto \sqrt{-2 \cdot \frac{\color{blue}{2 \cdot \left(n \cdot \left({\ell}^{2} \cdot U\right)\right)}}{Om}} \]
   14. Step-by-step derivation

       1. associate-*r* 26.0%

          \[\leadsto \sqrt{-2 \cdot \frac{2 \cdot \color{blue}{\left(\left(n \cdot {\ell}^{2}\right) \cdot U\right)}}{Om}} \]

       2. unpow2 26.0%

          \[\leadsto \sqrt{-2 \cdot \frac{2 \cdot \left(\left(n \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot U\right)}{Om}} \]

   15. Simplified 26.0%

       \[\leadsto \sqrt{-2 \cdot \frac{\color{blue}{2 \cdot \left(\left(n \cdot \left(\ell \cdot \ell\right)\right) \cdot U\right)}}{Om}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \frac{2 \cdot \left(U \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}}\\ \end{array} \]

Alternative 15: 36.2% 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 52.8%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Step-by-step derivation

   1. associate-*l* 47.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

   2. sub-neg 47.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

   3. associate--l+ 47.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \color{blue}{\left(t + \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\right)} \]

   4. *-commutative 47.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left(-\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot 2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   5. distribute-rgt-neg-in 47.1%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell \cdot \ell}{Om} \cdot \left(-2\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   6. associate-*l/ 49.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} \cdot \left(-2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   7. associate-*l* 49.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\ell \cdot \left(-2\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   8. *-commutative 49.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(-2\right) \cdot \ell\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)} \]

   9. *-commutative 49.8%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)\right)\right)} \]

   10. associate-*l* 45.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{{\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}\right)\right)\right)} \]

   11. unpow2 45.3%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)} \]

   12. associate-*l* 46.6%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}\right)\right)\right)} \]

3. Simplified 51.8%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in t around inf 33.8%

   \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(t \cdot U\right)}} \]

5. Taylor expanded in n around 0 33.8%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* 38.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot t\right) \cdot U\right)}} \]

7. Simplified 38.8%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(n \cdot t\right) \cdot U\right)}} \]

8. Final simplification 38.8%

   \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

Reproduce

herbie shell --seed 2023187 
(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*))))))