Toniolo and Linder, Equation (13)

Percentage Accurate: 50.0% → 64.1%

Time: 24.0s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_2 \leq 10^{-99}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) + t_1\right)\right)}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+147}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{2} \cdot \frac{\sqrt{U \cdot U*} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_2 (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) t_1)))))
   (if (<= t_2 1e-99)
     (sqrt (* (* 2.0 n) (* U (+ (fma (* l (/ l Om)) -2.0 t) t_1))))
     (if (<= t_2 5e+147)
       t_2
       (fabs (* (sqrt 2.0) (/ (* (sqrt (* U U*)) (* n l)) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 1e-99) {
		tmp = sqrt(((2.0 * n) * (U * (fma((l * (l / Om)), -2.0, t) + t_1))));
	} else if (t_2 <= 5e+147) {
		tmp = t_2;
	} else {
		tmp = fabs((sqrt(2.0) * ((sqrt((U * U_42_)) * (n * l)) / Om)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_1)))
	tmp = 0.0
	if (t_2 <= 1e-99)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(fma(Float64(l * Float64(l / Om)), -2.0, t) + t_1))));
	elseif (t_2 <= 5e+147)
		tmp = t_2;
	else
		tmp = abs(Float64(sqrt(2.0) * Float64(Float64(sqrt(Float64(U * U_42_)) * Float64(n * l)) / Om)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[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] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1e-99], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+147], t$95$2, N[Abs[N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(n * l), $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*))))) < 1e-99

   1. Initial program 43.4%

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

   3. Simplified 66.9%

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

      1. add-sqr-sqrt 43.8%

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

      2. pow2 43.8%

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

      3. sqrt-prod 25.1%

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

      4. unpow2 25.1%

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

      5. sqrt-prod 18.2%

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

      6. add-sqr-sqrt 25.1%

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

   5. Applied egg-rr 25.1%

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

      1. unpow-prod-down 25.1%

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

      2. pow2 25.1%

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

      3. add-sqr-sqrt 66.9%

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

      4. associate-*r* 70.5%

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

      5. cancel-sign-sub-inv 70.5%

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

      6. +-commutative 70.5%

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

      7. *-commutative 70.5%

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

      8. fma-def 70.5%

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

      9. div-inv 70.5%

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

      10. clear-num 70.5%

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

   7. Applied egg-rr 70.5%

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

   if 1e-99 < (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*))))) < 5.0000000000000002e147

   1. Initial program 97.5%

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

   if 5.0000000000000002e147 < (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 18.8%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in U* around inf 19.9%

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

      1. add-sqr-sqrt 19.9%

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

      2. rem-sqrt-square 19.9%

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

      3. sqrt-prod 19.9%

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

      4. sqrt-div 19.9%

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

   6. Applied egg-rr 41.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{-99}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{2} \cdot \frac{\sqrt{U \cdot U*} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \end{array} \]

Alternative 2: 63.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_1 \leq 10^{-99}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{2} \cdot \frac{\sqrt{U \cdot U*} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \end{array} \end{array} \]

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 1e-99)
     (sqrt (* (* 2.0 n) (* U (+ t (* -2.0 (/ (pow l 2.0) Om))))))
     (if (<= t_1 5e+147)
       t_1
       (fabs (* (sqrt 2.0) (/ (* (sqrt (* U U*)) (* n l)) 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 <= 1e-99) {
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (pow(l, 2.0) / Om))))));
	} else if (t_1 <= 5e+147) {
		tmp = t_1;
	} else {
		tmp = fabs((sqrt(2.0) * ((sqrt((U * U_42_)) * (n * 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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) + ((n * ((l / om) ** 2.0d0)) * (u_42 - u)))))
    if (t_1 <= 1d-99) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((-2.0d0) * ((l ** 2.0d0) / om))))))
    else if (t_1 <= 5d+147) then
        tmp = t_1
    else
        tmp = abs((sqrt(2.0d0) * ((sqrt((u * u_42)) * (n * 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 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 <= 1e-99) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (Math.pow(l, 2.0) / Om))))));
	} else if (t_1 <= 5e+147) {
		tmp = t_1;
	} else {
		tmp = Math.abs((Math.sqrt(2.0) * ((Math.sqrt((U * U_42_)) * (n * l)) / 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 <= 1e-99:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (math.pow(l, 2.0) / Om))))))
	elif t_1 <= 5e+147:
		tmp = t_1
	else:
		tmp = math.fabs((math.sqrt(2.0) * ((math.sqrt((U * U_42_)) * (n * l)) / 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 <= 1e-99)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(-2.0 * Float64((l ^ 2.0) / Om))))));
	elseif (t_1 <= 5e+147)
		tmp = t_1;
	else
		tmp = abs(Float64(sqrt(2.0) * Float64(Float64(sqrt(Float64(U * U_42_)) * Float64(n * l)) / 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 <= 1e-99)
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * ((l ^ 2.0) / Om))))));
	elseif (t_1 <= 5e+147)
		tmp = t_1;
	else
		tmp = abs((sqrt(2.0) * ((sqrt((U * U_42_)) * (n * l)) / 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, 1e-99], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e+147], t$95$1, N[Abs[N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(n * l), $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*))))) < 1e-99

   1. Initial program 43.4%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in Om around inf 68.7%

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

   if 1e-99 < (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*))))) < 5.0000000000000002e147

   1. Initial program 97.5%

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

   if 5.0000000000000002e147 < (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 18.8%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in U* around inf 19.9%

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

      1. add-sqr-sqrt 19.9%

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

      2. rem-sqrt-square 19.9%

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

      3. sqrt-prod 19.9%

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

      4. sqrt-div 19.9%

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

   6. Applied egg-rr 41.4%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{-99}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{2} \cdot \frac{\sqrt{U \cdot U*} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \end{array} \]

Alternative 3: 60.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\frac{Om}{\ell}}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_4 := \left(n \cdot t_3\right) \cdot \left(U* - U\right)\\ t_5 := t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_4\right)\\ \mathbf{if}\;t_5 \leq 10^{-198}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot t_1\right) + n \cdot \left(t_3 \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(t_4 + \left(t - 2 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right)}{Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ l (/ Om l)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (pow (/ l Om) 2.0))
        (t_4 (* (* n t_3) (- U* U)))
        (t_5 (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) t_4))))
   (if (<= t_5 1e-198)
     (sqrt (* (* 2.0 n) (* U (+ (+ t (* -2.0 t_1)) (* n (* t_3 (- U* U)))))))
     (if (<= t_5 INFINITY)
       (sqrt (* t_2 (+ t_4 (- t (* 2.0 t_1)))))
       (/ (* (sqrt (* U U*)) (* l (* n (sqrt 2.0)))) Om)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l / (Om / l);
	double t_2 = (2.0 * n) * U;
	double t_3 = pow((l / Om), 2.0);
	double t_4 = (n * t_3) * (U_42_ - U);
	double t_5 = t_2 * ((t - (2.0 * ((l * l) / Om))) + t_4);
	double tmp;
	if (t_5 <= 1e-198) {
		tmp = sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + (n * (t_3 * (U_42_ - U)))))));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * (t_4 + (t - (2.0 * t_1)))));
	} else {
		tmp = (sqrt((U * U_42_)) * (l * (n * sqrt(2.0)))) / Om;
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l / (Om / l);
	double t_2 = (2.0 * n) * U;
	double t_3 = Math.pow((l / Om), 2.0);
	double t_4 = (n * t_3) * (U_42_ - U);
	double t_5 = t_2 * ((t - (2.0 * ((l * l) / Om))) + t_4);
	double tmp;
	if (t_5 <= 1e-198) {
		tmp = Math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + (n * (t_3 * (U_42_ - U)))))));
	} else if (t_5 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * (t_4 + (t - (2.0 * t_1)))));
	} else {
		tmp = (Math.sqrt((U * U_42_)) * (l * (n * Math.sqrt(2.0)))) / Om;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l / (Om / l)
	t_2 = (2.0 * n) * U
	t_3 = math.pow((l / Om), 2.0)
	t_4 = (n * t_3) * (U_42_ - U)
	t_5 = t_2 * ((t - (2.0 * ((l * l) / Om))) + t_4)
	tmp = 0
	if t_5 <= 1e-198:
		tmp = math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + (n * (t_3 * (U_42_ - U)))))))
	elif t_5 <= math.inf:
		tmp = math.sqrt((t_2 * (t_4 + (t - (2.0 * t_1)))))
	else:
		tmp = (math.sqrt((U * U_42_)) * (l * (n * math.sqrt(2.0)))) / Om
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l / Float64(Om / l))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(l / Om) ^ 2.0
	t_4 = Float64(Float64(n * t_3) * Float64(U_42_ - U))
	t_5 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_4))
	tmp = 0.0
	if (t_5 <= 1e-198)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t + Float64(-2.0 * t_1)) + Float64(n * Float64(t_3 * Float64(U_42_ - U)))))));
	elseif (t_5 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(t_4 + Float64(t - Float64(2.0 * t_1)))));
	else
		tmp = Float64(Float64(sqrt(Float64(U * U_42_)) * Float64(l * Float64(n * sqrt(2.0)))) / Om);
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l / (Om / l);
	t_2 = (2.0 * n) * U;
	t_3 = (l / Om) ^ 2.0;
	t_4 = (n * t_3) * (U_42_ - U);
	t_5 = t_2 * ((t - (2.0 * ((l * l) / Om))) + t_4);
	tmp = 0.0;
	if (t_5 <= 1e-198)
		tmp = sqrt(((2.0 * n) * (U * ((t + (-2.0 * t_1)) + (n * (t_3 * (U_42_ - U)))))));
	elseif (t_5 <= Inf)
		tmp = sqrt((t_2 * (t_4 + (t - (2.0 * t_1)))));
	else
		tmp = (sqrt((U * U_42_)) * (l * (n * sqrt(2.0)))) / Om;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(n * t$95$3), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 1e-198], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(n * N[(t$95$3 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[Sqrt[N[(t$95$2 * N[(t$95$4 + N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(l * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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*)))) < 9.9999999999999991e-199

   1. Initial program 39.7%

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

   3. Simplified 64.7%

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

   if 9.9999999999999991e-199 < (*.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 68.2%

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

      1. associate-/l* 70.9%

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

      2. add-sqr-sqrt 34.4%

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

      3. *-un-lft-identity 34.4%

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

      4. times-frac 34.4%

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

   3. Applied egg-rr 34.4%

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

      1. /-rgt-identity 34.4%

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

      2. associate-*r/ 34.4%

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

      3. rem-square-sqrt 70.9%

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

   5. Simplified 70.9%

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

   if +inf.0 < (*.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%

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

   3. Simplified 0.5%

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

   4. Taylor expanded in U* around inf 18.5%

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

   5. Taylor expanded in l around 0 8.9%

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

      1. *-commutative 8.9%

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

      2. associate-*r/ 11.2%

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

   7. Simplified 11.2%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{-198}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right)}{Om}\\ \end{array} \]

Alternative 4: 54.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 20500:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 20500.0)
   (sqrt
    (*
     (* 2.0 n)
     (*
      U
      (+
       (+ t (* -2.0 (/ l (/ Om l))))
       (* n (* (pow (/ l Om) 2.0) (- U* U)))))))
   (pow (* (* 2.0 (* n U)) (- t (/ (* 2.0 (pow l 2.0)) Om))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 20500.0) {
		tmp = sqrt(((2.0 * n) * (U * ((t + (-2.0 * (l / (Om / l)))) + (n * (pow((l / Om), 2.0) * (U_42_ - U)))))));
	} else {
		tmp = pow(((2.0 * (n * U)) * (t - ((2.0 * pow(l, 2.0)) / Om))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= 20500.0d0) then
        tmp = sqrt(((2.0d0 * n) * (u * ((t + ((-2.0d0) * (l / (om / l)))) + (n * (((l / om) ** 2.0d0) * (u_42 - u)))))))
    else
        tmp = ((2.0d0 * (n * u)) * (t - ((2.0d0 * (l ** 2.0d0)) / om))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 20500.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * (l / (Om / l)))) + (n * (Math.pow((l / Om), 2.0) * (U_42_ - U)))))));
	} else {
		tmp = Math.pow(((2.0 * (n * U)) * (t - ((2.0 * Math.pow(l, 2.0)) / Om))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 20500.0:
		tmp = math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * (l / (Om / l)))) + (n * (math.pow((l / Om), 2.0) * (U_42_ - U)))))))
	else:
		tmp = math.pow(((2.0 * (n * U)) * (t - ((2.0 * math.pow(l, 2.0)) / Om))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 20500.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l)))) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U_42_ - U)))))));
	else
		tmp = Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(2.0 * (l ^ 2.0)) / Om))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 20500.0)
		tmp = sqrt(((2.0 * n) * (U * ((t + (-2.0 * (l / (Om / l)))) + (n * (((l / Om) ^ 2.0) * (U_42_ - U)))))));
	else
		tmp = ((2.0 * (n * U)) * (t - ((2.0 * (l ^ 2.0)) / Om))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 20500.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 20500

   1. Initial program 43.4%

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

   3. Simplified 51.8%

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

   if 20500 < U

   1. Initial program 75.8%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in Om around inf 72.2%

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

      1. pow1/2 74.1%

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

      2. associate-*r* 74.1%

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

      3. associate-*r/ 74.1%

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

   6. Applied egg-rr 74.1%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 20500:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 5: 49.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -2.4 \cdot 10^{-77} \lor \neg \left(U \leq 4.3 \cdot 10^{-83}\right):\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= U -2.4e-77) (not (<= U 4.3e-83)))
   (pow (* (* 2.0 (* n U)) (- t (/ (* 2.0 (pow l 2.0)) Om))) 0.5)
   (sqrt (* (* 2.0 n) (* U (+ t (* -2.0 (/ (pow l 2.0) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U <= -2.4e-77) || !(U <= 4.3e-83)) {
		tmp = pow(((2.0 * (n * U)) * (t - ((2.0 * pow(l, 2.0)) / Om))), 0.5);
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (pow(l, 2.0) / Om))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((u <= (-2.4d-77)) .or. (.not. (u <= 4.3d-83))) then
        tmp = ((2.0d0 * (n * u)) * (t - ((2.0d0 * (l ** 2.0d0)) / om))) ** 0.5d0
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((-2.0d0) * ((l ** 2.0d0) / om))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U <= -2.4e-77) || !(U <= 4.3e-83)) {
		tmp = Math.pow(((2.0 * (n * U)) * (t - ((2.0 * Math.pow(l, 2.0)) / Om))), 0.5);
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (Math.pow(l, 2.0) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (U <= -2.4e-77) or not (U <= 4.3e-83):
		tmp = math.pow(((2.0 * (n * U)) * (t - ((2.0 * math.pow(l, 2.0)) / Om))), 0.5)
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (math.pow(l, 2.0) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((U <= -2.4e-77) || !(U <= 4.3e-83))
		tmp = Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(2.0 * (l ^ 2.0)) / Om))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(-2.0 * Float64((l ^ 2.0) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((U <= -2.4e-77) || ~((U <= 4.3e-83)))
		tmp = ((2.0 * (n * U)) * (t - ((2.0 * (l ^ 2.0)) / Om))) ^ 0.5;
	else
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * ((l ^ 2.0) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[U, -2.4e-77], N[Not[LessEqual[U, 4.3e-83]], $MachinePrecision]], N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -2.3999999999999999e-77 or 4.30000000000000033e-83 < U

   1. Initial program 65.6%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in Om around inf 60.0%

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

      1. pow1/2 63.4%

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

      2. associate-*r* 63.4%

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

      3. associate-*r/ 63.4%

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

   6. Applied egg-rr 63.4%

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

   if -2.3999999999999999e-77 < U < 4.30000000000000033e-83

   1. Initial program 35.6%

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

   3. Simplified 48.7%

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

   4. Taylor expanded in Om around inf 44.3%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2.4 \cdot 10^{-77} \lor \neg \left(U \leq 4.3 \cdot 10^{-83}\right):\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \end{array} \]

Alternative 6: 44.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+252}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 2.9e+252)
   (sqrt (* (* 2.0 n) (* U (+ t (* -2.0 (/ (pow l 2.0) Om))))))
   (sqrt (* 2.0 (fabs (* U (* n t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 2.9e+252) {
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * (pow(l, 2.0) / Om))))));
	} else {
		tmp = sqrt((2.0 * fabs((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 (t <= 2.9d+252) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((-2.0d0) * ((l ** 2.0d0) / om))))))
    else
        tmp = sqrt((2.0d0 * abs((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 (t <= 2.9e+252) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (Math.pow(l, 2.0) / Om))))));
	} else {
		tmp = Math.sqrt((2.0 * Math.abs((U * (n * t)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 2.9e+252:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (-2.0 * (math.pow(l, 2.0) / Om))))))
	else:
		tmp = math.sqrt((2.0 * math.fabs((U * (n * t)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 2.9e+252)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(-2.0 * Float64((l ^ 2.0) / Om))))));
	else
		tmp = sqrt(Float64(2.0 * abs(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 (t <= 2.9e+252)
		tmp = sqrt(((2.0 * n) * (U * (t + (-2.0 * ((l ^ 2.0) / Om))))));
	else
		tmp = sqrt((2.0 * abs((U * (n * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 2.9e+252], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[Abs[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.89999999999999996e252

   1. Initial program 50.3%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in Om around inf 49.6%

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

   if 2.89999999999999996e252 < t

   1. Initial program 52.0%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in t around inf 56.5%

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

      1. associate-*r* 43.5%

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

      2. *-commutative 43.5%

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

   6. Simplified 43.5%

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

      1. add-sqr-sqrt 43.3%

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

      2. sqrt-unprod 48.1%

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

      3. pow2 48.1%

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

      4. *-commutative 48.1%

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

   8. Applied egg-rr 48.1%

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

      1. unpow2 48.1%

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

      2. rem-sqrt-square 52.6%

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

      3. *-commutative 52.6%

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

      4. *-commutative 52.6%

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

      5. associate-*r* 65.6%

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

   10. Simplified 65.6%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+252}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \end{array} \]

Alternative 7: 44.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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 - (2.0d0 * ((l ** 2.0d0) / om)))))))
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 - (2.0 * (Math.pow(l, 2.0) / Om)))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

3. Simplified 49.2%

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

4. Taylor expanded in n around 0 49.1%

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

5. Final simplification 49.1%

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

Alternative 8: 39.2% accurate, 1.1× 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 (fabs (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * fabs((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 * abs((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 * Math.abs((U * (n * t)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

3. Simplified 49.2%

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

4. Taylor expanded in t around inf 38.7%

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

   1. associate-*r* 35.3%

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

   2. *-commutative 35.3%

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

6. Simplified 35.3%

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

   1. add-sqr-sqrt 35.2%

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

   2. sqrt-unprod 27.2%

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

   3. pow2 27.2%

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

   4. *-commutative 27.2%

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

8. Applied egg-rr 27.2%

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

   1. unpow2 27.2%

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

   2. rem-sqrt-square 37.7%

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

   3. *-commutative 37.7%

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

   4. *-commutative 37.7%

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

   5. associate-*r* 41.5%

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

10. Simplified 41.5%

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

11. Final simplification 41.5%

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

Alternative 9: 36.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (pow (* 2.0 (* n (* U t))) 0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow((2.0 * (n * (U * t))), 0.5);
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = (2.0d0 * (n * (u * t))) ** 0.5d0
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.pow((2.0 * (n * (U * t))), 0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow((2.0 * (n * (U * t))), 0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = (2.0 * (n * (U * t))) ^ 0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 50.5%

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

3. Simplified 54.4%

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

4. Taylor expanded in t around inf 38.3%

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

   1. pow1/2 39.9%

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

   2. associate-*l* 39.9%

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

6. Applied egg-rr 39.9%

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

7. Final simplification 39.9%

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

Alternative 10: 36.3% 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 50.5%

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

3. Simplified 49.2%

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

4. Taylor expanded in t around inf 38.7%

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

5. Final simplification 38.7%

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

Reproduce

herbie shell --seed 2023310 
(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*))))))