Toniolo and Linder, Equation (13)

Percentage Accurate: 49.5% → 71.5%

Time: 28.5s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.5% 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: 71.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\right)}\\ t_3 := \sqrt{2 \cdot \left|n\right|}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;t\_3 \cdot \sqrt{\left|U \cdot \left(t + t\_1 \cdot \left(n \cdot U*\right)\right)\right|}\\ \mathbf{elif}\;t\_2 \leq 10^{+147}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(\sqrt{\left|U\right|} \cdot \sqrt{\left|\mathsf{fma}\left(n, t\_1 \cdot U*, t\right)\right|}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l Om) 2.0))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 (/ (* l l) Om))) (* (* n t_1) (- U* U))))))
        (t_3 (sqrt (* 2.0 (fabs n)))))
   (if (<= t_2 0.0)
     (* t_3 (sqrt (fabs (* U (+ t (* t_1 (* n U*)))))))
     (if (<= t_2 1e+147)
       t_2
       (* t_3 (* (sqrt (fabs U)) (sqrt (fabs (fma n (* t_1 U*) t)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((l / Om), 2.0);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * t_1) * (U_42_ - U)))));
	double t_3 = sqrt((2.0 * fabs(n)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = t_3 * sqrt(fabs((U * (t + (t_1 * (n * U_42_))))));
	} else if (t_2 <= 1e+147) {
		tmp = t_2;
	} else {
		tmp = t_3 * (sqrt(fabs(U)) * sqrt(fabs(fma(n, (t_1 * U_42_), t))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l / Om) ^ 2.0
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * t_1) * Float64(U_42_ - U)))))
	t_3 = sqrt(Float64(2.0 * abs(n)))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(t_3 * sqrt(abs(Float64(U * Float64(t + Float64(t_1 * Float64(n * U_42_)))))));
	elseif (t_2 <= 1e+147)
		tmp = t_2;
	else
		tmp = Float64(t_3 * Float64(sqrt(abs(U)) * sqrt(abs(fma(n, Float64(t_1 * U_42_), t)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $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] + N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(2.0 * N[Abs[n], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(t$95$3 * N[Sqrt[N[Abs[N[(U * N[(t + N[(t$95$1 * N[(n * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+147], t$95$2, N[(t$95$3 * N[(N[Sqrt[N[Abs[U], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[N[(n * N[(t$95$1 * U$42$), $MachinePrecision] + t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.4%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in U* around inf 24.5%

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

      1. mul-1-neg 24.5%

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

      2. associate-/l* 24.5%

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

      3. distribute-rgt-neg-in 24.5%

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

      4. distribute-neg-frac2 24.5%

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

      5. *-commutative 24.5%

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

   7. Simplified 24.5%

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

      1. add-sqr-sqrt 24.5%

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

      2. pow1/2 24.5%

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

      3. pow1/2 24.5%

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

      4. pow-prod-down 17.9%

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

   9. Applied egg-rr 14.7%

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

       1. unpow1/2 14.7%

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

       2. unpow2 14.7%

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

       3. rem-sqrt-square 14.7%

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

       4. associate-*l* 24.5%

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

       5. associate-/l* 24.5%

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

       6. associate-*r/ 24.6%

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

       7. distribute-frac-neg2 24.6%

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

       8. unpow2 24.6%

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

       9. unpow2 24.6%

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

       10. times-frac 28.6%

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

       11. unpow2 28.6%

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

   11. Simplified 28.6%

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

       1. pow1/2 28.6%

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

       2. fabs-mul 28.6%

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

       3. unpow-prod-down 81.7%

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

       4. associate-*r* 81.7%

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

   13. Applied egg-rr 81.7%

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

   15. Simplified 81.7%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 9.9999999999999998e146

   1. Initial program 99.3%

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

   if 9.9999999999999998e146 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 26.2%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in U* around inf 34.8%

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

      1. mul-1-neg 34.8%

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

      2. associate-/l* 34.8%

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

      3. distribute-rgt-neg-in 34.8%

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

      4. distribute-neg-frac2 34.8%

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

      5. *-commutative 34.8%

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

   7. Simplified 34.8%

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

      1. add-sqr-sqrt 34.8%

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

      2. pow1/2 34.8%

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

      3. pow1/2 35.3%

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

      4. pow-prod-down 32.8%

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

   9. Applied egg-rr 30.6%

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

       1. unpow1/2 30.6%

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

       2. unpow2 30.6%

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

       3. rem-sqrt-square 33.1%

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

       4. associate-*l* 35.3%

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

       5. associate-/l* 35.4%

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

       6. associate-*r/ 34.6%

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

       7. distribute-frac-neg2 34.6%

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

       8. unpow2 34.6%

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

       9. unpow2 34.6%

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

       10. times-frac 38.0%

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

       11. unpow2 38.0%

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

   11. Simplified 38.0%

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

       1. pow1/2 38.0%

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

       2. fabs-mul 38.0%

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

       3. unpow-prod-down 43.9%

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

       4. associate-*r* 41.8%

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

   13. Applied egg-rr 41.8%

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

   15. Simplified 41.8%

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

       1. pow1/2 41.8%

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

       2. fabs-mul 41.8%

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

       3. unpow-prod-down 46.7%

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

       4. +-commutative 46.7%

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

       5. associate-*l* 52.3%

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

       6. fma-define 52.3%

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

   17. Applied egg-rr 52.3%

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

       1. unpow1/2 52.3%

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

       2. unpow1/2 52.3%

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

   19. Simplified 52.3%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left|n\right|} \cdot \sqrt{\left|U \cdot \left(t + {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot U*\right)\right)\right|}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{+147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|n\right|} \cdot \left(\sqrt{\left|U\right|} \cdot \sqrt{\left|\mathsf{fma}\left(n, {\left(\frac{\ell}{Om}\right)}^{2} \cdot U*, t\right)\right|}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\right)}\\ t_3 := \sqrt{2 \cdot \left|n\right|}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;t\_3 \cdot \sqrt{\left|U \cdot \left(t + t\_1 \cdot \left(n \cdot U*\right)\right)\right|}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+147}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \sqrt{\left|n \cdot \left(U \cdot \left(t\_1 \cdot U* + \frac{t}{n}\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l Om) 2.0))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 (/ (* l l) Om))) (* (* n t_1) (- U* U))))))
        (t_3 (sqrt (* 2.0 (fabs n)))))
   (if (<= t_2 0.0)
     (* t_3 (sqrt (fabs (* U (+ t (* t_1 (* n U*)))))))
     (if (<= t_2 2e+147)
       t_2
       (* t_3 (sqrt (fabs (* n (* U (+ (* t_1 U*) (/ t n)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((l / Om), 2.0);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * t_1) * (U_42_ - U)))));
	double t_3 = sqrt((2.0 * fabs(n)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = t_3 * sqrt(fabs((U * (t + (t_1 * (n * U_42_))))));
	} else if (t_2 <= 2e+147) {
		tmp = t_2;
	} else {
		tmp = t_3 * sqrt(fabs((n * (U * ((t_1 * U_42_) + (t / n))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (l / om) ** 2.0d0
    t_2 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) + ((n * t_1) * (u_42 - u)))))
    t_3 = sqrt((2.0d0 * abs(n)))
    if (t_2 <= 0.0d0) then
        tmp = t_3 * sqrt(abs((u * (t + (t_1 * (n * u_42))))))
    else if (t_2 <= 2d+147) then
        tmp = t_2
    else
        tmp = t_3 * sqrt(abs((n * (u * ((t_1 * u_42) + (t / n))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.pow((l / Om), 2.0);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * t_1) * (U_42_ - U)))));
	double t_3 = Math.sqrt((2.0 * Math.abs(n)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = t_3 * Math.sqrt(Math.abs((U * (t + (t_1 * (n * U_42_))))));
	} else if (t_2 <= 2e+147) {
		tmp = t_2;
	} else {
		tmp = t_3 * Math.sqrt(Math.abs((n * (U * ((t_1 * U_42_) + (t / n))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.pow((l / Om), 2.0)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * t_1) * (U_42_ - U)))))
	t_3 = math.sqrt((2.0 * math.fabs(n)))
	tmp = 0
	if t_2 <= 0.0:
		tmp = t_3 * math.sqrt(math.fabs((U * (t + (t_1 * (n * U_42_))))))
	elif t_2 <= 2e+147:
		tmp = t_2
	else:
		tmp = t_3 * math.sqrt(math.fabs((n * (U * ((t_1 * U_42_) + (t / n))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l / Om) ^ 2.0
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * t_1) * Float64(U_42_ - U)))))
	t_3 = sqrt(Float64(2.0 * abs(n)))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(t_3 * sqrt(abs(Float64(U * Float64(t + Float64(t_1 * Float64(n * U_42_)))))));
	elseif (t_2 <= 2e+147)
		tmp = t_2;
	else
		tmp = Float64(t_3 * sqrt(abs(Float64(n * Float64(U * Float64(Float64(t_1 * U_42_) + Float64(t / n)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l / Om) ^ 2.0;
	t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * t_1) * (U_42_ - U)))));
	t_3 = sqrt((2.0 * abs(n)));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = t_3 * sqrt(abs((U * (t + (t_1 * (n * U_42_))))));
	elseif (t_2 <= 2e+147)
		tmp = t_2;
	else
		tmp = t_3 * sqrt(abs((n * (U * ((t_1 * U_42_) + (t / n))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $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] + N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(2.0 * N[Abs[n], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(t$95$3 * N[Sqrt[N[Abs[N[(U * N[(t + N[(t$95$1 * N[(n * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+147], t$95$2, N[(t$95$3 * N[Sqrt[N[Abs[N[(n * N[(U * N[(N[(t$95$1 * U$42$), $MachinePrecision] + N[(t / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.4%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in U* around inf 24.5%

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

      1. mul-1-neg 24.5%

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

      2. associate-/l* 24.5%

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

      3. distribute-rgt-neg-in 24.5%

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

      4. distribute-neg-frac2 24.5%

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

      5. *-commutative 24.5%

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

   7. Simplified 24.5%

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

      1. add-sqr-sqrt 24.5%

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

      2. pow1/2 24.5%

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

      3. pow1/2 24.5%

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

      4. pow-prod-down 17.9%

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

   9. Applied egg-rr 14.7%

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

       1. unpow1/2 14.7%

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

       2. unpow2 14.7%

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

       3. rem-sqrt-square 14.7%

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

       4. associate-*l* 24.5%

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

       5. associate-/l* 24.5%

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

       6. associate-*r/ 24.6%

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

       7. distribute-frac-neg2 24.6%

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

       8. unpow2 24.6%

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

       9. unpow2 24.6%

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

       10. times-frac 28.6%

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

       11. unpow2 28.6%

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

   11. Simplified 28.6%

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

       1. pow1/2 28.6%

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

       2. fabs-mul 28.6%

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

       3. unpow-prod-down 81.7%

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

       4. associate-*r* 81.7%

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

   13. Applied egg-rr 81.7%

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

   15. Simplified 81.7%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2e147

   1. Initial program 99.3%

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

   if 2e147 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 25.5%

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

   3. Simplified 36.1%

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

   5. Taylor expanded in U* around inf 35.0%

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

      1. mul-1-neg 35.0%

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

      2. associate-/l* 35.1%

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

      3. distribute-rgt-neg-in 35.1%

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

      4. distribute-neg-frac2 35.1%

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

      5. *-commutative 35.1%

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

   7. Simplified 35.1%

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

      1. add-sqr-sqrt 35.1%

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

      2. pow1/2 35.1%

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

      3. pow1/2 35.6%

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

      4. pow-prod-down 33.1%

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

   9. Applied egg-rr 30.8%

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

       1. unpow1/2 30.8%

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

       2. unpow2 30.8%

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

       3. rem-sqrt-square 32.6%

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

       4. associate-*l* 35.6%

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

       5. associate-/l* 35.6%

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

       6. associate-*r/ 34.9%

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

       7. distribute-frac-neg2 34.9%

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

       8. unpow2 34.9%

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

       9. unpow2 34.9%

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

       10. times-frac 38.3%

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

       11. unpow2 38.3%

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

   11. Simplified 38.3%

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

       1. pow1/2 38.3%

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

       2. fabs-mul 38.3%

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

       3. unpow-prod-down 44.2%

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

       4. associate-*r* 42.1%

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

   13. Applied egg-rr 42.1%

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

   15. Simplified 42.1%

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

   16. Taylor expanded in n around inf 38.7%

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

       1. associate-/l* 38.5%

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

       2. associate-/l* 38.7%

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

       3. distribute-lft-out 38.7%

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

       4. associate-/l* 40.6%

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

       5. unpow2 40.6%

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

       6. unpow2 40.6%

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

       7. times-frac 46.3%

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

       8. unpow2 46.3%

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

   18. Simplified 46.3%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left|n\right|} \cdot \sqrt{\left|U \cdot \left(t + {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot U*\right)\right)\right|}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|n\right|} \cdot \sqrt{\left|n \cdot \left(U \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot U* + \frac{t}{n}\right)\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left|n\right|}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := n \cdot t\_2\\ t_4 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_3 \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;t\_1 \cdot \sqrt{\left|U \cdot \left(t + t\_2 \cdot \left(n \cdot U*\right)\right)\right|}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{t\_4}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left|U* \cdot \left(U \cdot \left(t\_3 + \frac{t}{U*}\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (fabs n))))
        (t_2 (pow (/ l Om) 2.0))
        (t_3 (* n t_2))
        (t_4
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 (/ (* l l) Om))) (* t_3 (- U* U))))))
   (if (<= t_4 0.0)
     (* t_1 (sqrt (fabs (* U (+ t (* t_2 (* n U*)))))))
     (if (<= t_4 2e+299)
       (sqrt t_4)
       (* t_1 (sqrt (fabs (* U* (* U (+ t_3 (/ t U*)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * fabs(n)));
	double t_2 = pow((l / Om), 2.0);
	double t_3 = n * t_2;
	double t_4 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_3 * (U_42_ - U)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = t_1 * sqrt(fabs((U * (t + (t_2 * (n * U_42_))))));
	} else if (t_4 <= 2e+299) {
		tmp = sqrt(t_4);
	} else {
		tmp = t_1 * sqrt(fabs((U_42_ * (U * (t_3 + (t / 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * abs(n)))
    t_2 = (l / om) ** 2.0d0
    t_3 = n * t_2
    t_4 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) + (t_3 * (u_42 - u)))
    if (t_4 <= 0.0d0) then
        tmp = t_1 * sqrt(abs((u * (t + (t_2 * (n * u_42))))))
    else if (t_4 <= 2d+299) then
        tmp = sqrt(t_4)
    else
        tmp = t_1 * sqrt(abs((u_42 * (u * (t_3 + (t / 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 t_1 = Math.sqrt((2.0 * Math.abs(n)));
	double t_2 = Math.pow((l / Om), 2.0);
	double t_3 = n * t_2;
	double t_4 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_3 * (U_42_ - U)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = t_1 * Math.sqrt(Math.abs((U * (t + (t_2 * (n * U_42_))))));
	} else if (t_4 <= 2e+299) {
		tmp = Math.sqrt(t_4);
	} else {
		tmp = t_1 * Math.sqrt(Math.abs((U_42_ * (U * (t_3 + (t / U_42_))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((2.0 * math.fabs(n)))
	t_2 = math.pow((l / Om), 2.0)
	t_3 = n * t_2
	t_4 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_3 * (U_42_ - U)))
	tmp = 0
	if t_4 <= 0.0:
		tmp = t_1 * math.sqrt(math.fabs((U * (t + (t_2 * (n * U_42_))))))
	elif t_4 <= 2e+299:
		tmp = math.sqrt(t_4)
	else:
		tmp = t_1 * math.sqrt(math.fabs((U_42_ * (U * (t_3 + (t / U_42_))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * abs(n)))
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(n * t_2)
	t_4 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_3 * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(t_1 * sqrt(abs(Float64(U * Float64(t + Float64(t_2 * Float64(n * U_42_)))))));
	elseif (t_4 <= 2e+299)
		tmp = sqrt(t_4);
	else
		tmp = Float64(t_1 * sqrt(abs(Float64(U_42_ * Float64(U * Float64(t_3 + Float64(t / U_42_)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((2.0 * abs(n)));
	t_2 = (l / Om) ^ 2.0;
	t_3 = n * t_2;
	t_4 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_3 * (U_42_ - U)));
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = t_1 * sqrt(abs((U * (t + (t_2 * (n * U_42_))))));
	elseif (t_4 <= 2e+299)
		tmp = sqrt(t_4);
	else
		tmp = t_1 * sqrt(abs((U_42_ * (U * (t_3 + (t / U_42_))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * N[Abs[n], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(n * t$95$2), $MachinePrecision]}, Block[{t$95$4 = 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[(t$95$3 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(t$95$1 * N[Sqrt[N[Abs[N[(U * N[(t + N[(t$95$2 * N[(n * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+299], N[Sqrt[t$95$4], $MachinePrecision], N[(t$95$1 * N[Sqrt[N[Abs[N[(U$42$ * N[(U * N[(t$95$3 + N[(t / U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 14.0%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in U* around inf 22.4%

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

      1. mul-1-neg 22.4%

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

      2. associate-/l* 22.4%

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

      3. distribute-rgt-neg-in 22.4%

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

      4. distribute-neg-frac2 22.4%

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

      5. *-commutative 22.4%

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

   7. Simplified 22.4%

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

      1. add-sqr-sqrt 22.4%

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

      2. pow1/2 22.4%

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

      3. pow1/2 22.4%

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

      4. pow-prod-down 16.4%

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

   9. Applied egg-rr 13.5%

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

       1. unpow1/2 13.5%

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

       2. unpow2 13.5%

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

       3. rem-sqrt-square 13.5%

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

       4. associate-*l* 22.4%

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

       5. associate-/l* 22.4%

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

       6. associate-*r/ 22.5%

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

       7. distribute-frac-neg2 22.5%

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

       8. unpow2 22.5%

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

       9. unpow2 22.5%

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

       10. times-frac 26.5%

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

       11. unpow2 26.5%

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

   11. Simplified 26.5%

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

       1. pow1/2 26.5%

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

       2. fabs-mul 26.5%

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

       3. unpow-prod-down 77.6%

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

       4. associate-*r* 77.6%

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

   13. Applied egg-rr 77.6%

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

   15. Simplified 77.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.0000000000000001e299

   1. Initial program 99.3%

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

   if 2.0000000000000001e299 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 25.5%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in U* around inf 35.4%

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

      1. mul-1-neg 35.4%

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

      2. associate-/l* 35.4%

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

      3. distribute-rgt-neg-in 35.4%

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

      4. distribute-neg-frac2 35.4%

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

      5. *-commutative 35.4%

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

   7. Simplified 35.4%

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

      1. add-sqr-sqrt 35.4%

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

      2. pow1/2 35.4%

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

      3. pow1/2 35.9%

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

      4. pow-prod-down 34.2%

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

   9. Applied egg-rr 31.9%

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

       1. unpow1/2 31.9%

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

       2. unpow2 31.9%

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

       3. rem-sqrt-square 32.8%

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

       4. associate-*l* 35.9%

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

       5. associate-/l* 36.0%

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

       6. associate-*r/ 35.2%

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

       7. distribute-frac-neg2 35.2%

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

       8. unpow2 35.2%

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

       9. unpow2 35.2%

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

       10. times-frac 38.6%

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

       11. unpow2 38.6%

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

   11. Simplified 38.6%

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

       1. pow1/2 38.6%

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

       2. fabs-mul 38.6%

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

       3. unpow-prod-down 43.9%

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

       4. associate-*r* 41.7%

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

   13. Applied egg-rr 41.7%

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

   15. Simplified 41.7%

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

   16. Taylor expanded in U* around inf 40.2%

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

       1. associate-/l* 39.2%

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

       2. associate-/l* 38.4%

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

       3. distribute-lft-out 38.4%

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

       4. *-commutative 38.4%

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

       5. associate-/l* 38.5%

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

       6. unpow2 38.5%

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

       7. unpow2 38.5%

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

       8. times-frac 42.9%

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

       9. unpow2 42.9%

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

   18. Simplified 42.9%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left|n\right|} \cdot \sqrt{\left|U \cdot \left(t + {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot U*\right)\right)\right|}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|n\right|} \cdot \sqrt{\left|U* \cdot \left(U \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2} + \frac{t}{U*}\right)\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.8% accurate, 0.3× speedup

Math

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

FPCore

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.4%

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

   3. Simplified 25.1%

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

      1. associate-*r* 28.6%

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

      2. sub-neg 28.6%

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

      3. distribute-lft-in 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. distribute-lft-out 28.6%

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

      2. sub-neg 28.6%

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

   8. Simplified 28.6%

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

   9. Taylor expanded in t around inf 25.3%

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

       1. pow1/2 25.3%

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

       2. *-commutative 25.3%

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

       3. unpow-prod-down 43.5%

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

       4. pow1/2 43.5%

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

       5. pow1/2 43.5%

          \[\leadsto \sqrt{U \cdot t} \cdot \color{blue}{\sqrt{2 \cdot n}} \]

   11. Applied egg-rr 43.5%

       \[\leadsto \color{blue}{\sqrt{U \cdot t} \cdot \sqrt{2 \cdot n}} \]

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

   1. Initial program 75.5%

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

   3. Simplified 78.6%

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

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

   1. Initial program 0.0%

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

   3. Simplified 6.2%

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

   5. Taylor expanded in U* around inf 22.4%

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

      1. mul-1-neg 22.4%

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

      2. associate-/l* 22.4%

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

      3. distribute-rgt-neg-in 22.4%

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

      4. distribute-neg-frac2 22.4%

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

      5. *-commutative 22.4%

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

   7. Simplified 22.4%

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

      1. add-sqr-sqrt 22.4%

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

      2. pow1/2 22.4%

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

      3. pow1/2 23.7%

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

      4. pow-prod-down 23.8%

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

   9. Applied egg-rr 22.4%

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

       1. unpow1/2 22.4%

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

       2. unpow2 22.4%

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

       3. rem-sqrt-square 22.4%

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

       4. associate-*l* 23.8%

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

       5. associate-/l* 23.8%

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

       6. associate-*r/ 23.7%

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

       7. distribute-frac-neg2 23.7%

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

       8. unpow2 23.7%

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

       9. unpow2 23.7%

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

       10. times-frac 27.1%

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

       11. unpow2 27.1%

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

   11. Simplified 27.1%

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

   12. Taylor expanded in U around 0 23.8%

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

       1. associate-*r* 22.4%

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

       2. sub-neg 22.4%

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

       3. mul-1-neg 22.4%

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

       4. remove-double-neg 22.4%

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

       5. associate-/l* 22.4%

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

       6. *-commutative 22.4%

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

       7. associate-/l* 22.4%

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

       8. unpow2 22.4%

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

       9. unpow2 22.4%

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

       10. times-frac 25.7%

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

       11. unpow2 25.7%

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

   14. Simplified 25.7%

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

4. Final simplification 66.7%

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

Alternative 5: 61.1% accurate, 0.3× speedup

Math

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

FPCore

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.4%

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

   3. Simplified 25.1%

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

      1. associate-*r* 28.6%

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

      2. sub-neg 28.6%

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

      3. distribute-lft-in 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. distribute-lft-out 28.6%

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

      2. sub-neg 28.6%

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

   8. Simplified 28.6%

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

   9. Taylor expanded in t around inf 25.3%

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

       1. pow1/2 25.3%

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

       2. *-commutative 25.3%

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

       3. unpow-prod-down 43.5%

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

       4. pow1/2 43.5%

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

       5. pow1/2 43.5%

          \[\leadsto \sqrt{U \cdot t} \cdot \color{blue}{\sqrt{2 \cdot n}} \]

   11. Applied egg-rr 43.5%

       \[\leadsto \color{blue}{\sqrt{U \cdot t} \cdot \sqrt{2 \cdot n}} \]

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

   1. Initial program 75.5%

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

   3. Simplified 78.6%

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

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

   1. Initial program 0.0%

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

   3. Simplified 6.2%

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

   5. Taylor expanded in U* around inf 22.4%

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

      1. mul-1-neg 22.4%

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

      2. associate-/l* 22.4%

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

      3. distribute-rgt-neg-in 22.4%

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

      4. distribute-neg-frac2 22.4%

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

      5. *-commutative 22.4%

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

   7. Simplified 22.4%

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

      1. add-sqr-sqrt 22.4%

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

      2. pow1/2 22.4%

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

      3. pow1/2 23.7%

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

      4. pow-prod-down 23.8%

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

   9. Applied egg-rr 22.4%

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

       1. unpow1/2 22.4%

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

       2. unpow2 22.4%

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

       3. rem-sqrt-square 22.4%

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

       4. associate-*l* 23.8%

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

       5. associate-/l* 23.8%

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

       6. associate-*r/ 23.7%

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

       7. distribute-frac-neg2 23.7%

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

       8. unpow2 23.7%

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

       9. unpow2 23.7%

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

       10. times-frac 27.1%

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

       11. unpow2 27.1%

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

   11. Simplified 27.1%

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

       1. pow1 27.1%

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

       2. associate-*r* 25.7%

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

       3. associate-*r* 25.4%

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

   13. Applied egg-rr 25.4%

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

   15. Simplified 27.1%

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

4. Final simplification 67.0%

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

Alternative 6: 60.9% accurate, 0.3× speedup

Math

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

FPCore

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

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);
	double t_2 = t_1 * (U_42_ - U);
	double t_3 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((U * t)) * sqrt((2.0 * n));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t + (t_1 * U_42_)))));
	}
	return tmp;
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.4%

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

   3. Simplified 25.1%

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

      1. associate-*r* 28.6%

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

      2. sub-neg 28.6%

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

      3. distribute-lft-in 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. distribute-lft-out 28.6%

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

      2. sub-neg 28.6%

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

   8. Simplified 28.6%

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

   9. Taylor expanded in t around inf 25.3%

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

       1. pow1/2 25.3%

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

       2. *-commutative 25.3%

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

       3. unpow-prod-down 43.5%

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

       4. pow1/2 43.5%

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

       5. pow1/2 43.5%

          \[\leadsto \sqrt{U \cdot t} \cdot \color{blue}{\sqrt{2 \cdot n}} \]

   11. Applied egg-rr 43.5%

       \[\leadsto \color{blue}{\sqrt{U \cdot t} \cdot \sqrt{2 \cdot n}} \]

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

   1. Initial program 75.5%

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

   3. Simplified 78.6%

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

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

   1. Initial program 0.0%

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

   3. Simplified 6.2%

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

   5. Taylor expanded in U* around inf 22.4%

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

      1. mul-1-neg 22.4%

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

      2. associate-/l* 22.4%

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

      3. distribute-rgt-neg-in 22.4%

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

      4. distribute-neg-frac2 22.4%

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

      5. *-commutative 22.4%

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

   7. Simplified 22.4%

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

      1. add-sqr-sqrt 22.4%

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

      2. pow1/2 22.4%

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

      3. pow1/2 23.7%

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

      4. pow-prod-down 23.8%

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

   9. Applied egg-rr 22.4%

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

       1. unpow1/2 22.4%

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

       2. unpow2 22.4%

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

       3. rem-sqrt-square 22.4%

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

       4. associate-*l* 23.8%

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

       5. associate-/l* 23.8%

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

       6. associate-*r/ 23.7%

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

       7. distribute-frac-neg2 23.7%

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

       8. unpow2 23.7%

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

       9. unpow2 23.7%

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

       10. times-frac 27.1%

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

       11. unpow2 27.1%

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

   11. Simplified 27.1%

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

       1. *-un-lft-identity 27.1%

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

       2. associate-*r* 25.7%

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

       3. associate-*r* 25.4%

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

   13. Applied egg-rr 25.4%

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

   15. Simplified 25.7%

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

4. Final simplification 66.7%

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

Alternative 7: 66.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left|n\right|} \cdot \sqrt{\left|U \cdot t\right|}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\right)}^{0.3333333333333333}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_2 (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) t_1))))
   (if (<= t_2 0.0)
     (* (sqrt (* 2.0 (fabs n))) (sqrt (fabs (* U t))))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l (/ l Om)))))))
       (pow
        (pow
         (* (sqrt (* U U*)) (* l (* n (/ (sqrt 2.0) Om))))
         0.3333333333333333)
        3.0)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((2.0 * fabs(n))) * sqrt(fabs((U * t)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = pow(pow((sqrt((U * U_42_)) * (l * (n * (sqrt(2.0) / Om)))), 0.3333333333333333), 3.0);
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt((2.0 * Math.abs(n))) * Math.sqrt(Math.abs((U * t)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.pow(Math.pow((Math.sqrt((U * U_42_)) * (l * (n * (Math.sqrt(2.0) / Om)))), 0.3333333333333333), 3.0);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1)
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt((2.0 * math.fabs(n))) * math.sqrt(math.fabs((U * t)))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.pow(math.pow((math.sqrt((U * U_42_)) * (l * (n * (math.sqrt(2.0) / Om)))), 0.3333333333333333), 3.0)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_1))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * abs(n))) * sqrt(abs(Float64(U * t))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = (Float64(sqrt(Float64(U * U_42_)) * Float64(l * Float64(n * Float64(sqrt(2.0) / Om)))) ^ 0.3333333333333333) ^ 3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt((2.0 * abs(n))) * sqrt(abs((U * t)));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	else
		tmp = ((sqrt((U * U_42_)) * (l * (n * (sqrt(2.0) / Om)))) ^ 0.3333333333333333) ^ 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(2.0 * N[Abs[n], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[N[(U * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[Power[N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(l * N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 3.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 14.0%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in U* around inf 22.4%

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

      1. mul-1-neg 22.4%

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

      2. associate-/l* 22.4%

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

      3. distribute-rgt-neg-in 22.4%

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

      4. distribute-neg-frac2 22.4%

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

      5. *-commutative 22.4%

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

   7. Simplified 22.4%

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

      1. add-sqr-sqrt 22.4%

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

      2. pow1/2 22.4%

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

      3. pow1/2 22.4%

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

      4. pow-prod-down 16.4%

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

   9. Applied egg-rr 13.5%

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

       1. unpow1/2 13.5%

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

       2. unpow2 13.5%

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

       3. rem-sqrt-square 13.5%

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

       4. associate-*l* 22.4%

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

       5. associate-/l* 22.4%

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

       6. associate-*r/ 22.5%

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

       7. distribute-frac-neg2 22.5%

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

       8. unpow2 22.5%

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

       9. unpow2 22.5%

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

       10. times-frac 26.5%

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

       11. unpow2 26.5%

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

   11. Simplified 26.5%

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

       1. pow1/2 26.5%

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

       2. fabs-mul 26.5%

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

       3. unpow-prod-down 77.6%

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

       4. associate-*r* 77.6%

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

   13. Applied egg-rr 77.6%

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

   15. Simplified 77.6%

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

   16. Taylor expanded in t around inf 72.0%

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

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

   1. Initial program 75.5%

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

   3. Simplified 78.6%

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

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

   1. Initial program 0.0%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in U* around inf 21.2%

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

      1. add-cube-cbrt 21.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}} \cdot \sqrt[3]{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}}\right) \cdot \sqrt[3]{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}}} \]

      2. pow3 21.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}}\right)}^{3}} \]

      3. *-commutative 21.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\sqrt{U \cdot U*} \cdot \frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om}}}\right)}^{3} \]

      4. associate-/l* 21.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{U \cdot U*} \cdot \color{blue}{\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right)}}\right)}^{3} \]

   7. Applied egg-rr 21.2%

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

      1. pow1/3 26.5%

         \[\leadsto {\color{blue}{\left({\left(\sqrt{U \cdot U*} \cdot \left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\right)}^{0.3333333333333333}\right)}}^{3} \]

      2. associate-/l* 26.5%

         \[\leadsto {\left({\left(\sqrt{U \cdot U*} \cdot \left(\ell \cdot \color{blue}{\left(n \cdot \frac{\sqrt{2}}{Om}\right)}\right)\right)}^{0.3333333333333333}\right)}^{3} \]

   9. Applied egg-rr 26.5%

      \[\leadsto {\color{blue}{\left({\left(\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\right)}^{0.3333333333333333}\right)}}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left|n\right|} \cdot \sqrt{\left|U \cdot t\right|}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\right)}^{0.3333333333333333}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\\ t_3 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_2\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left|n\right|} \cdot \sqrt{\left|U \cdot \left(t + t\_1 \cdot \left(n \cdot U*\right)\right)\right|}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_2 - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\right)}^{0.3333333333333333}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l Om) 2.0))
        (t_2 (* (* n t_1) (- U* U)))
        (t_3 (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) t_2))))
   (if (<= t_3 0.0)
     (* (sqrt (* 2.0 (fabs n))) (sqrt (fabs (* U (+ t (* t_1 (* n U*)))))))
     (if (<= t_3 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_2 (* 2.0 (* l (/ l Om)))))))
       (pow
        (pow
         (* (sqrt (* U U*)) (* l (* n (/ (sqrt 2.0) Om))))
         0.3333333333333333)
        3.0)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((l / Om), 2.0);
	double t_2 = (n * t_1) * (U_42_ - U);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_2);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * fabs(n))) * sqrt(fabs((U * (t + (t_1 * (n * U_42_))))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = pow(pow((sqrt((U * U_42_)) * (l * (n * (sqrt(2.0) / Om)))), 0.3333333333333333), 3.0);
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.pow((l / Om), 2.0);
	double t_2 = (n * t_1) * (U_42_ - U);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_2);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt((2.0 * Math.abs(n))) * Math.sqrt(Math.abs((U * (t + (t_1 * (n * U_42_))))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.pow(Math.pow((Math.sqrt((U * U_42_)) * (l * (n * (Math.sqrt(2.0) / Om)))), 0.3333333333333333), 3.0);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.pow((l / Om), 2.0)
	t_2 = (n * t_1) * (U_42_ - U)
	t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_2)
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.sqrt((2.0 * math.fabs(n))) * math.sqrt(math.fabs((U * (t + (t_1 * (n * U_42_))))))
	elif t_3 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.pow(math.pow((math.sqrt((U * U_42_)) * (l * (n * (math.sqrt(2.0) / Om)))), 0.3333333333333333), 3.0)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l / Om) ^ 2.0
	t_2 = Float64(Float64(n * t_1) * Float64(U_42_ - U))
	t_3 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_2))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * abs(n))) * sqrt(abs(Float64(U * Float64(t + Float64(t_1 * Float64(n * U_42_)))))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_2 - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = (Float64(sqrt(Float64(U * U_42_)) * Float64(l * Float64(n * Float64(sqrt(2.0) / Om)))) ^ 0.3333333333333333) ^ 3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l / Om) ^ 2.0;
	t_2 = (n * t_1) * (U_42_ - U);
	t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_2);
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = sqrt((2.0 * abs(n))) * sqrt(abs((U * (t + (t_1 * (n * U_42_))))));
	elseif (t_3 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l * (l / Om)))))));
	else
		tmp = ((sqrt((U * U_42_)) * (l * (n * (sqrt(2.0) / Om)))) ^ 0.3333333333333333) ^ 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(2.0 * N[Abs[n], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[N[(U * N[(t + N[(t$95$1 * N[(n * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$2 - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[Power[N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(l * N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 3.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 14.0%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in U* around inf 22.4%

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

      1. mul-1-neg 22.4%

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

      2. associate-/l* 22.4%

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

      3. distribute-rgt-neg-in 22.4%

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

      4. distribute-neg-frac2 22.4%

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

      5. *-commutative 22.4%

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

   7. Simplified 22.4%

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

      1. add-sqr-sqrt 22.4%

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

      2. pow1/2 22.4%

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

      3. pow1/2 22.4%

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

      4. pow-prod-down 16.4%

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

   9. Applied egg-rr 13.5%

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

       1. unpow1/2 13.5%

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

       2. unpow2 13.5%

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

       3. rem-sqrt-square 13.5%

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

       4. associate-*l* 22.4%

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

       5. associate-/l* 22.4%

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

       6. associate-*r/ 22.5%

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

       7. distribute-frac-neg2 22.5%

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

       8. unpow2 22.5%

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

       9. unpow2 22.5%

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

       10. times-frac 26.5%

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

       11. unpow2 26.5%

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

   11. Simplified 26.5%

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

       1. pow1/2 26.5%

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

       2. fabs-mul 26.5%

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

       3. unpow-prod-down 77.6%

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

       4. associate-*r* 77.6%

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

   13. Applied egg-rr 77.6%

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

   15. Simplified 77.6%

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

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

   1. Initial program 75.5%

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

   3. Simplified 78.6%

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

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

   1. Initial program 0.0%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in U* around inf 21.2%

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

      1. add-cube-cbrt 21.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}} \cdot \sqrt[3]{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}}\right) \cdot \sqrt[3]{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}}} \]

      2. pow3 21.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}}\right)}^{3}} \]

      3. *-commutative 21.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\sqrt{U \cdot U*} \cdot \frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om}}}\right)}^{3} \]

      4. associate-/l* 21.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{U \cdot U*} \cdot \color{blue}{\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right)}}\right)}^{3} \]

   7. Applied egg-rr 21.2%

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

      1. pow1/3 26.5%

         \[\leadsto {\color{blue}{\left({\left(\sqrt{U \cdot U*} \cdot \left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\right)}^{0.3333333333333333}\right)}}^{3} \]

      2. associate-/l* 26.5%

         \[\leadsto {\left({\left(\sqrt{U \cdot U*} \cdot \left(\ell \cdot \color{blue}{\left(n \cdot \frac{\sqrt{2}}{Om}\right)}\right)\right)}^{0.3333333333333333}\right)}^{3} \]

   9. Applied egg-rr 26.5%

      \[\leadsto {\color{blue}{\left({\left(\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\right)}^{0.3333333333333333}\right)}}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left|n\right|} \cdot \sqrt{\left|U \cdot \left(t + {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot U*\right)\right)\right|}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\right)}^{0.3333333333333333}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 0.4× speedup

Math

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

FPCore

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 14.0%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in U* around inf 22.4%

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

      1. mul-1-neg 22.4%

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

      2. associate-/l* 22.4%

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

      3. distribute-rgt-neg-in 22.4%

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

      4. distribute-neg-frac2 22.4%

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

      5. *-commutative 22.4%

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

   7. Simplified 22.4%

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

      1. add-sqr-sqrt 22.4%

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

      2. pow1/2 22.4%

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

      3. pow1/2 22.4%

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

      4. pow-prod-down 16.4%

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

   9. Applied egg-rr 13.5%

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

       1. unpow1/2 13.5%

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

       2. unpow2 13.5%

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

       3. rem-sqrt-square 13.5%

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

       4. associate-*l* 22.4%

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

       5. associate-/l* 22.4%

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

       6. associate-*r/ 22.5%

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

       7. distribute-frac-neg2 22.5%

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

       8. unpow2 22.5%

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

       9. unpow2 22.5%

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

       10. times-frac 26.5%

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

       11. unpow2 26.5%

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

   11. Simplified 26.5%

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

       1. pow1/2 26.5%

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

       2. fabs-mul 26.5%

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

       3. unpow-prod-down 77.6%

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

       4. associate-*r* 77.6%

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

   13. Applied egg-rr 77.6%

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

   15. Simplified 77.6%

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

   16. Taylor expanded in t around inf 72.0%

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

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

   1. Initial program 75.5%

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

   3. Simplified 78.6%

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

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

   1. Initial program 0.0%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in U* around inf 24.2%

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

      1. mul-1-neg 24.2%

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

      2. associate-/l* 24.2%

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

      3. distribute-rgt-neg-in 24.2%

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

      4. distribute-neg-frac2 24.2%

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

      5. *-commutative 24.2%

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

   7. Simplified 24.2%

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

      1. add-sqr-sqrt 24.2%

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

      2. pow1/2 24.2%

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

      3. pow1/2 25.5%

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

      4. pow-prod-down 25.7%

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

   9. Applied egg-rr 24.2%

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

       1. unpow1/2 24.2%

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

       2. unpow2 24.2%

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

       3. rem-sqrt-square 24.2%

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

       4. associate-*l* 25.7%

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

       5. associate-/l* 25.7%

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

       6. associate-*r/ 25.6%

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

       7. distribute-frac-neg2 25.6%

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

       8. unpow2 25.6%

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

       9. unpow2 25.6%

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

       10. times-frac 28.9%

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

       11. unpow2 28.9%

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

   11. Simplified 28.9%

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

       1. pow1 28.9%

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

       2. associate-*r* 24.6%

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

       3. associate-*r* 24.3%

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

   13. Applied egg-rr 24.3%

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

   15. Simplified 28.9%

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

4. Final simplification 71.0%

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

Alternative 10: 54.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -1 \cdot 10^{+19} \lor \neg \left(Om \leq 2.9 \cdot 10^{-102}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -1e+19) (not (<= Om 2.9e-102)))
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l (/ l Om))))))))
   (sqrt (* (* 2.0 n) (* U (+ t (* (* n (pow (/ l Om) 2.0)) U*)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1e+19) || !(Om <= 2.9e-102)) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t + ((n * pow((l / Om), 2.0)) * 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 <= (-1d+19)) .or. (.not. (om <= 2.9d-102))) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l * (l / om))))))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((n * ((l / om) ** 2.0d0)) * 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 <= -1e+19) || !(Om <= 2.9e-102)) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((n * Math.pow((l / Om), 2.0)) * U_42_)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (Om <= -1e+19) or not (Om <= 2.9e-102):
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((n * math.pow((l / Om), 2.0)) * U_42_)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -1e+19) || !(Om <= 2.9e-102))
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om))))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * U_42_)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -1e+19) || ~((Om <= 2.9e-102)))
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t + ((n * ((l / Om) ^ 2.0)) * U_42_)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -1e+19], N[Not[LessEqual[Om, 2.9e-102]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1e19 or 2.89999999999999986e-102 < Om

   1. Initial program 62.2%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in n around 0 57.1%

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

      1. unpow2 57.1%

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

      2. associate-*r/ 61.2%

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

      3. *-commutative 61.2%

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

   7. Applied egg-rr 61.2%

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

   if -1e19 < Om < 2.89999999999999986e-102

   1. Initial program 48.7%

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

   3. Simplified 47.6%

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

   5. Taylor expanded in U* around inf 47.5%

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

      1. mul-1-neg 47.5%

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

      2. associate-/l* 47.6%

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

      3. distribute-rgt-neg-in 47.6%

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

      4. distribute-neg-frac2 47.6%

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

      5. *-commutative 47.6%

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

   7. Simplified 47.6%

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

      1. add-sqr-sqrt 47.6%

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

      2. pow1/2 47.6%

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

      3. pow1/2 47.7%

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

      4. pow-prod-down 38.3%

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

   9. Applied egg-rr 37.9%

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

       1. unpow1/2 37.9%

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

       2. unpow2 37.9%

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

       3. rem-sqrt-square 47.6%

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

       4. associate-*l* 48.0%

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

       5. associate-/l* 48.0%

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

       6. associate-*r/ 47.6%

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

       7. distribute-frac-neg2 47.6%

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

       8. unpow2 47.6%

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

       9. unpow2 47.6%

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

       10. times-frac 55.8%

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

       11. unpow2 55.8%

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

   11. Simplified 55.8%

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

       1. *-un-lft-identity 55.8%

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

       2. associate-*r* 56.5%

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

       3. associate-*r* 57.4%

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

   13. Applied egg-rr 57.4%

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

   15. Simplified 55.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1 \cdot 10^{+19} \lor \neg \left(Om \leq 2.9 \cdot 10^{-102}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U*\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq 6.8 \cdot 10^{+249}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U \cdot U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 6.8e+249)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l (/ l Om))))))))
   (sqrt (* (* 2.0 n) (* (* n (pow (/ l Om) 2.0)) (* U U*))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 6.8e+249) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else {
		tmp = sqrt(((2.0 * n) * ((n * pow((l / Om), 2.0)) * (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 (u_42 <= 6.8d+249) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l * (l / om))))))))
    else
        tmp = sqrt(((2.0d0 * n) * ((n * ((l / om) ** 2.0d0)) * (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 (U_42_ <= 6.8e+249) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * ((n * Math.pow((l / Om), 2.0)) * (U * U_42_))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 6.8e+249:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))))
	else:
		tmp = math.sqrt(((2.0 * n) * ((n * math.pow((l / Om), 2.0)) * (U * U_42_))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 6.8e+249)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om))))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U * U_42_))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 6.8e+249)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	else
		tmp = sqrt(((2.0 * n) * ((n * ((l / Om) ^ 2.0)) * (U * U_42_))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 6.8e+249], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 6.80000000000000026e249

   1. Initial program 56.3%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in n around 0 49.8%

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

      1. unpow2 49.8%

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

      2. associate-*r/ 52.7%

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

      3. *-commutative 52.7%

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

   7. Applied egg-rr 52.7%

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

   if 6.80000000000000026e249 < U*

   1. Initial program 75.9%

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

   3. Simplified 69.8%

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

      1. associate-*r* 69.9%

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

      2. sub-neg 69.9%

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

      3. distribute-lft-in 19.6%

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

   6. Applied egg-rr 19.6%

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

      1. distribute-lft-out 69.9%

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

      2. sub-neg 69.9%

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

   8. Simplified 69.9%

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

   9. Taylor expanded in U* around inf 69.6%

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

       1. associate-/l* 69.6%

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

       2. *-commutative 69.6%

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

   11. Applied egg-rr 69.6%

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

       1. associate-*r/ 69.6%

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

       2. associate-*l* 69.8%

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

       3. associate-/l* 70.0%

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

       4. associate-/l* 70.0%

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

       5. unpow2 70.0%

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

       6. unpow2 70.0%

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

       7. times-frac 73.7%

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

       8. unpow2 73.7%

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

   13. Simplified 73.7%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 6.8 \cdot 10^{+249}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U \cdot U*\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.42 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{n \cdot \left(2 \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.42e+162)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l (/ l Om))))))))
   (* (sqrt t) (sqrt (* n (* 2.0 U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.42e+162) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else {
		tmp = sqrt(t) * sqrt((n * (2.0 * U)));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 1.42d+162) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l * (l / om))))))))
    else
        tmp = sqrt(t) * sqrt((n * (2.0d0 * u)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.42e+162) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt((n * (2.0 * U)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 1.42e+162:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))))
	else:
		tmp = math.sqrt(t) * math.sqrt((n * (2.0 * U)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.42e+162)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om))))))));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(n * Float64(2.0 * U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 1.42e+162)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	else
		tmp = sqrt(t) * sqrt((n * (2.0 * U)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.42e+162], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.4199999999999999e162

   1. Initial program 59.0%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in n around 0 49.6%

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

      1. unpow2 49.6%

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

      2. associate-*r/ 52.7%

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

      3. *-commutative 52.7%

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

   7. Applied egg-rr 52.7%

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

   if 1.4199999999999999e162 < t

   1. Initial program 46.4%

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

   3. Simplified 42.2%

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

   6. Applied egg-rr 58.4%

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

      1. associate-*r/ 58.4%

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

      2. *-commutative 58.4%

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

      3. *-commutative 58.4%

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

   8. Simplified 58.4%

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

   9. Taylor expanded in t around inf 61.8%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.42 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{n \cdot \left(2 \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.5e-94)
   (sqrt (* (* (* 2.0 n) U) t))
   (sqrt (* 2.0 (* U (* n (- 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 <= 4.5e-94) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt((2.0 * (U * (n * (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 <= 4.5d-94) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt((2.0d0 * (u * (n * (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 <= 4.5e-94) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.5e-94:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.5e-94)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * 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 <= 4.5e-94)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.5e-94], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.5000000000000002e-94

   1. Initial program 68.1%

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

   3. Taylor expanded in t around inf 54.2%

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

   if 4.5000000000000002e-94 < l

   1. Initial program 40.1%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in n around 0 36.6%

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

      1. unpow2 36.6%

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

      2. associate-*r/ 42.7%

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

      3. *-commutative 42.7%

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

   7. Applied egg-rr 42.7%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.4 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.4e-89)
   (sqrt (* (* (* 2.0 n) U) t))
   (pow (* (* 2.0 U) (* n t)) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.4e-89) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = pow(((2.0 * U) * (n * t)), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 6.4d-89) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = ((2.0d0 * u) * (n * t)) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.4e-89) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.pow(((2.0 * U) * (n * t)), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.4e-89:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.pow(((2.0 * U) * (n * t)), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.4e-89)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = Float64(Float64(2.0 * U) * Float64(n * t)) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6.4e-89)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = ((2.0 * U) * (n * t)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.4e-89], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.39999999999999997e-89

   1. Initial program 67.7%

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

   3. Taylor expanded in t around inf 53.9%

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

   if 6.39999999999999997e-89 < l

   1. Initial program 40.5%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in t around inf 26.4%

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

      1. pow1/2 27.4%

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

      2. associate-*r* 27.4%

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

   7. Applied egg-rr 27.4%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.4 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 9.5e-87) (sqrt (* (* (* 2.0 n) U) t)) (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 <= 9.5e-87) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} 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 <= 9.5d-87) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    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 <= 9.5e-87) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} 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 <= 9.5e-87:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	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 <= 9.5e-87)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	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 <= 9.5e-87)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 9.5e-87], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $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 < 9.5e-87

   1. Initial program 67.7%

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

   3. Taylor expanded in t around inf 53.9%

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

   if 9.5e-87 < l

   1. Initial program 40.5%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in t around inf 26.4%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 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 57.5%

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

3. Simplified 55.5%

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

5. Taylor expanded in t around inf 41.2%

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

6. Final simplification 41.2%

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

Reproduce

herbie shell --seed 2024110 
(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*))))))