Toniolo and Linder, Equation (13)

Percentage Accurate: 49.5% → 71.5%

Time: 23.5s

Alternatives: 13

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

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

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 (* (* n U*) t_1))))))
     (if (<= t_2 1e+147)
       t_2
       (* t_3 (* (sqrt (fabs U)) (sqrt (fabs (fma n (* U* t_1) 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 + ((n * U_42_) * t_1)))));
	} else if (t_2 <= 1e+147) {
		tmp = t_2;
	} else {
		tmp = t_3 * (sqrt(fabs(U)) * sqrt(fabs(fma(n, (U_42_ * t_1), 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(Float64(n * U_42_) * t_1))))));
	elseif (t_2 <= 1e+147)
		tmp = t_2;
	else
		tmp = Float64(t_3 * Float64(sqrt(abs(U)) * sqrt(abs(fma(n, Float64(U_42_ * t_1), 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[(N[(n * U$42$), $MachinePrecision] * t$95$1), $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[(U$42$ * t$95$1), $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%

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

Alternative 2: 65.7% accurate, 0.2× speedup

Math

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

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)))))))
   (if (or (<= t_2 0.0) (not (<= t_2 2e+147)))
     (* (sqrt (* 2.0 (fabs n))) (sqrt (fabs (* U (+ t (* (* n U*) t_1))))))
     t_2)))

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 tmp;
	if ((t_2 <= 0.0) || !(t_2 <= 2e+147)) {
		tmp = sqrt((2.0 * fabs(n))) * sqrt(fabs((U * (t + ((n * U_42_) * t_1)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (l / om) ** 2.0d0
    t_2 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) + ((n * t_1) * (u_42 - u)))))
    if ((t_2 <= 0.0d0) .or. (.not. (t_2 <= 2d+147))) then
        tmp = sqrt((2.0d0 * abs(n))) * sqrt(abs((u * (t + ((n * u_42) * t_1)))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.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 tmp;
	if ((t_2 <= 0.0) || !(t_2 <= 2e+147)) {
		tmp = Math.sqrt((2.0 * Math.abs(n))) * Math.sqrt(Math.abs((U * (t + ((n * U_42_) * t_1)))));
	} else {
		tmp = t_2;
	}
	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)))))
	tmp = 0
	if (t_2 <= 0.0) or not (t_2 <= 2e+147):
		tmp = math.sqrt((2.0 * math.fabs(n))) * math.sqrt(math.fabs((U * (t + ((n * U_42_) * t_1)))))
	else:
		tmp = t_2
	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)))))
	tmp = 0.0
	if ((t_2 <= 0.0) || !(t_2 <= 2e+147))
		tmp = Float64(sqrt(Float64(2.0 * abs(n))) * sqrt(abs(Float64(U * Float64(t + Float64(Float64(n * U_42_) * t_1))))));
	else
		tmp = t_2;
	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)))));
	tmp = 0.0;
	if ((t_2 <= 0.0) || ~((t_2 <= 2e+147)))
		tmp = sqrt((2.0 * abs(n))) * sqrt(abs((U * (t + ((n * U_42_) * t_1)))));
	else
		tmp = t_2;
	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]}, If[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, 2e+147]], $MachinePrecision]], N[(N[Sqrt[N[(2.0 * N[Abs[n], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[N[(U * N[(t + N[(N[(n * U$42$), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]

Derivation

1. Split input into 2 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 or 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 23.4%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in U* around inf 32.9%

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

      1. mul-1-neg 32.9%

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

      2. associate-/l* 32.9%

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

      3. distribute-rgt-neg-in 32.9%

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

      4. distribute-neg-frac2 32.9%

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

      5. *-commutative 32.9%

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

   7. Simplified 32.9%

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

      1. add-sqr-sqrt 32.9%

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

      2. pow1/2 32.9%

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

      3. pow1/2 33.3%

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

      4. pow-prod-down 30.0%

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

   9. Applied egg-rr 27.5%

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

       1. unpow1/2 27.5%

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

       2. unpow2 27.5%

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

       3. rem-sqrt-square 28.9%

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

       4. associate-*l* 33.3%

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

       5. associate-/l* 33.4%

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

       6. associate-*r/ 32.8%

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

       7. distribute-frac-neg2 32.8%

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

       8. unpow2 32.8%

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

       9. unpow2 32.8%

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

       10. times-frac 36.3%

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

       11. unpow2 36.3%

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

   11. Simplified 36.3%

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

       1. pow1/2 36.3%

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

       2. fabs-mul 36.3%

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

       3. unpow-prod-down 52.0%

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

       4. associate-*r* 50.3%

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

   13. Applied egg-rr 50.3%

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

   15. Simplified 50.3%

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

   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
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

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

Alternative 3: 61.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((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.sqrt((2.0 * (((n * U) * (Math.pow(l, 2.0) * (-2.0 + (n * ((U_42_ - U) / Om))))) / Om)));
	}
	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 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1)))
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((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.sqrt((2.0 * (((n * U) * (math.pow(l, 2.0) * (-2.0 + (n * ((U_42_ - U) / Om))))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_1)))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(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 = sqrt(Float64(2.0 * Float64(Float64(Float64(n * U) * Float64((l ^ 2.0) * Float64(-2.0 + Float64(n * Float64(Float64(U_42_ - U) / Om))))) / Om)));
	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 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((2.0 * (((n * U) * ((l ^ 2.0) * (-2.0 + (n * ((U_42_ - U) / Om))))) / Om)));
	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[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $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[Sqrt[N[(2.0 * N[(N[(N[(n * U), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[(-2.0 + N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $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 Om around -inf 8.6%

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

      1. mul-1-neg 8.6%

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

      2. distribute-neg-frac2 8.6%

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

      3. mul-1-neg 8.6%

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

      4. unsub-neg 8.6%

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

      5. *-commutative 8.6%

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

      6. associate-/l* 8.6%

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

   7. Simplified 8.6%

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

   8. Taylor expanded in t around 0 18.8%

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

      1. associate-*r* 13.4%

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

      2. sub-neg 13.4%

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

      3. *-commutative 13.4%

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

      4. associate-/l* 13.4%

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

      5. associate-*r/ 13.4%

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

      6. distribute-rgt-neg-in 13.4%

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

      7. distribute-lft-in 27.8%

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

      8. sub-neg 27.8%

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

   10. Simplified 27.8%

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

4. Final simplification 67.1%

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

Alternative 4: 65.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- 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)))))))
       (sqrt
        (*
         -2.0
         (*
          (* U (pow l 2.0))
          (* n (- (/ 2.0 Om) (/ (* n (- U* U)) (pow Om 2.0)))))))))))

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 = sqrt((-2.0 * ((U * pow(l, 2.0)) * (n * ((2.0 / Om) - ((n * (U_42_ - U)) / pow(Om, 2.0)))))));
	}
	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.sqrt((-2.0 * ((U * Math.pow(l, 2.0)) * (n * ((2.0 / Om) - ((n * (U_42_ - U)) / Math.pow(Om, 2.0)))))));
	}
	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.sqrt((-2.0 * ((U * math.pow(l, 2.0)) * (n * ((2.0 / Om) - ((n * (U_42_ - U)) / math.pow(Om, 2.0)))))))
	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 = sqrt(Float64(-2.0 * Float64(Float64(U * (l ^ 2.0)) * Float64(n * Float64(Float64(2.0 / Om) - Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)))))));
	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((-2.0 * ((U * (l ^ 2.0)) * (n * ((2.0 / Om) - ((n * (U_42_ - U)) / (Om ^ 2.0)))))));
	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[Sqrt[N[(-2.0 * N[(N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $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%

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

   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 Om around -inf 9.4%

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

      1. mul-1-neg 9.4%

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

      2. distribute-neg-frac2 9.4%

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

      3. mul-1-neg 9.4%

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

      4. unsub-neg 9.4%

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

      5. *-commutative 9.4%

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

      6. associate-/l* 9.3%

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

   7. Simplified 9.3%

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

   8. Taylor expanded in l around inf 27.8%

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

      1. associate-*r* 30.5%

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

      2. associate-*r/ 30.5%

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

      3. metadata-eval 30.5%

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

   10. Simplified 30.5%

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

4. Final simplification 71.2%

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

Alternative 5: 62.0% accurate, 0.4× speedup

Math

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

FPCore

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

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 * n)) * sqrt((U * (t + ((pow(l, 2.0) * (-2.0 + (n * ((U_42_ - U) / Om)))) / Om))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((-2.0 * ((U * pow(l, 2.0)) * (n * ((2.0 / Om) - ((n * (U_42_ - U)) / pow(Om, 2.0)))))));
	}
	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 * n)) * Math.sqrt((U * (t + ((Math.pow(l, 2.0) * (-2.0 + (n * ((U_42_ - U) / Om)))) / Om))));
	} 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.sqrt((-2.0 * ((U * Math.pow(l, 2.0)) * (n * ((2.0 / Om) - ((n * (U_42_ - U)) / Math.pow(Om, 2.0)))))));
	}
	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 * n)) * math.sqrt((U * (t + ((math.pow(l, 2.0) * (-2.0 + (n * ((U_42_ - U) / Om)))) / Om))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt((-2.0 * ((U * math.pow(l, 2.0)) * (n * ((2.0 / Om) - ((n * (U_42_ - U)) / math.pow(Om, 2.0)))))))
	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 * n)) * sqrt(Float64(U * Float64(t + Float64(Float64((l ^ 2.0) * Float64(-2.0 + Float64(n * Float64(Float64(U_42_ - U) / Om)))) / Om)))));
	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 = sqrt(Float64(-2.0 * Float64(Float64(U * (l ^ 2.0)) * Float64(n * Float64(Float64(2.0 / Om) - Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)))))));
	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 * n)) * sqrt((U * (t + (((l ^ 2.0) * (-2.0 + (n * ((U_42_ - U) / Om)))) / Om))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((-2.0 * ((U * (l ^ 2.0)) * (n * ((2.0 / Om) - ((n * (U_42_ - U)) / (Om ^ 2.0)))))));
	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), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(-2.0 + N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $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[Sqrt[N[(-2.0 * N[(N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $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 Om around -inf 22.9%

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

      1. mul-1-neg 22.9%

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

      2. distribute-neg-frac2 22.9%

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

      3. mul-1-neg 22.9%

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

      4. unsub-neg 22.9%

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

      5. *-commutative 22.9%

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

      6. associate-/l* 22.9%

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

   7. Simplified 22.9%

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

      1. sqrt-prod 45.1%

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

      2. *-commutative 45.1%

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

      3. distribute-lft-out-- 45.1%

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

      4. associate-/l* 42.0%

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

   9. Applied egg-rr 42.0%

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

       1. *-commutative 42.0%

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

       2. sub-neg 42.0%

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

       3. distribute-frac-neg2 42.0%

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

       4. remove-double-neg 42.0%

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

   11. Simplified 42.0%

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

   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 Om around -inf 9.4%

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

      1. mul-1-neg 9.4%

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

      2. distribute-neg-frac2 9.4%

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

      3. mul-1-neg 9.4%

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

      4. unsub-neg 9.4%

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

      5. *-commutative 9.4%

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

      6. associate-/l* 9.3%

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

   7. Simplified 9.3%

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

   8. Taylor expanded in l around inf 27.8%

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

      1. associate-*r* 30.5%

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

      2. associate-*r/ 30.5%

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

      3. metadata-eval 30.5%

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

   10. Simplified 30.5%

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

4. Final simplification 67.4%

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

Alternative 6: 61.1% accurate, 0.4× speedup

Math

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

FPCore

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

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 * n)) * sqrt((U * t));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((-2.0 * ((U * pow(l, 2.0)) * (n * ((2.0 / Om) - ((n * (U_42_ - U)) / pow(Om, 2.0)))))));
	}
	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 * n)) * Math.sqrt((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.sqrt((-2.0 * ((U * Math.pow(l, 2.0)) * (n * ((2.0 / Om) - ((n * (U_42_ - U)) / Math.pow(Om, 2.0)))))));
	}
	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 * n)) * math.sqrt((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.sqrt((-2.0 * ((U * math.pow(l, 2.0)) * (n * ((2.0 / Om) - ((n * (U_42_ - U)) / math.pow(Om, 2.0)))))))
	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 * n)) * sqrt(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 = sqrt(Float64(-2.0 * Float64(Float64(U * (l ^ 2.0)) * Float64(n * Float64(Float64(2.0 / Om) - Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)))))));
	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 * n)) * sqrt((U * t));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((-2.0 * ((U * (l ^ 2.0)) * (n * ((2.0 / Om) - ((n * (U_42_ - U)) / (Om ^ 2.0)))))));
	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), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $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[Sqrt[N[(-2.0 * N[(N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $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. Step-by-step derivation

      1. associate-*r* 26.5%

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

      2. sub-neg 26.5%

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

      3. distribute-lft-in 26.5%

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

   6. Applied egg-rr 26.5%

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

      1. distribute-lft-out 26.5%

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

      2. sub-neg 26.5%

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

   8. Simplified 26.5%

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

   9. Taylor expanded in t around inf 23.2%

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

       1. pow1/2 23.5%

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

       2. *-commutative 23.5%

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

       3. unpow-prod-down 39.5%

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

       4. pow1/2 39.5%

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

       5. pow1/2 39.5%

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

   11. Applied egg-rr 39.5%

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

   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 Om around -inf 9.4%

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

      1. mul-1-neg 9.4%

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

      2. distribute-neg-frac2 9.4%

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

      3. mul-1-neg 9.4%

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

      4. unsub-neg 9.4%

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

      5. *-commutative 9.4%

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

      6. associate-/l* 9.3%

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

   7. Simplified 9.3%

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

   8. Taylor expanded in l around inf 27.8%

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

      1. associate-*r* 30.5%

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

      2. associate-*r/ 30.5%

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

      3. metadata-eval 30.5%

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

   10. Simplified 30.5%

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

4. Final simplification 67.1%

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

Alternative 7: 44.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t - {\ell}^{2} \cdot \frac{n \cdot \frac{U - U*}{Om} - -2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.15e-103)
   (sqrt (* (* (* 2.0 n) U) t))
   (sqrt
    (*
     U
     (*
      (* 2.0 n)
      (- t (* (pow l 2.0) (/ (- (* n (/ (- U U*) Om)) -2.0) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e-103) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt((U * ((2.0 * n) * (t - (pow(l, 2.0) * (((n * ((U - U_42_) / Om)) - -2.0) / Om))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.15d-103) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt((u * ((2.0d0 * n) * (t - ((l ** 2.0d0) * (((n * ((u - u_42) / om)) - (-2.0d0)) / om))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e-103) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt((U * ((2.0 * n) * (t - (Math.pow(l, 2.0) * (((n * ((U - U_42_) / Om)) - -2.0) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.15e-103:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt((U * ((2.0 * n) * (t - (math.pow(l, 2.0) * (((n * ((U - U_42_) / Om)) - -2.0) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.15e-103)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * Float64(t - Float64((l ^ 2.0) * Float64(Float64(Float64(n * Float64(Float64(U - U_42_) / Om)) - -2.0) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.15e-103)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt((U * ((2.0 * n) * (t - ((l ^ 2.0) * (((n * ((U - U_42_) / Om)) - -2.0) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.15e-103], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(t - N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.15e-103

   1. Initial program 68.3%

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

   3. Taylor expanded in t around inf 55.2%

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

   if 1.15e-103 < l

   1. Initial program 41.1%

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

   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 Om around -inf 35.3%

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

      1. mul-1-neg 35.3%

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

      2. distribute-neg-frac2 35.3%

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

      3. mul-1-neg 35.3%

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

      4. unsub-neg 35.3%

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

      5. *-commutative 35.3%

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

      6. associate-/l* 39.1%

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

   7. Simplified 39.1%

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

      1. *-un-lft-identity 39.1%

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

      2. associate-*r* 44.7%

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

      3. *-commutative 44.7%

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

      4. *-commutative 44.7%

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

      5. distribute-lft-out-- 48.2%

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

      6. associate-/l* 49.2%

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

   9. Applied egg-rr 49.2%

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

       1. *-lft-identity 49.2%

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

       2. associate-*l* 50.5%

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

       3. *-commutative 50.5%

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

       4. associate-/l* 50.5%

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

   11. Simplified 50.5%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t - {\ell}^{2} \cdot \frac{n \cdot \frac{U - U*}{Om} - -2}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot \left(-2 + n \cdot \frac{U* - U}{Om}\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.15e-103)
   (sqrt (* (* (* 2.0 n) U) t))
   (sqrt
    (*
     U
     (*
      (* 2.0 n)
      (+ t (/ (* (pow l 2.0) (+ -2.0 (* n (/ (- U* U) Om)))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e-103) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt((U * ((2.0 * n) * (t + ((pow(l, 2.0) * (-2.0 + (n * ((U_42_ - U) / Om)))) / Om)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.15d-103) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt((u * ((2.0d0 * n) * (t + (((l ** 2.0d0) * ((-2.0d0) + (n * ((u_42 - u) / om)))) / om)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e-103) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt((U * ((2.0 * n) * (t + ((Math.pow(l, 2.0) * (-2.0 + (n * ((U_42_ - U) / Om)))) / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.15e-103:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt((U * ((2.0 * n) * (t + ((math.pow(l, 2.0) * (-2.0 + (n * ((U_42_ - U) / Om)))) / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.15e-103)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * Float64(t + Float64(Float64((l ^ 2.0) * Float64(-2.0 + Float64(n * Float64(Float64(U_42_ - U) / Om)))) / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.15e-103)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt((U * ((2.0 * n) * (t + (((l ^ 2.0) * (-2.0 + (n * ((U_42_ - U) / Om)))) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.15e-103], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(t + N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(-2.0 + N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.15e-103

   1. Initial program 68.3%

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

   3. Taylor expanded in t around inf 55.2%

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

   if 1.15e-103 < l

   1. Initial program 41.1%

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

   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 Om around -inf 35.3%

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

      1. mul-1-neg 35.3%

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

      2. distribute-neg-frac2 35.3%

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

      3. mul-1-neg 35.3%

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

      4. unsub-neg 35.3%

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

      5. *-commutative 35.3%

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

      6. associate-/l* 39.1%

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

   7. Simplified 39.1%

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

      1. *-un-lft-identity 39.1%

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

      2. associate-*r* 44.7%

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

      3. *-commutative 44.7%

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

      4. *-commutative 44.7%

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

      5. distribute-lft-out-- 48.2%

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

      6. associate-/l* 49.2%

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

   9. Applied egg-rr 49.2%

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

       1. *-lft-identity 49.2%

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

       2. associate-*l* 50.5%

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

       3. sub-neg 50.5%

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

       4. distribute-frac-neg2 50.5%

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

       5. remove-double-neg 50.5%

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

       6. *-commutative 50.5%

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

   11. Simplified 50.5%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(t + \frac{{\ell}^{2} \cdot \left(-2 + n \cdot \frac{U* - U}{Om}\right)}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -7.2 \cdot 10^{+18} \lor \neg \left(Om \leq 1.9 \cdot 10^{-14}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -7.2e+18) (not (<= Om 1.9e-14)))
   (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 <= -7.2e+18) || !(Om <= 1.9e-14)) {
		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 <= (-7.2d+18)) .or. (.not. (om <= 1.9d-14))) 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 <= -7.2e+18) || !(Om <= 1.9e-14)) {
		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 <= -7.2e+18) or not (Om <= 1.9e-14):
		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 <= -7.2e+18) || !(Om <= 1.9e-14))
		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 <= -7.2e+18) || ~((Om <= 1.9e-14)))
		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, -7.2e+18], N[Not[LessEqual[Om, 1.9e-14]], $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 < -7.2e18 or 1.9000000000000001e-14 < Om

   1. Initial program 61.9%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in n around 0 56.4%

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

      1. unpow2 56.4%

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

      2. associate-*r/ 60.9%

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

      3. *-commutative 60.9%

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

   7. Applied egg-rr 60.9%

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

   if -7.2e18 < Om < 1.9000000000000001e-14

   1. Initial program 50.6%

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

   3. Simplified 49.6%

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

   5. Taylor expanded in U* around inf 49.6%

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

      1. mul-1-neg 49.6%

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

      2. associate-/l* 49.6%

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

      3. distribute-rgt-neg-in 49.6%

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

      4. distribute-neg-frac2 49.6%

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

      5. *-commutative 49.6%

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

   7. Simplified 49.6%

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

      1. add-sqr-sqrt 49.6%

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

      2. pow1/2 49.6%

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

      3. pow1/2 49.8%

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

      4. pow-prod-down 39.1%

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

   9. Applied egg-rr 38.7%

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

       1. unpow1/2 38.7%

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

       2. unpow2 38.7%

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

       3. rem-sqrt-square 49.6%

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

       4. associate-*l* 50.0%

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

       5. associate-/l* 50.0%

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

       6. associate-*r/ 49.7%

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

       7. distribute-frac-neg2 49.7%

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

       8. unpow2 49.7%

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

       9. unpow2 49.7%

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

       10. times-frac 57.0%

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

       11. unpow2 57.0%

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

   11. Simplified 57.0%

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

       1. *-un-lft-identity 57.0%

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

       2. associate-*r* 57.5%

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

       3. associate-*r* 58.2%

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

   13. Applied egg-rr 58.2%

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

   15. Simplified 56.6%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -7.2 \cdot 10^{+18} \lor \neg \left(Om \leq 1.9 \cdot 10^{-14}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U*\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.2e-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 <= 5.2e-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 <= 5.2d-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 <= 5.2e-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 <= 5.2e-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 <= 5.2e-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 <= 5.2e-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, 5.2e-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 < 5.19999999999999988e-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 5.19999999999999988e-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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.5e-94)
   (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 <= 4.5e-94) {
		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 <= 4.5d-94) 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 <= 4.5e-94) {
		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 <= 4.5e-94:
		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 <= 4.5e-94)
		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 <= 4.5e-94)
		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, 4.5e-94], 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 < 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 t around inf 26.2%

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

      1. pow1/2 27.2%

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

      2. associate-*r* 27.2%

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

   7. Applied egg-rr 27.2%

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

Alternative 12: 36.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.8 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.8e-94) (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 <= 4.8e-94) {
		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 <= 4.8d-94) 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 <= 4.8e-94) {
		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 <= 4.8e-94:
		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 <= 4.8e-94)
		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 <= 4.8e-94)
		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, 4.8e-94], 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 < 4.8e-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 4.8e-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 t around inf 26.2%

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

Alternative 13: 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. 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*))))))