Toniolo and Linder, Equation (13)

Percentage Accurate: 48.8% → 63.4%

Time: 26.6s

Alternatives: 18

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: 48.8% 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: 63.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := t\_1 \cdot \left(U* - U\right)\\ t_3 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_2\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{{l\_m}^{2}}{Om}, t\_1 \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_2 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l_m Om) 2.0)))
        (t_2 (* t_1 (- U* U)))
        (t_3
         (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_2)))))
   (if (<= t_3 0.0)
     (*
      (sqrt (* U (- t (fma 2.0 (/ (pow l_m 2.0) Om) (* t_1 (- U U*))))))
      (sqrt (* 2.0 n)))
     (if (<= t_3 5e+145)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_2 (* 2.0 (* l_m (/ l_m Om)))))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* U (* n (- (* U* (/ n (pow Om 2.0))) (/ 2.0 Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * pow((l_m / Om), 2.0);
	double t_2 = t_1 * (U_42_ - U);
	double t_3 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((U * (t - fma(2.0, (pow(l_m, 2.0) / Om), (t_1 * (U - U_42_)))))) * sqrt((2.0 * n));
	} else if (t_3 <= 5e+145) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * ((U_42_ * (n / pow(Om, 2.0))) - (2.0 / Om)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(n * (Float64(l_m / Om) ^ 2.0))
	t_2 = Float64(t_1 * Float64(U_42_ - U))
	t_3 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_2)))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(U * Float64(t - fma(2.0, Float64((l_m ^ 2.0) / Om), Float64(t_1 * Float64(U - U_42_)))))) * sqrt(Float64(2.0 * n)));
	elseif (t_3 <= 5e+145)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_2 - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(U_42_ * Float64(n / (Om ^ 2.0))) - Float64(2.0 / Om))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision] + N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+145], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$2 - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(U$42$ * N[(n / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.5%

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

   3. Simplified 39.9%

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

      1. *-commutative 39.9%

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

      2. sqrt-prod 44.4%

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

      3. associate-*r/ 44.4%

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

      4. pow2 44.4%

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

   6. Applied egg-rr 44.4%

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

      1. associate-*r* 47.2%

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

   8. Simplified 47.2%

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

   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*))))) < 4.99999999999999967e145

   1. Initial program 97.9%

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

   3. Simplified 97.9%

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

   if 4.99999999999999967e145 < (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 22.2%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in U around 0 22.9%

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

      1. mul-1-neg 22.9%

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

      2. associate-/l* 22.1%

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

      3. unpow2 22.1%

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

      4. unpow2 22.1%

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

      5. times-frac 30.7%

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

      6. unpow2 30.7%

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

   7. Simplified 30.7%

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

      1. clear-num 30.7%

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

      2. inv-pow 30.7%

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

   9. Applied egg-rr 30.7%

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

       1. unpow-1 30.7%

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

   11. Simplified 30.7%

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

       1. distribute-rgt-neg-out 30.7%

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

       2. fmm-undef 30.7%

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

       3. clear-num 30.7%

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

       4. clear-num 30.7%

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

       5. inv-pow 30.7%

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

       6. pow-pow 30.7%

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

       7. metadata-eval 30.7%

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

   13. Applied egg-rr 30.7%

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

   14. Taylor expanded in l around inf 25.7%

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

       1. *-commutative 25.7%

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

       2. associate-/l* 25.7%

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

       3. associate-*r/ 25.7%

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

       4. metadata-eval 25.7%

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

   16. Simplified 25.7%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\\ t_2 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_2\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \left(t\_1 - n \cdot \left(U* \cdot {\left(\frac{Om}{l\_m}\right)}^{-2}\right)\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_2 - t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* l_m (/ l_m Om))))
        (t_2 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_3
         (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_2)))))
   (if (<= t_3 0.0)
     (*
      (sqrt (* 2.0 n))
      (sqrt (* U (- t (- t_1 (* n (* U* (pow (/ Om l_m) -2.0))))))))
     (if (<= t_3 5e+145)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_2 t_1))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* U (* n (- (* U* (/ n (pow Om 2.0))) (/ 2.0 Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (l_m * (l_m / Om));
	double t_2 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - (t_1 - (n * (U_42_ * pow((Om / l_m), -2.0)))))));
	} else if (t_3 <= 5e+145) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - t_1))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * ((U_42_ * (n / pow(Om, 2.0))) - (2.0 / Om)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = 2.0d0 * (l_m * (l_m / om))
    t_2 = (n * ((l_m / om) ** 2.0d0)) * (u_42 - u)
    t_3 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) + t_2)))
    if (t_3 <= 0.0d0) then
        tmp = sqrt((2.0d0 * n)) * sqrt((u * (t - (t_1 - (n * (u_42 * ((om / l_m) ** (-2.0d0))))))))
    else if (t_3 <= 5d+145) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (t_2 - t_1))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt((u * (n * ((u_42 * (n / (om ** 2.0d0))) - (2.0d0 / om)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (l_m * (l_m / Om));
	double t_2 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_3 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t - (t_1 - (n * (U_42_ * Math.pow((Om / l_m), -2.0)))))));
	} else if (t_3 <= 5e+145) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_2 - t_1))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((U * (n * ((U_42_ * (n / Math.pow(Om, 2.0))) - (2.0 / Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = 2.0 * (l_m * (l_m / Om))
	t_2 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_3 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)))
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * (t - (t_1 - (n * (U_42_ * math.pow((Om / l_m), -2.0)))))))
	elif t_3 <= 5e+145:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_2 - t_1))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((U * (n * ((U_42_ * (n / math.pow(Om, 2.0))) - (2.0 / Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(l_m * Float64(l_m / Om)))
	t_2 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_2)))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t - Float64(t_1 - Float64(n * Float64(U_42_ * (Float64(Om / l_m) ^ -2.0))))))));
	elseif (t_3 <= 5e+145)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_2 - t_1))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(U_42_ * Float64(n / (Om ^ 2.0))) - Float64(2.0 / Om))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = 2.0 * (l_m * (l_m / Om));
	t_2 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)));
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - (t_1 - (n * (U_42_ * ((Om / l_m) ^ -2.0)))))));
	elseif (t_3 <= 5e+145)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - t_1))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * ((U_42_ * (n / (Om ^ 2.0))) - (2.0 / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(t$95$1 - N[(n * N[(U$42$ * N[Power[N[(Om / l$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+145], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$2 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(U$42$ * N[(n / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.5%

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

   3. Simplified 39.9%

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

   5. Taylor expanded in U around 0 28.3%

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

      1. mul-1-neg 28.3%

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

      2. associate-/l* 31.2%

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

      3. unpow2 31.2%

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

      4. unpow2 31.2%

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

      5. times-frac 39.9%

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

      6. unpow2 39.9%

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

   7. Simplified 39.9%

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

      1. clear-num 39.9%

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

      2. inv-pow 39.9%

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

   9. Applied egg-rr 39.9%

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

       1. unpow-1 39.9%

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

   11. Simplified 39.9%

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

   13. Applied egg-rr 44.4%

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

   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*))))) < 4.99999999999999967e145

   1. Initial program 97.9%

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

   3. Simplified 97.9%

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

   if 4.99999999999999967e145 < (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 22.2%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in U around 0 22.9%

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

      1. mul-1-neg 22.9%

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

      2. associate-/l* 22.1%

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

      3. unpow2 22.1%

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

      4. unpow2 22.1%

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

      5. times-frac 30.7%

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

      6. unpow2 30.7%

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

   7. Simplified 30.7%

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

      1. clear-num 30.7%

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

      2. inv-pow 30.7%

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

   9. Applied egg-rr 30.7%

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

       1. unpow-1 30.7%

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

   11. Simplified 30.7%

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

       1. distribute-rgt-neg-out 30.7%

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

       2. fmm-undef 30.7%

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

       3. clear-num 30.7%

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

       4. clear-num 30.7%

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

       5. inv-pow 30.7%

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

       6. pow-pow 30.7%

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

       7. metadata-eval 30.7%

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

   13. Applied egg-rr 30.7%

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

   14. Taylor expanded in l around inf 25.7%

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

       1. *-commutative 25.7%

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

       2. associate-/l* 25.7%

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

       3. associate-*r/ 25.7%

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

       4. metadata-eval 25.7%

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

   16. Simplified 25.7%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - n \cdot \left(U* \cdot {\left(\frac{Om}{\ell}\right)}^{-2}\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)}\\ t_3 := 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left({\left(\frac{Om}{l\_m}\right)}^{-2} \cdot \left(n \cdot U*\right) - t\_3\right)\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - t\_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2
         (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
        (t_3 (* 2.0 (* l_m (/ l_m Om)))))
   (if (<= t_2 0.0)
     (sqrt
      (* (* 2.0 n) (* U (+ t (- (* (pow (/ Om l_m) -2.0) (* n U*)) t_3)))))
     (if (<= t_2 5e+145)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 t_3))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* U (* n (- (* U* (/ n (pow Om 2.0))) (/ 2.0 Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double t_3 = 2.0 * (l_m * (l_m / Om));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((pow((Om / l_m), -2.0) * (n * U_42_)) - t_3)))));
	} else if (t_2 <= 5e+145) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - t_3))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * ((U_42_ * (n / pow(Om, 2.0))) - (2.0 / Om)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (n * ((l_m / om) ** 2.0d0)) * (u_42 - u)
    t_2 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) + t_1)))
    t_3 = 2.0d0 * (l_m * (l_m / om))
    if (t_2 <= 0.0d0) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((((om / l_m) ** (-2.0d0)) * (n * u_42)) - t_3)))))
    else if (t_2 <= 5d+145) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (t_1 - t_3))))
    else
        tmp = (l_m * sqrt(2.0d0)) * sqrt((u * (n * ((u_42 * (n / (om ** 2.0d0))) - (2.0d0 / om)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double t_3 = 2.0 * (l_m * (l_m / Om));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((Math.pow((Om / l_m), -2.0) * (n * U_42_)) - t_3)))));
	} else if (t_2 <= 5e+145) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - t_3))));
	} else {
		tmp = (l_m * Math.sqrt(2.0)) * Math.sqrt((U * (n * ((U_42_ * (n / Math.pow(Om, 2.0))) - (2.0 / Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)))
	t_3 = 2.0 * (l_m * (l_m / Om))
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((math.pow((Om / l_m), -2.0) * (n * U_42_)) - t_3)))))
	elif t_2 <= 5e+145:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - t_3))))
	else:
		tmp = (l_m * math.sqrt(2.0)) * math.sqrt((U * (n * ((U_42_ * (n / math.pow(Om, 2.0))) - (2.0 / Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / 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_m * l_m) / Om))) + t_1)))
	t_3 = Float64(2.0 * Float64(l_m * Float64(l_m / Om)))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64((Float64(Om / l_m) ^ -2.0) * Float64(n * U_42_)) - t_3)))));
	elseif (t_2 <= 5e+145)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - t_3))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(U * Float64(n * Float64(Float64(U_42_ * Float64(n / (Om ^ 2.0))) - Float64(2.0 / Om))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	t_3 = 2.0 * (l_m * (l_m / Om));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt(((2.0 * n) * (U * (t + ((((Om / l_m) ^ -2.0) * (n * U_42_)) - t_3)))));
	elseif (t_2 <= 5e+145)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - t_3))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * ((U_42_ * (n / (Om ^ 2.0))) - (2.0 / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / 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$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[Power[N[(Om / l$95$m), $MachinePrecision], -2.0], $MachinePrecision] * N[(n * U$42$), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+145], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(N[(U$42$ * N[(n / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.5%

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

   3. Simplified 39.9%

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

   5. Taylor expanded in U around 0 28.3%

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

      1. mul-1-neg 28.3%

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

      2. associate-/l* 31.2%

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

      3. unpow2 31.2%

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

      4. unpow2 31.2%

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

      5. times-frac 39.9%

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

      6. unpow2 39.9%

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

   7. Simplified 39.9%

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

      1. clear-num 39.9%

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

      2. inv-pow 39.9%

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

   9. Applied egg-rr 39.9%

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

       1. unpow-1 39.9%

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

   11. Simplified 39.9%

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

       1. distribute-rgt-neg-out 39.9%

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

       2. fmm-undef 39.9%

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

       3. clear-num 39.9%

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

       4. clear-num 39.9%

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

       5. inv-pow 39.9%

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

       6. pow-pow 39.9%

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

       7. metadata-eval 39.9%

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

   13. Applied egg-rr 39.9%

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

       1. pow1 39.9%

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

       2. associate-*r* 40.7%

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

   15. Applied egg-rr 40.7%

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

       1. unpow1 40.7%

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

       2. *-commutative 40.7%

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

   17. Simplified 40.7%

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

   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*))))) < 4.99999999999999967e145

   1. Initial program 97.9%

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

   3. Simplified 97.9%

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

   if 4.99999999999999967e145 < (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 22.2%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in U around 0 22.9%

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

      1. mul-1-neg 22.9%

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

      2. associate-/l* 22.1%

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

      3. unpow2 22.1%

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

      4. unpow2 22.1%

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

      5. times-frac 30.7%

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

      6. unpow2 30.7%

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

   7. Simplified 30.7%

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

      1. clear-num 30.7%

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

      2. inv-pow 30.7%

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

   9. Applied egg-rr 30.7%

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

       1. unpow-1 30.7%

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

   11. Simplified 30.7%

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

       1. distribute-rgt-neg-out 30.7%

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

       2. fmm-undef 30.7%

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

       3. clear-num 30.7%

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

       4. clear-num 30.7%

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

       5. inv-pow 30.7%

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

       6. pow-pow 30.7%

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

       7. metadata-eval 30.7%

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

   13. Applied egg-rr 30.7%

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

   14. Taylor expanded in l around inf 25.7%

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

       1. *-commutative 25.7%

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

       2. associate-/l* 25.7%

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

       3. associate-*r/ 25.7%

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

       4. metadata-eval 25.7%

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

   16. Simplified 25.7%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left({\left(\frac{Om}{\ell}\right)}^{-2} \cdot \left(n \cdot U*\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(U* \cdot \frac{n}{{Om}^{2}} - \frac{2}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.0% accurate, 0.3× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2
         (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
        (t_3 (* 2.0 (* l_m (/ l_m Om)))))
   (if (<= t_2 0.0)
     (sqrt
      (* (* 2.0 n) (* U (+ t (- (* (pow (/ Om l_m) -2.0) (* n U*)) t_3)))))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 t_3))))
       (sqrt
        (*
         2.0
         (-
          (* U (* n t))
          (*
           (pow l_m 2.0)
           (* n (/ (+ (* 2.0 U) (/ (* U (* n (- U U*))) Om)) Om))))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.5%

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

   3. Simplified 39.9%

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

   5. Taylor expanded in U around 0 28.3%

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

      1. mul-1-neg 28.3%

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

      2. associate-/l* 31.2%

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

      3. unpow2 31.2%

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

      4. unpow2 31.2%

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

      5. times-frac 39.9%

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

      6. unpow2 39.9%

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

   7. Simplified 39.9%

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

      1. clear-num 39.9%

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

      2. inv-pow 39.9%

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

   9. Applied egg-rr 39.9%

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

       1. unpow-1 39.9%

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

   11. Simplified 39.9%

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

       1. distribute-rgt-neg-out 39.9%

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

       2. fmm-undef 39.9%

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

       3. clear-num 39.9%

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

       4. clear-num 39.9%

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

       5. inv-pow 39.9%

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

       6. pow-pow 39.9%

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

       7. metadata-eval 39.9%

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

   13. Applied egg-rr 39.9%

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

       1. pow1 39.9%

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

       2. associate-*r* 40.7%

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

   15. Applied egg-rr 40.7%

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

       1. unpow1 40.7%

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

       2. *-commutative 40.7%

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

   17. Simplified 40.7%

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

   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*))))) < +inf.0

   1. Initial program 70.6%

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

   3. Simplified 72.3%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   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 11.5%

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

   5. Taylor expanded in Om around -inf 23.0%

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

      1. +-commutative 23.0%

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

      2. mul-1-neg 23.0%

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

      3. unsub-neg 23.0%

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

      4. +-commutative 23.0%

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

      5. mul-1-neg 23.0%

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

      6. unsub-neg 23.0%

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

      7. associate-*r* 23.0%

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

      8. associate-/l* 19.2%

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

      9. associate-/l* 19.0%

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

   7. Simplified 19.0%

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

   8. Taylor expanded in l around 0 28.2%

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

   9. Taylor expanded in Om around inf 38.9%

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

4. Final simplification 61.5%

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

Alternative 5: 59.5% accurate, 0.3× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2
         (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
        (t_3 (* 2.0 (* l_m (/ l_m Om)))))
   (if (<= t_2 0.0)
     (sqrt
      (* (* 2.0 n) (* U (+ t (- (* (pow (/ Om l_m) -2.0) (* n U*)) t_3)))))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 t_3))))
       (* (sqrt (* U U*)) (* l_m (* n (/ (sqrt 2.0) (- Om)))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / 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$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[Power[N[(Om / l$95$m), $MachinePrecision], -2.0], $MachinePrecision] * N[(n * U$42$), $MachinePrecision]), $MachinePrecision] - t$95$3), $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 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / (-Om)), $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 13.5%

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

   3. Simplified 39.9%

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

   5. Taylor expanded in U around 0 28.3%

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

      1. mul-1-neg 28.3%

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

      2. associate-/l* 31.2%

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

      3. unpow2 31.2%

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

      4. unpow2 31.2%

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

      5. times-frac 39.9%

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

      6. unpow2 39.9%

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

   7. Simplified 39.9%

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

      1. clear-num 39.9%

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

      2. inv-pow 39.9%

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

   9. Applied egg-rr 39.9%

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

       1. unpow-1 39.9%

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

   11. Simplified 39.9%

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

       1. distribute-rgt-neg-out 39.9%

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

       2. fmm-undef 39.9%

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

       3. clear-num 39.9%

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

       4. clear-num 39.9%

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

       5. inv-pow 39.9%

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

       6. pow-pow 39.9%

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

       7. metadata-eval 39.9%

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

   13. Applied egg-rr 39.9%

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

       1. pow1 39.9%

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

       2. associate-*r* 40.7%

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

   15. Applied egg-rr 40.7%

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

       1. unpow1 40.7%

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

       2. *-commutative 40.7%

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

   17. Simplified 40.7%

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

   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*))))) < +inf.0

   1. Initial program 70.6%

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

   3. Simplified 72.3%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   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 11.5%

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

   5. Taylor expanded in Om around -inf 23.0%

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

      1. +-commutative 23.0%

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

      2. mul-1-neg 23.0%

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

      3. unsub-neg 23.0%

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

      4. +-commutative 23.0%

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

      5. mul-1-neg 23.0%

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

      6. unsub-neg 23.0%

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

      7. associate-*r* 23.0%

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

      8. associate-/l* 19.2%

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

      9. associate-/l* 19.0%

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

   7. Simplified 19.0%

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

   8. Taylor expanded in U* around inf 22.2%

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

      1. associate-*r* 19.8%

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

   10. Simplified 19.8%

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

   11. Taylor expanded in n around -inf 21.5%

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

       1. mul-1-neg 21.5%

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

       2. *-commutative 21.5%

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

       3. associate-*r* 21.6%

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

       4. distribute-rgt-neg-in 21.6%

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

       5. *-commutative 21.6%

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

       6. associate-*r* 21.5%

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

       7. associate-*r/ 23.4%

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

       8. associate-/l* 23.5%

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

   13. Simplified 23.5%

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

4. Final simplification 58.5%

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

Alternative 6: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{-263} \lor \neg \left(n \leq 1.08 \cdot 10^{-243}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(U* \cdot {\left(\frac{Om}{l\_m}\right)}^{-2}\right) - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + -2 \cdot \left(U \cdot \frac{n \cdot {l\_m}^{2}}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= n -8.2e-263) (not (<= n 1.08e-243)))
   (sqrt
    (*
     (* 2.0 n)
     (*
      U
      (+
       t
       (- (* n (* U* (pow (/ Om l_m) -2.0))) (* 2.0 (* l_m (/ l_m Om))))))))
   (sqrt (* 2.0 (+ (* U (* n t)) (* -2.0 (* U (/ (* n (pow l_m 2.0)) Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((n <= -8.2e-263) || !(n <= 1.08e-243)) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((n * (U_42_ * pow((Om / l_m), -2.0))) - (2.0 * (l_m * (l_m / Om))))))));
	} else {
		tmp = sqrt((2.0 * ((U * (n * t)) + (-2.0 * (U * ((n * pow(l_m, 2.0)) / Om))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((n <= (-8.2d-263)) .or. (.not. (n <= 1.08d-243))) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((n * (u_42 * ((om / l_m) ** (-2.0d0)))) - (2.0d0 * (l_m * (l_m / om))))))))
    else
        tmp = sqrt((2.0d0 * ((u * (n * t)) + ((-2.0d0) * (u * ((n * (l_m ** 2.0d0)) / om))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((n <= -8.2e-263) || !(n <= 1.08e-243)) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((n * (U_42_ * Math.pow((Om / l_m), -2.0))) - (2.0 * (l_m * (l_m / Om))))))));
	} else {
		tmp = Math.sqrt((2.0 * ((U * (n * t)) + (-2.0 * (U * ((n * Math.pow(l_m, 2.0)) / Om))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (n <= -8.2e-263) or not (n <= 1.08e-243):
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((n * (U_42_ * math.pow((Om / l_m), -2.0))) - (2.0 * (l_m * (l_m / Om))))))))
	else:
		tmp = math.sqrt((2.0 * ((U * (n * t)) + (-2.0 * (U * ((n * math.pow(l_m, 2.0)) / Om))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((n <= -8.2e-263) || !(n <= 1.08e-243))
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(n * Float64(U_42_ * (Float64(Om / l_m) ^ -2.0))) - Float64(2.0 * Float64(l_m * Float64(l_m / Om))))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(U * Float64(n * t)) + Float64(-2.0 * Float64(U * Float64(Float64(n * (l_m ^ 2.0)) / Om))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((n <= -8.2e-263) || ~((n <= 1.08e-243)))
		tmp = sqrt(((2.0 * n) * (U * (t + ((n * (U_42_ * ((Om / l_m) ^ -2.0))) - (2.0 * (l_m * (l_m / Om))))))));
	else
		tmp = sqrt((2.0 * ((U * (n * t)) + (-2.0 * (U * ((n * (l_m ^ 2.0)) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[n, -8.2e-263], N[Not[LessEqual[n, 1.08e-243]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(n * N[(U$42$ * N[Power[N[(Om / l$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(U * N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -8.1999999999999994e-263 or 1.08e-243 < n

   1. Initial program 49.1%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in U around 0 46.2%

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

      1. mul-1-neg 46.2%

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

      2. associate-/l* 47.2%

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

      3. unpow2 47.2%

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

      4. unpow2 47.2%

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

      5. times-frac 54.3%

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

      6. unpow2 54.3%

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

   7. Simplified 54.3%

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

      1. clear-num 54.3%

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

      2. inv-pow 54.3%

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

   9. Applied egg-rr 54.3%

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

       1. unpow-1 54.3%

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

   11. Simplified 54.3%

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

       1. distribute-rgt-neg-out 54.3%

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

       2. fmm-undef 54.3%

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

       3. clear-num 54.3%

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

       4. clear-num 54.3%

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

       5. inv-pow 54.3%

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

       6. pow-pow 54.3%

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

       7. metadata-eval 54.3%

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

   13. Applied egg-rr 54.3%

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

   if -8.1999999999999994e-263 < n < 1.08e-243

   1. Initial program 48.7%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in Om around -inf 38.6%

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

      1. +-commutative 38.6%

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

      2. mul-1-neg 38.6%

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

      3. unsub-neg 38.6%

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

      4. +-commutative 38.6%

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

      5. mul-1-neg 38.6%

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

      6. unsub-neg 38.6%

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

      7. associate-*r* 38.6%

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

      8. associate-/l* 38.6%

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

      9. associate-/l* 38.5%

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

   7. Simplified 38.5%

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

   8. Taylor expanded in l around 0 51.4%

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

   9. Taylor expanded in n around 0 63.7%

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

       1. associate-/l* 64.5%

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

   11. Simplified 64.5%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{-263} \lor \neg \left(n \leq 1.08 \cdot 10^{-243}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(U* \cdot {\left(\frac{Om}{\ell}\right)}^{-2}\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + -2 \cdot \left(U \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := n \cdot \left(U* \cdot {\left(\frac{Om}{l\_m}\right)}^{-2}\right)\\ \mathbf{if}\;n \leq -2.05 \cdot 10^{-263}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(t\_1 + 2 \cdot \left(l\_m \cdot \frac{-1}{\frac{Om}{l\_m}}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + -2 \cdot \left(U \cdot \frac{n \cdot {l\_m}^{2}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right) - t\_1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* n (* U* (pow (/ Om l_m) -2.0)))))
   (if (<= n -2.05e-263)
     (sqrt
      (* (* 2.0 n) (* U (+ t (+ t_1 (* 2.0 (* l_m (/ -1.0 (/ Om l_m)))))))))
     (if (<= n 7.2e-244)
       (sqrt
        (* 2.0 (+ (* U (* n t)) (* -2.0 (* U (/ (* n (pow l_m 2.0)) Om))))))
       (sqrt (* (* 2.0 n) (* U (- t (- (* 2.0 (* l_m (/ l_m Om))) t_1)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * (U_42_ * pow((Om / l_m), -2.0));
	double tmp;
	if (n <= -2.05e-263) {
		tmp = sqrt(((2.0 * n) * (U * (t + (t_1 + (2.0 * (l_m * (-1.0 / (Om / l_m)))))))));
	} else if (n <= 7.2e-244) {
		tmp = sqrt((2.0 * ((U * (n * t)) + (-2.0 * (U * ((n * pow(l_m, 2.0)) / Om))))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t - ((2.0 * (l_m * (l_m / Om))) - t_1)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = n * (u_42 * ((om / l_m) ** (-2.0d0)))
    if (n <= (-2.05d-263)) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + (t_1 + (2.0d0 * (l_m * ((-1.0d0) / (om / l_m)))))))))
    else if (n <= 7.2d-244) then
        tmp = sqrt((2.0d0 * ((u * (n * t)) + ((-2.0d0) * (u * ((n * (l_m ** 2.0d0)) / om))))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t - ((2.0d0 * (l_m * (l_m / om))) - t_1)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * (U_42_ * Math.pow((Om / l_m), -2.0));
	double tmp;
	if (n <= -2.05e-263) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (t_1 + (2.0 * (l_m * (-1.0 / (Om / l_m)))))))));
	} else if (n <= 7.2e-244) {
		tmp = Math.sqrt((2.0 * ((U * (n * t)) + (-2.0 * (U * ((n * Math.pow(l_m, 2.0)) / Om))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - ((2.0 * (l_m * (l_m / Om))) - t_1)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = n * (U_42_ * math.pow((Om / l_m), -2.0))
	tmp = 0
	if n <= -2.05e-263:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (t_1 + (2.0 * (l_m * (-1.0 / (Om / l_m)))))))))
	elif n <= 7.2e-244:
		tmp = math.sqrt((2.0 * ((U * (n * t)) + (-2.0 * (U * ((n * math.pow(l_m, 2.0)) / Om))))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t - ((2.0 * (l_m * (l_m / Om))) - t_1)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(n * Float64(U_42_ * (Float64(Om / l_m) ^ -2.0)))
	tmp = 0.0
	if (n <= -2.05e-263)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(t_1 + Float64(2.0 * Float64(l_m * Float64(-1.0 / Float64(Om / l_m)))))))));
	elseif (n <= 7.2e-244)
		tmp = sqrt(Float64(2.0 * Float64(Float64(U * Float64(n * t)) + Float64(-2.0 * Float64(U * Float64(Float64(n * (l_m ^ 2.0)) / Om))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(Float64(2.0 * Float64(l_m * Float64(l_m / Om))) - t_1)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = n * (U_42_ * ((Om / l_m) ^ -2.0));
	tmp = 0.0;
	if (n <= -2.05e-263)
		tmp = sqrt(((2.0 * n) * (U * (t + (t_1 + (2.0 * (l_m * (-1.0 / (Om / l_m)))))))));
	elseif (n <= 7.2e-244)
		tmp = sqrt((2.0 * ((U * (n * t)) + (-2.0 * (U * ((n * (l_m ^ 2.0)) / Om))))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t - ((2.0 * (l_m * (l_m / Om))) - t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U$42$ * N[Power[N[(Om / l$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.05e-263], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(t$95$1 + N[(2.0 * N[(l$95$m * N[(-1.0 / N[(Om / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 7.2e-244], N[Sqrt[N[(2.0 * N[(N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(U * N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.0499999999999998e-263

   1. Initial program 48.9%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in U around 0 46.0%

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

      1. mul-1-neg 46.0%

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

      2. associate-/l* 46.0%

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

      3. unpow2 46.0%

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

      4. unpow2 46.0%

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

      5. times-frac 54.8%

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

      6. unpow2 54.8%

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

   7. Simplified 54.8%

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

      1. clear-num 54.9%

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

      2. inv-pow 54.9%

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

   9. Applied egg-rr 54.9%

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

       1. unpow-1 54.9%

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

   11. Simplified 54.9%

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

       1. distribute-rgt-neg-out 54.9%

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

       2. fmm-undef 54.9%

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

       3. clear-num 54.8%

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

       4. clear-num 54.8%

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

       5. inv-pow 54.8%

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

       6. pow-pow 54.8%

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

       7. metadata-eval 54.8%

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

   13. Applied egg-rr 54.8%

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

       1. clear-num 54.9%

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

       2. inv-pow 54.9%

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

   15. Applied egg-rr 54.9%

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

       1. unpow-1 54.9%

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

   17. Simplified 54.9%

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

   if -2.0499999999999998e-263 < n < 7.1999999999999995e-244

   1. Initial program 48.7%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in Om around -inf 38.6%

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

      1. +-commutative 38.6%

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

      2. mul-1-neg 38.6%

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

      3. unsub-neg 38.6%

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

      4. +-commutative 38.6%

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

      5. mul-1-neg 38.6%

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

      6. unsub-neg 38.6%

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

      7. associate-*r* 38.6%

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

      8. associate-/l* 38.6%

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

      9. associate-/l* 38.5%

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

   7. Simplified 38.5%

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

   8. Taylor expanded in l around 0 51.4%

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

   9. Taylor expanded in n around 0 63.7%

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

       1. associate-/l* 64.5%

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

   11. Simplified 64.5%

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

   if 7.1999999999999995e-244 < n

   1. Initial program 49.5%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in U around 0 46.5%

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

      1. mul-1-neg 46.5%

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

      2. associate-/l* 48.9%

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

      3. unpow2 48.9%

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

      4. unpow2 48.9%

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

      5. times-frac 53.5%

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

      6. unpow2 53.5%

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

   7. Simplified 53.5%

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

      1. clear-num 53.5%

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

      2. inv-pow 53.5%

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

   9. Applied egg-rr 53.5%

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

       1. unpow-1 53.5%

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

   11. Simplified 53.5%

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

       1. distribute-rgt-neg-out 53.5%

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

       2. fmm-undef 53.5%

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

       3. clear-num 53.5%

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

       4. clear-num 53.5%

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

       5. inv-pow 53.5%

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

       6. pow-pow 53.6%

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

       7. metadata-eval 53.6%

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

   13. Applied egg-rr 53.6%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.05 \cdot 10^{-263}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left(U* \cdot {\left(\frac{Om}{\ell}\right)}^{-2}\right) + 2 \cdot \left(\ell \cdot \frac{-1}{\frac{Om}{\ell}}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right) + -2 \cdot \left(U \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) - n \cdot \left(U* \cdot {\left(\frac{Om}{\ell}\right)}^{-2}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := t - \frac{2 \cdot {l\_m}^{2}}{Om}\\ \mathbf{if}\;U \leq -4.1 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\_1}\\ \mathbf{elif}\;U \leq -1.3 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(l\_m \cdot \left(n \cdot \frac{\sqrt{2}}{-Om}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (- t (/ (* 2.0 (pow l_m 2.0)) Om))))
   (if (<= U -4.1e-5)
     (sqrt (* (* 2.0 (* n U)) t_1))
     (if (<= U -1.3e-58)
       (* (sqrt (* U U*)) (* l_m (* n (/ (sqrt 2.0) (- Om)))))
       (sqrt (* (* 2.0 n) (* U t_1)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = t - ((2.0 * pow(l_m, 2.0)) / Om);
	double tmp;
	if (U <= -4.1e-5) {
		tmp = sqrt(((2.0 * (n * U)) * t_1));
	} else if (U <= -1.3e-58) {
		tmp = sqrt((U * U_42_)) * (l_m * (n * (sqrt(2.0) / -Om)));
	} else {
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t - ((2.0d0 * (l_m ** 2.0d0)) / om)
    if (u <= (-4.1d-5)) then
        tmp = sqrt(((2.0d0 * (n * u)) * t_1))
    else if (u <= (-1.3d-58)) then
        tmp = sqrt((u * u_42)) * (l_m * (n * (sqrt(2.0d0) / -om)))
    else
        tmp = sqrt(((2.0d0 * n) * (u * t_1)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = t - ((2.0 * Math.pow(l_m, 2.0)) / Om);
	double tmp;
	if (U <= -4.1e-5) {
		tmp = Math.sqrt(((2.0 * (n * U)) * t_1));
	} else if (U <= -1.3e-58) {
		tmp = Math.sqrt((U * U_42_)) * (l_m * (n * (Math.sqrt(2.0) / -Om)));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * t_1)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = t - ((2.0 * math.pow(l_m, 2.0)) / Om)
	tmp = 0
	if U <= -4.1e-5:
		tmp = math.sqrt(((2.0 * (n * U)) * t_1))
	elif U <= -1.3e-58:
		tmp = math.sqrt((U * U_42_)) * (l_m * (n * (math.sqrt(2.0) / -Om)))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * t_1)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(t - Float64(Float64(2.0 * (l_m ^ 2.0)) / Om))
	tmp = 0.0
	if (U <= -4.1e-5)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * t_1));
	elseif (U <= -1.3e-58)
		tmp = Float64(sqrt(Float64(U * U_42_)) * Float64(l_m * Float64(n * Float64(sqrt(2.0) / Float64(-Om)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t_1)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = t - ((2.0 * (l_m ^ 2.0)) / Om);
	tmp = 0.0;
	if (U <= -4.1e-5)
		tmp = sqrt(((2.0 * (n * U)) * t_1));
	elseif (U <= -1.3e-58)
		tmp = sqrt((U * U_42_)) * (l_m * (n * (sqrt(2.0) / -Om)));
	else
		tmp = sqrt(((2.0 * n) * (U * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -4.1e-5], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, -1.3e-58], N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / (-Om)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if U < -4.10000000000000005e-5

   1. Initial program 58.5%

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

   3. Simplified 58.6%

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

   5. Taylor expanded in Om around inf 51.6%

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

      1. associate-*r/ 51.6%

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

   7. Simplified 51.6%

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

   if -4.10000000000000005e-5 < U < -1.30000000000000003e-58

   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 26.9%

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

   5. Taylor expanded in Om around -inf 33.9%

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

      1. +-commutative 33.9%

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

      2. mul-1-neg 33.9%

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

      3. unsub-neg 33.9%

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

      4. +-commutative 33.9%

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

      5. mul-1-neg 33.9%

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

      6. unsub-neg 33.9%

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

      7. associate-*r* 33.9%

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

      8. associate-/l* 33.9%

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

      9. associate-/l* 33.9%

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

   7. Simplified 33.9%

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

   8. Taylor expanded in U* around inf 40.4%

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

      1. associate-*r* 40.4%

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

   10. Simplified 40.4%

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

   11. Taylor expanded in n around -inf 31.6%

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

       1. mul-1-neg 31.6%

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

       2. *-commutative 31.6%

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

       3. associate-*r* 31.7%

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

       4. distribute-rgt-neg-in 31.7%

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

       5. *-commutative 31.7%

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

       6. associate-*r* 31.6%

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

       7. associate-*r/ 38.8%

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

       8. associate-/l* 38.9%

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

   13. Simplified 38.9%

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

   if -1.30000000000000003e-58 < U

   1. Initial program 48.0%

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

   3. Simplified 53.9%

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

   5. Taylor expanded in Om around inf 44.1%

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

      1. metadata-eval 44.1%

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

      2. cancel-sign-sub-inv 44.1%

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

      3. associate-*r/ 44.1%

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

   7. Simplified 44.1%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -4.1 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;U \leq -1.3 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{-Om}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.02 \cdot 10^{+194}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot {l\_m}^{2}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(l\_m \cdot \left(n \cdot \frac{\sqrt{2}}{-Om}\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.02e+194)
   (sqrt (* (* 2.0 n) (* U (- t (/ (* 2.0 (pow l_m 2.0)) Om)))))
   (* (sqrt (* U U*)) (* l_m (* n (/ (sqrt 2.0) (- Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.02e+194) {
		tmp = sqrt(((2.0 * n) * (U * (t - ((2.0 * pow(l_m, 2.0)) / Om)))));
	} else {
		tmp = sqrt((U * U_42_)) * (l_m * (n * (sqrt(2.0) / -Om)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 1.02d+194) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - ((2.0d0 * (l_m ** 2.0d0)) / om)))))
    else
        tmp = sqrt((u * u_42)) * (l_m * (n * (sqrt(2.0d0) / -om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.02e+194) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - ((2.0 * Math.pow(l_m, 2.0)) / Om)))));
	} else {
		tmp = Math.sqrt((U * U_42_)) * (l_m * (n * (Math.sqrt(2.0) / -Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.02e+194:
		tmp = math.sqrt(((2.0 * n) * (U * (t - ((2.0 * math.pow(l_m, 2.0)) / Om)))))
	else:
		tmp = math.sqrt((U * U_42_)) * (l_m * (n * (math.sqrt(2.0) / -Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.02e+194)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(Float64(2.0 * (l_m ^ 2.0)) / Om)))));
	else
		tmp = Float64(sqrt(Float64(U * U_42_)) * Float64(l_m * Float64(n * Float64(sqrt(2.0) / Float64(-Om)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 1.02e+194)
		tmp = sqrt(((2.0 * n) * (U * (t - ((2.0 * (l_m ^ 2.0)) / Om)))));
	else
		tmp = sqrt((U * U_42_)) * (l_m * (n * (sqrt(2.0) / -Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.02e+194], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / (-Om)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.02e194

   1. Initial program 52.9%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in Om around inf 45.5%

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

      1. metadata-eval 45.5%

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

      2. cancel-sign-sub-inv 45.5%

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

      3. associate-*r/ 45.5%

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

   7. Simplified 45.5%

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

   if 1.02e194 < l

   1. Initial program 6.3%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in Om around -inf 1.4%

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

      1. +-commutative 1.4%

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

      2. mul-1-neg 1.4%

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

      3. unsub-neg 1.4%

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

      4. +-commutative 1.4%

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

      5. mul-1-neg 1.4%

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

      6. unsub-neg 1.4%

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

      7. associate-*r* 1.4%

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

      8. associate-/l* 1.4%

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

      9. associate-/l* 1.2%

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

   7. Simplified 1.2%

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

   8. Taylor expanded in U* around inf 11.5%

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

      1. associate-*r* 11.3%

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

   10. Simplified 11.3%

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

   11. Taylor expanded in n around -inf 25.0%

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

       1. mul-1-neg 25.0%

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

       2. *-commutative 25.0%

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

       3. associate-*r* 25.2%

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

       4. distribute-rgt-neg-in 25.2%

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

       5. *-commutative 25.2%

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

       6. associate-*r* 25.0%

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

       7. associate-*r/ 29.5%

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

       8. associate-/l* 29.6%

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

   13. Simplified 29.6%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.02 \cdot 10^{+194}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{-Om}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 6.5 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(l\_m \cdot \left(n \cdot \frac{\sqrt{2}}{-Om}\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 6.5e+41)
   (sqrt (fabs (* (* 2.0 n) (* U t))))
   (* (sqrt (* U U*)) (* l_m (* n (/ (sqrt 2.0) (- Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 6.5e+41) {
		tmp = sqrt(fabs(((2.0 * n) * (U * t))));
	} else {
		tmp = sqrt((U * U_42_)) * (l_m * (n * (sqrt(2.0) / -Om)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 6.5d+41) then
        tmp = sqrt(abs(((2.0d0 * n) * (u * t))))
    else
        tmp = sqrt((u * u_42)) * (l_m * (n * (sqrt(2.0d0) / -om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 6.5e+41) {
		tmp = Math.sqrt(Math.abs(((2.0 * n) * (U * t))));
	} else {
		tmp = Math.sqrt((U * U_42_)) * (l_m * (n * (Math.sqrt(2.0) / -Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 6.5e+41:
		tmp = math.sqrt(math.fabs(((2.0 * n) * (U * t))))
	else:
		tmp = math.sqrt((U * U_42_)) * (l_m * (n * (math.sqrt(2.0) / -Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 6.5e+41)
		tmp = sqrt(abs(Float64(Float64(2.0 * n) * Float64(U * t))));
	else
		tmp = Float64(sqrt(Float64(U * U_42_)) * Float64(l_m * Float64(n * Float64(sqrt(2.0) / Float64(-Om)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 6.5e+41)
		tmp = sqrt(abs(((2.0 * n) * (U * t))));
	else
		tmp = sqrt((U * U_42_)) * (l_m * (n * (sqrt(2.0) / -Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 6.5e+41], N[Sqrt[N[Abs[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / (-Om)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.49999999999999975e41

   1. Initial program 53.6%

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

   3. Simplified 56.0%

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

   5. Taylor expanded in t around inf 38.9%

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

      1. add-sqr-sqrt 38.9%

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

      2. pow1/2 38.9%

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

      3. pow1/2 39.4%

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

      4. pow-prod-down 28.1%

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

      5. pow2 28.1%

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

   7. Applied egg-rr 28.1%

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

      1. unpow1/2 28.1%

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

      2. unpow2 28.1%

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

      3. rem-sqrt-square 39.9%

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

      4. *-commutative 39.9%

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

   9. Simplified 39.9%

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

   if 6.49999999999999975e41 < l

   1. Initial program 29.9%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in Om around -inf 26.0%

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

      1. +-commutative 26.0%

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

      2. mul-1-neg 26.0%

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

      3. unsub-neg 26.0%

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

      4. +-commutative 26.0%

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

      5. mul-1-neg 26.0%

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

      6. unsub-neg 26.0%

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

      7. associate-*r* 26.0%

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

      8. associate-/l* 26.0%

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

      9. associate-/l* 29.9%

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

   7. Simplified 29.9%

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

   8. Taylor expanded in U* around inf 22.5%

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

      1. associate-*r* 22.1%

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

   10. Simplified 22.1%

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

   11. Taylor expanded in n around -inf 27.4%

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

       1. mul-1-neg 27.4%

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

       2. *-commutative 27.4%

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

       3. associate-*r* 27.5%

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

       4. distribute-rgt-neg-in 27.5%

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

       5. *-commutative 27.5%

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

       6. associate-*r* 27.4%

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

       7. associate-*r/ 29.4%

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

       8. associate-/l* 29.5%

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

   13. Simplified 29.5%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{-Om}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 7 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(l\_m \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 7e+97)
   (sqrt (fabs (* (* 2.0 n) (* U t))))
   (* (sqrt (* U U*)) (* l_m (* n (/ (sqrt 2.0) Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 7e+97) {
		tmp = sqrt(fabs(((2.0 * n) * (U * t))));
	} else {
		tmp = sqrt((U * U_42_)) * (l_m * (n * (sqrt(2.0) / Om)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 7d+97) then
        tmp = sqrt(abs(((2.0d0 * n) * (u * t))))
    else
        tmp = sqrt((u * u_42)) * (l_m * (n * (sqrt(2.0d0) / om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 7e+97) {
		tmp = Math.sqrt(Math.abs(((2.0 * n) * (U * t))));
	} else {
		tmp = Math.sqrt((U * U_42_)) * (l_m * (n * (Math.sqrt(2.0) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 7e+97:
		tmp = math.sqrt(math.fabs(((2.0 * n) * (U * t))))
	else:
		tmp = math.sqrt((U * U_42_)) * (l_m * (n * (math.sqrt(2.0) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 7e+97)
		tmp = sqrt(abs(Float64(Float64(2.0 * n) * Float64(U * t))));
	else
		tmp = Float64(sqrt(Float64(U * U_42_)) * Float64(l_m * Float64(n * Float64(sqrt(2.0) / Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 7e+97)
		tmp = sqrt(abs(((2.0 * n) * (U * t))));
	else
		tmp = sqrt((U * U_42_)) * (l_m * (n * (sqrt(2.0) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 7e+97], N[Sqrt[N[Abs[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.0000000000000001e97

   1. Initial program 52.4%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in t around inf 38.0%

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

      1. add-sqr-sqrt 38.0%

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

      2. pow1/2 38.0%

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

      3. pow1/2 38.4%

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

      4. pow-prod-down 27.3%

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

      5. pow2 27.3%

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

   7. Applied egg-rr 27.3%

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

      1. unpow1/2 27.3%

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

      2. unpow2 27.3%

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

      3. rem-sqrt-square 39.0%

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

      4. *-commutative 39.0%

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

   9. Simplified 39.0%

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

   if 7.0000000000000001e97 < l

   1. Initial program 30.1%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in U around 0 25.3%

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

      1. mul-1-neg 25.3%

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

      2. associate-/l* 25.3%

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

      3. unpow2 25.3%

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

      4. unpow2 25.3%

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

      5. times-frac 37.0%

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

      6. unpow2 37.0%

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

   7. Simplified 37.0%

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

      1. clear-num 36.9%

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

      2. inv-pow 36.9%

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

   9. Applied egg-rr 36.9%

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

       1. unpow-1 36.9%

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

   11. Simplified 36.9%

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

   12. Taylor expanded in n around inf 11.9%

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

       1. *-commutative 11.9%

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

       2. *-commutative 11.9%

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

       3. associate-*r/ 19.5%

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

       4. associate-/l* 19.5%

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

   14. Simplified 19.5%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 4.4 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 4.4e-308)
   (sqrt (fabs (* (* 2.0 n) (* U t))))
   (* (sqrt (* 2.0 n)) (sqrt (* U t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 4.4e-308) {
		tmp = sqrt(fabs(((2.0 * n) * (U * t))));
	} else {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 4.4d-308) then
        tmp = sqrt(abs(((2.0d0 * n) * (u * t))))
    else
        tmp = sqrt((2.0d0 * n)) * sqrt((u * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 4.4e-308) {
		tmp = Math.sqrt(Math.abs(((2.0 * n) * (U * t))));
	} else {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 4.4e-308:
		tmp = math.sqrt(math.fabs(((2.0 * n) * (U * t))))
	else:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 4.4e-308)
		tmp = sqrt(abs(Float64(Float64(2.0 * n) * Float64(U * t))));
	else
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (n <= 4.4e-308)
		tmp = sqrt(abs(((2.0 * n) * (U * t))));
	else
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, 4.4e-308], N[Sqrt[N[Abs[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 4.3999999999999999e-308

   1. Initial program 49.1%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in t around inf 35.0%

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

      1. add-sqr-sqrt 35.0%

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

      2. pow1/2 35.0%

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

      3. pow1/2 35.7%

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

      4. pow-prod-down 26.5%

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

      5. pow2 26.5%

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

   7. Applied egg-rr 26.5%

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

      1. unpow1/2 26.5%

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

      2. unpow2 26.5%

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

      3. rem-sqrt-square 36.3%

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

      4. *-commutative 36.3%

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

   9. Simplified 36.3%

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

   if 4.3999999999999999e-308 < n

   1. Initial program 49.0%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in t around inf 31.0%

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

      1. pow1/2 31.9%

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

      2. *-commutative 31.9%

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

      3. unpow-prod-down 39.9%

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

      4. pow1/2 39.0%

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

      5. pow1/2 39.0%

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

   7. Applied egg-rr 39.0%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.4 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -5.8e-226)
   (sqrt (fabs (* (* 2.0 n) (* U t))))
   (sqrt (fabs (* 2.0 (* U (* n t)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -5.8e-226) {
		tmp = sqrt(fabs(((2.0 * n) * (U * t))));
	} else {
		tmp = sqrt(fabs((2.0 * (U * (n * t)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-5.8d-226)) then
        tmp = sqrt(abs(((2.0d0 * n) * (u * t))))
    else
        tmp = sqrt(abs((2.0d0 * (u * (n * t)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -5.8e-226) {
		tmp = Math.sqrt(Math.abs(((2.0 * n) * (U * t))));
	} else {
		tmp = Math.sqrt(Math.abs((2.0 * (U * (n * t)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -5.8e-226:
		tmp = math.sqrt(math.fabs(((2.0 * n) * (U * t))))
	else:
		tmp = math.sqrt(math.fabs((2.0 * (U * (n * t)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -5.8e-226)
		tmp = sqrt(abs(Float64(Float64(2.0 * n) * Float64(U * t))));
	else
		tmp = sqrt(abs(Float64(2.0 * Float64(U * Float64(n * t)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -5.8e-226)
		tmp = sqrt(abs(((2.0 * n) * (U * t))));
	else
		tmp = sqrt(abs((2.0 * (U * (n * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -5.8e-226], N[Sqrt[N[Abs[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.80000000000000003e-226

   1. Initial program 45.9%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in t around inf 36.4%

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

      1. add-sqr-sqrt 36.4%

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

      2. pow1/2 36.4%

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

      3. pow1/2 37.3%

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

      4. pow-prod-down 24.2%

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

      5. pow2 24.2%

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

   7. Applied egg-rr 24.2%

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

      1. unpow1/2 24.2%

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

      2. unpow2 24.2%

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

      3. rem-sqrt-square 37.8%

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

      4. *-commutative 37.8%

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

   9. Simplified 37.8%

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

   if -5.80000000000000003e-226 < t

   1. Initial program 51.8%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in l around 0 34.0%

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

      1. associate-*r* 34.0%

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

   7. Simplified 34.0%

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

   8. Taylor expanded in U around 0 34.0%

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

      1. associate-*r* 34.0%

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

      2. associate-*r* 30.5%

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

   10. Simplified 30.5%

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

       1. pow1 30.5%

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

       2. metadata-eval 30.5%

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

       3. pow-prod-up 31.2%

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

       4. pow-prod-down 27.5%

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

       5. pow2 27.5%

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

       6. associate-*l* 25.7%

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

       7. associate-*l* 25.7%

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

   12. Applied egg-rr 25.7%

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

       1. unpow1/2 25.7%

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

       2. unpow2 25.7%

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

       3. rem-sqrt-square 35.6%

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

   14. Simplified 35.6%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-226}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-224}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -3.8e-224)
   (pow (* (* 2.0 n) (* U t)) 0.5)
   (sqrt (fabs (* 2.0 (* U (* n t)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -3.8e-224) {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = sqrt(fabs((2.0 * (U * (n * t)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-3.8d-224)) then
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    else
        tmp = sqrt(abs((2.0d0 * (u * (n * t)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -3.8e-224) {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = Math.sqrt(Math.abs((2.0 * (U * (n * t)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -3.8e-224:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	else:
		tmp = math.sqrt(math.fabs((2.0 * (U * (n * t)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -3.8e-224)
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	else
		tmp = sqrt(abs(Float64(2.0 * Float64(U * Float64(n * t)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -3.8e-224)
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	else
		tmp = sqrt(abs((2.0 * (U * (n * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -3.8e-224], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[Abs[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.80000000000000002e-224

   1. Initial program 45.9%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in t around inf 36.4%

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

      1. pow1/2 37.3%

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

   7. Applied egg-rr 37.3%

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

   if -3.80000000000000002e-224 < t

   1. Initial program 51.8%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in l around 0 34.0%

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

      1. associate-*r* 34.0%

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

   7. Simplified 34.0%

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

   8. Taylor expanded in U around 0 34.0%

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

      1. associate-*r* 34.0%

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

      2. associate-*r* 30.5%

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

   10. Simplified 30.5%

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

       1. pow1 30.5%

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

       2. metadata-eval 30.5%

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

       3. pow-prod-up 31.2%

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

       4. pow-prod-down 27.5%

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

       5. pow2 27.5%

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

       6. associate-*l* 25.7%

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

       7. associate-*l* 25.7%

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

   12. Applied egg-rr 25.7%

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

       1. unpow1/2 25.7%

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

       2. unpow2 25.7%

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

       3. rem-sqrt-square 35.6%

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

   14. Simplified 35.6%

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

Alternative 15: 36.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-226}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -9.5e-226)
   (pow (* (* 2.0 n) (* U t)) 0.5)
   (pow (* (* n t) (* 2.0 U)) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -9.5e-226) {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = pow(((n * t) * (2.0 * U)), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-9.5d-226)) then
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    else
        tmp = ((n * t) * (2.0d0 * u)) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -9.5e-226) {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	} else {
		tmp = Math.pow(((n * t) * (2.0 * U)), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -9.5e-226:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	else:
		tmp = math.pow(((n * t) * (2.0 * U)), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -9.5e-226)
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	else
		tmp = Float64(Float64(n * t) * Float64(2.0 * U)) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -9.5e-226)
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	else
		tmp = ((n * t) * (2.0 * U)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -9.5e-226], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.5000000000000007e-226

   1. Initial program 45.9%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in t around inf 36.4%

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

      1. pow1/2 37.3%

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

   7. Applied egg-rr 37.3%

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

   if -9.5000000000000007e-226 < t

   1. Initial program 51.8%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in l around 0 34.0%

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

      1. associate-*r* 34.0%

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

   7. Simplified 34.0%

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

      1. pow1/2 34.8%

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

   9. Applied egg-rr 34.8%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-226}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 36.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.15 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om -1.15e-51)
   (sqrt (* (* 2.0 n) (* U t)))
   (pow (* (* n t) (* 2.0 U)) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= -1.15e-51) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = pow(((n * t) * (2.0 * U)), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= (-1.15d-51)) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else
        tmp = ((n * t) * (2.0d0 * u)) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= -1.15e-51) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = Math.pow(((n * t) * (2.0 * U)), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if Om <= -1.15e-51:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	else:
		tmp = math.pow(((n * t) * (2.0 * U)), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Om <= -1.15e-51)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	else
		tmp = Float64(Float64(n * t) * Float64(2.0 * U)) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (Om <= -1.15e-51)
		tmp = sqrt(((2.0 * n) * (U * t)));
	else
		tmp = ((n * t) * (2.0 * U)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[Om, -1.15e-51], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.15000000000000001e-51

   1. Initial program 48.0%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in t around inf 40.6%

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

   if -1.15000000000000001e-51 < Om

   1. Initial program 49.5%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in l around 0 32.8%

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

      1. associate-*r* 32.8%

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

   7. Simplified 32.8%

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

      1. pow1/2 33.9%

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

   9. Applied egg-rr 33.9%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.15 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 34.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -3.2e-225)
   (sqrt (* (* 2.0 n) (* U t)))
   (sqrt (* (* n t) (* 2.0 U)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -3.2e-225) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = sqrt(((n * t) * (2.0 * U)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-3.2d-225)) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else
        tmp = sqrt(((n * t) * (2.0d0 * u)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -3.2e-225) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = Math.sqrt(((n * t) * (2.0 * U)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -3.2e-225:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	else:
		tmp = math.sqrt(((n * t) * (2.0 * U)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -3.2e-225)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	else
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * U)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -3.2e-225)
		tmp = sqrt(((2.0 * n) * (U * t)));
	else
		tmp = sqrt(((n * t) * (2.0 * U)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -3.2e-225], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.19999999999999975e-225

   1. Initial program 45.9%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in t around inf 36.4%

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

   if -3.19999999999999975e-225 < t

   1. Initial program 51.8%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in l around 0 34.0%

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

      1. associate-*r* 34.0%

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

   7. Simplified 34.0%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* n t) (* 2.0 U))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt(((n * t) * (2.0 * U)));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.1%

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

3. Simplified 52.4%

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

5. Taylor expanded in l around 0 32.8%

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

   1. associate-*r* 32.8%

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

7. Simplified 32.8%

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

8. Final simplification 32.8%

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

Reproduce

herbie shell --seed 2024157 
(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*))))))