Toniolo and Linder, Equation (13)

Percentage Accurate: 50.5% → 59.3%

Time: 1.1min

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 59.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \frac{U \cdot \left({l\_m}^{2} \cdot \left({n}^{2} \cdot \left(U* - U\right)\right)\right)}{Om} - 4 \cdot \left(U \cdot \left(n \cdot {l\_m}^{2}\right)\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left(t\_1 \cdot \left(U* - U\right)\right) + 2 \cdot \left(l\_m \cdot \left(l\_m \cdot \frac{-1}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \left(\sqrt{\left|U \cdot U*\right|} \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l_m Om) 2.0))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 (/ (* l_m l_m) Om))) (* (* n t_1) (- U* U))))))
   (if (<= t_2 4e-279)
     (sqrt
      (+
       (/
        (-
         (* 2.0 (/ (* U (* (pow l_m 2.0) (* (pow n 2.0) (- U* U)))) Om))
         (* 4.0 (* U (* n (pow l_m 2.0)))))
        Om)
       (* 2.0 (* U (* n t)))))
     (if (<= t_2 INFINITY)
       (sqrt
        (*
         (* 2.0 (* n U))
         (+ t (+ (* n (* t_1 (- U* U))) (* 2.0 (* l_m (* l_m (/ -1.0 Om))))))))
       (* l_m (* (sqrt (fabs (* U U*))) (/ (* 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 = pow((l_m / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-279) {
		tmp = sqrt(((((2.0 * ((U * (pow(l_m, 2.0) * (pow(n, 2.0) * (U_42_ - U)))) / Om)) - (4.0 * (U * (n * pow(l_m, 2.0))))) / Om) + (2.0 * (U * (n * t)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))));
	} else {
		tmp = l_m * (sqrt(fabs((U * U_42_))) * ((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 = Math.pow((l_m / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-279) {
		tmp = Math.sqrt(((((2.0 * ((U * (Math.pow(l_m, 2.0) * (Math.pow(n, 2.0) * (U_42_ - U)))) / Om)) - (4.0 * (U * (n * Math.pow(l_m, 2.0))))) / Om) + (2.0 * (U * (n * t)))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))));
	} else {
		tmp = l_m * (Math.sqrt(Math.abs((U * U_42_))) * ((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 = math.pow((l_m / Om), 2.0)
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)))
	tmp = 0
	if t_2 <= 4e-279:
		tmp = math.sqrt(((((2.0 * ((U * (math.pow(l_m, 2.0) * (math.pow(n, 2.0) * (U_42_ - U)))) / Om)) - (4.0 * (U * (n * math.pow(l_m, 2.0))))) / Om) + (2.0 * (U * (n * t)))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))))
	else:
		tmp = l_m * (math.sqrt(math.fabs((U * U_42_))) * ((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(l_m / Om) ^ 2.0
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * t_1) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 4e-279)
		tmp = sqrt(Float64(Float64(Float64(Float64(2.0 * Float64(Float64(U * Float64((l_m ^ 2.0) * Float64((n ^ 2.0) * Float64(U_42_ - U)))) / Om)) - Float64(4.0 * Float64(U * Float64(n * (l_m ^ 2.0))))) / Om) + Float64(2.0 * Float64(U * Float64(n * t)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * Float64(t_1 * Float64(U_42_ - U))) + Float64(2.0 * Float64(l_m * Float64(l_m * Float64(-1.0 / Om))))))));
	else
		tmp = Float64(l_m * Float64(sqrt(abs(Float64(U * U_42_))) * Float64(Float64(n * 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_)
	t_1 = (l_m / Om) ^ 2.0;
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	tmp = 0.0;
	if (t_2 <= 4e-279)
		tmp = sqrt(((((2.0 * ((U * ((l_m ^ 2.0) * ((n ^ 2.0) * (U_42_ - U)))) / Om)) - (4.0 * (U * (n * (l_m ^ 2.0))))) / Om) + (2.0 * (U * (n * t)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))));
	else
		tmp = l_m * (sqrt(abs((U * U_42_))) * ((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[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = 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] + N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-279], N[Sqrt[N[(N[(N[(N[(2.0 * N[(N[(U * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[Power[n, 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] - N[(4.0 * N[(U * N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $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[(N[(n * N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l$95$m * N[(l$95$m * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[(N[Sqrt[N[Abs[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.00000000000000022e-279

   1. Initial program 18.7%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in Om around -inf 49.4%

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

   if 4.00000000000000022e-279 < (*.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 63.1%

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

   3. Simplified 68.1%

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

      1. sub-neg 68.1%

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

      2. distribute-lft-in 57.3%

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

   6. Applied egg-rr 57.3%

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

      1. distribute-lft-out 68.1%

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

      2. sub-neg 68.1%

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

      3. associate-*r* 69.1%

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

      4. *-commutative 69.1%

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

   8. Simplified 69.1%

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

      1. div-inv 69.1%

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

   10. Applied egg-rr 69.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.3%

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

      1. sub-neg 0.3%

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

      2. distribute-lft-in 0.3%

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

   6. Applied egg-rr 0.3%

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

      1. distribute-lft-out 0.3%

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

      2. sub-neg 0.3%

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

      3. associate-*r* 3.0%

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

      4. *-commutative 3.0%

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

   8. Simplified 3.0%

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

      1. div-inv 3.0%

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

   10. Applied egg-rr 3.0%

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

   11. Taylor expanded in U* around inf 30.7%

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

       1. associate-/l* 33.3%

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

       2. associate-*r/ 33.3%

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

       3. associate-*r* 33.3%

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

       4. *-commutative 33.3%

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

       5. associate-*r/ 33.3%

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

   13. Simplified 33.3%

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

       1. add-sqr-sqrt 33.3%

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

       2. pow1/2 33.3%

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

       3. pow1/2 33.3%

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

       4. pow-prod-down 28.4%

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

       5. pow2 28.4%

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

   15. Applied egg-rr 28.4%

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

       1. unpow1/2 28.4%

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

       2. unpow2 28.4%

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

       3. rem-sqrt-square 33.7%

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

   17. Simplified 33.7%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U* - U\right)\right)\right)}{Om} - 4 \cdot \left(U \cdot \left(n \cdot {\ell}^{2}\right)\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) + 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\left|U \cdot U*\right|} \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\frac{\frac{U \cdot \left({l\_m}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{Om} - 2 \cdot \left(U \cdot {l\_m}^{2}\right)}{Om} + U \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left(t\_1 \cdot \left(U* - U\right)\right) + 2 \cdot \left(l\_m \cdot \left(l\_m \cdot \frac{-1}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \left(\sqrt{\left|U \cdot U*\right|} \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l_m Om) 2.0))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 (/ (* l_m l_m) Om))) (* (* n t_1) (- U* U))))))
   (if (<= t_2 4e-279)
     (sqrt
      (*
       (* 2.0 n)
       (+
        (/
         (-
          (/ (* U (* (pow l_m 2.0) (* n (- U* U)))) Om)
          (* 2.0 (* U (pow l_m 2.0))))
         Om)
        (* U t))))
     (if (<= t_2 INFINITY)
       (sqrt
        (*
         (* 2.0 (* n U))
         (+ t (+ (* n (* t_1 (- U* U))) (* 2.0 (* l_m (* l_m (/ -1.0 Om))))))))
       (* l_m (* (sqrt (fabs (* U U*))) (/ (* 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 = pow((l_m / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-279) {
		tmp = sqrt(((2.0 * n) * (((((U * (pow(l_m, 2.0) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * pow(l_m, 2.0)))) / Om) + (U * t))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))));
	} else {
		tmp = l_m * (sqrt(fabs((U * U_42_))) * ((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 = Math.pow((l_m / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-279) {
		tmp = Math.sqrt(((2.0 * n) * (((((U * (Math.pow(l_m, 2.0) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * Math.pow(l_m, 2.0)))) / Om) + (U * t))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))));
	} else {
		tmp = l_m * (Math.sqrt(Math.abs((U * U_42_))) * ((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 = math.pow((l_m / Om), 2.0)
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)))
	tmp = 0
	if t_2 <= 4e-279:
		tmp = math.sqrt(((2.0 * n) * (((((U * (math.pow(l_m, 2.0) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * math.pow(l_m, 2.0)))) / Om) + (U * t))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))))
	else:
		tmp = l_m * (math.sqrt(math.fabs((U * U_42_))) * ((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(l_m / Om) ^ 2.0
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * t_1) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 4e-279)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(Float64(Float64(Float64(U * Float64((l_m ^ 2.0) * Float64(n * Float64(U_42_ - U)))) / Om) - Float64(2.0 * Float64(U * (l_m ^ 2.0)))) / Om) + Float64(U * t))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * Float64(t_1 * Float64(U_42_ - U))) + Float64(2.0 * Float64(l_m * Float64(l_m * Float64(-1.0 / Om))))))));
	else
		tmp = Float64(l_m * Float64(sqrt(abs(Float64(U * U_42_))) * Float64(Float64(n * 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_)
	t_1 = (l_m / Om) ^ 2.0;
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	tmp = 0.0;
	if (t_2 <= 4e-279)
		tmp = sqrt(((2.0 * n) * (((((U * ((l_m ^ 2.0) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * (l_m ^ 2.0)))) / Om) + (U * t))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))));
	else
		tmp = l_m * (sqrt(abs((U * U_42_))) * ((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[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = 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] + N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-279], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(N[(N[(N[(U * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(2.0 * N[(U * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n * N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l$95$m * N[(l$95$m * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[(N[Sqrt[N[Abs[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.00000000000000022e-279

   1. Initial program 18.7%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in Om around -inf 47.2%

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

   if 4.00000000000000022e-279 < (*.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 63.1%

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

   3. Simplified 68.1%

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

      1. sub-neg 68.1%

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

      2. distribute-lft-in 57.3%

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

   6. Applied egg-rr 57.3%

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

      1. distribute-lft-out 68.1%

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

      2. sub-neg 68.1%

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

      3. associate-*r* 69.1%

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

      4. *-commutative 69.1%

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

   8. Simplified 69.1%

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

      1. div-inv 69.1%

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

   10. Applied egg-rr 69.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.3%

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

      1. sub-neg 0.3%

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

      2. distribute-lft-in 0.3%

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

   6. Applied egg-rr 0.3%

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

      1. distribute-lft-out 0.3%

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

      2. sub-neg 0.3%

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

      3. associate-*r* 3.0%

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

      4. *-commutative 3.0%

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

   8. Simplified 3.0%

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

      1. div-inv 3.0%

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

   10. Applied egg-rr 3.0%

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

   11. Taylor expanded in U* around inf 30.7%

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

       1. associate-/l* 33.3%

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

       2. associate-*r/ 33.3%

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

       3. associate-*r* 33.3%

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

       4. *-commutative 33.3%

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

       5. associate-*r/ 33.3%

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

   13. Simplified 33.3%

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

       1. add-sqr-sqrt 33.3%

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

       2. pow1/2 33.3%

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

       3. pow1/2 33.3%

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

       4. pow-prod-down 28.4%

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

       5. pow2 28.4%

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

   15. Applied egg-rr 28.4%

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

       1. unpow1/2 28.4%

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

       2. unpow2 28.4%

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

       3. rem-sqrt-square 33.7%

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

   17. Simplified 33.7%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\frac{\frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{Om} - 2 \cdot \left(U \cdot {\ell}^{2}\right)}{Om} + U \cdot t\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) + 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\left|U \cdot U*\right|} \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{{l\_m}^{2}}{Om} \cdot -2\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left(t\_1 \cdot \left(U* - U\right)\right) + 2 \cdot \left(l\_m \cdot \left(l\_m \cdot \frac{-1}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \left(\sqrt{\left|U \cdot U*\right|} \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l_m Om) 2.0))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 (/ (* l_m l_m) Om))) (* (* n t_1) (- U* U))))))
   (if (<= t_2 4e-279)
     (sqrt (* (* 2.0 U) (* n (+ t (* (/ (pow l_m 2.0) Om) -2.0)))))
     (if (<= t_2 INFINITY)
       (sqrt
        (*
         (* 2.0 (* n U))
         (+ t (+ (* n (* t_1 (- U* U))) (* 2.0 (* l_m (* l_m (/ -1.0 Om))))))))
       (* l_m (* (sqrt (fabs (* U U*))) (/ (* 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 = pow((l_m / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-279) {
		tmp = sqrt(((2.0 * U) * (n * (t + ((pow(l_m, 2.0) / Om) * -2.0)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))));
	} else {
		tmp = l_m * (sqrt(fabs((U * U_42_))) * ((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 = Math.pow((l_m / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-279) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t + ((Math.pow(l_m, 2.0) / Om) * -2.0)))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))));
	} else {
		tmp = l_m * (Math.sqrt(Math.abs((U * U_42_))) * ((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 = math.pow((l_m / Om), 2.0)
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)))
	tmp = 0
	if t_2 <= 4e-279:
		tmp = math.sqrt(((2.0 * U) * (n * (t + ((math.pow(l_m, 2.0) / Om) * -2.0)))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))))
	else:
		tmp = l_m * (math.sqrt(math.fabs((U * U_42_))) * ((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(l_m / Om) ^ 2.0
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * t_1) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 4e-279)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t + Float64(Float64((l_m ^ 2.0) / Om) * -2.0)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * Float64(t_1 * Float64(U_42_ - U))) + Float64(2.0 * Float64(l_m * Float64(l_m * Float64(-1.0 / Om))))))));
	else
		tmp = Float64(l_m * Float64(sqrt(abs(Float64(U * U_42_))) * Float64(Float64(n * 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_)
	t_1 = (l_m / Om) ^ 2.0;
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	tmp = 0.0;
	if (t_2 <= 4e-279)
		tmp = sqrt(((2.0 * U) * (n * (t + (((l_m ^ 2.0) / Om) * -2.0)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))));
	else
		tmp = l_m * (sqrt(abs((U * U_42_))) * ((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[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = 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] + N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-279], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t + N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0), $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[(N[(n * N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l$95$m * N[(l$95$m * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[(N[Sqrt[N[Abs[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.00000000000000022e-279

   1. Initial program 18.7%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in U around 0 35.4%

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

      1. associate-/l* 33.2%

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

      2. unpow2 33.2%

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

      3. unpow2 33.2%

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

      4. times-frac 40.5%

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

      5. unpow2 40.5%

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

      6. neg-mul-1 40.5%

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

      7. distribute-lft-neg-out 40.5%

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

      8. *-commutative 40.5%

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

   7. Simplified 40.5%

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

   8. Taylor expanded in n around 0 47.2%

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

      1. associate-*r* 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. *-commutative 47.2%

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

   10. Simplified 47.2%

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

   if 4.00000000000000022e-279 < (*.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 63.1%

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

   3. Simplified 68.1%

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

      1. sub-neg 68.1%

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

      2. distribute-lft-in 57.3%

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

   6. Applied egg-rr 57.3%

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

      1. distribute-lft-out 68.1%

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

      2. sub-neg 68.1%

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

      3. associate-*r* 69.1%

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

      4. *-commutative 69.1%

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

   8. Simplified 69.1%

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

      1. div-inv 69.1%

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

   10. Applied egg-rr 69.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.3%

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

      1. sub-neg 0.3%

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

      2. distribute-lft-in 0.3%

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

   6. Applied egg-rr 0.3%

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

      1. distribute-lft-out 0.3%

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

      2. sub-neg 0.3%

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

      3. associate-*r* 3.0%

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

      4. *-commutative 3.0%

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

   8. Simplified 3.0%

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

      1. div-inv 3.0%

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

   10. Applied egg-rr 3.0%

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

   11. Taylor expanded in U* around inf 30.7%

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

       1. associate-/l* 33.3%

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

       2. associate-*r/ 33.3%

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

       3. associate-*r* 33.3%

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

       4. *-commutative 33.3%

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

       5. associate-*r/ 33.3%

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

   13. Simplified 33.3%

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

       1. add-sqr-sqrt 33.3%

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

       2. pow1/2 33.3%

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

       3. pow1/2 33.3%

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

       4. pow-prod-down 28.4%

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

       5. pow2 28.4%

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

   15. Applied egg-rr 28.4%

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

       1. unpow1/2 28.4%

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

       2. unpow2 28.4%

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

       3. rem-sqrt-square 33.7%

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

   17. Simplified 33.7%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) + 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{\left|U \cdot U*\right|} \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{{l\_m}^{2}}{Om} \cdot -2\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left(t\_1 \cdot \left(U* - U\right)\right) + 2 \cdot \left(l\_m \cdot \left(l\_m \cdot \frac{-1}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \left(\sqrt{U \cdot U*} \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l_m Om) 2.0))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 (/ (* l_m l_m) Om))) (* (* n t_1) (- U* U))))))
   (if (<= t_2 4e-279)
     (sqrt (* (* 2.0 U) (* n (+ t (* (/ (pow l_m 2.0) Om) -2.0)))))
     (if (<= t_2 INFINITY)
       (sqrt
        (*
         (* 2.0 (* n U))
         (+ t (+ (* n (* t_1 (- U* U))) (* 2.0 (* l_m (* l_m (/ -1.0 Om))))))))
       (* l_m (* (sqrt (* U U*)) (* 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 = pow((l_m / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-279) {
		tmp = sqrt(((2.0 * U) * (n * (t + ((pow(l_m, 2.0) / Om) * -2.0)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))));
	} else {
		tmp = l_m * (sqrt((U * U_42_)) * (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 = Math.pow((l_m / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-279) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t + ((Math.pow(l_m, 2.0) / Om) * -2.0)))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))));
	} else {
		tmp = l_m * (Math.sqrt((U * U_42_)) * (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 = math.pow((l_m / Om), 2.0)
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)))
	tmp = 0
	if t_2 <= 4e-279:
		tmp = math.sqrt(((2.0 * U) * (n * (t + ((math.pow(l_m, 2.0) / Om) * -2.0)))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))))
	else:
		tmp = l_m * (math.sqrt((U * U_42_)) * (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(l_m / Om) ^ 2.0
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * t_1) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 4e-279)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t + Float64(Float64((l_m ^ 2.0) / Om) * -2.0)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * Float64(t_1 * Float64(U_42_ - U))) + Float64(2.0 * Float64(l_m * Float64(l_m * Float64(-1.0 / Om))))))));
	else
		tmp = Float64(l_m * Float64(sqrt(Float64(U * U_42_)) * 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_)
	t_1 = (l_m / Om) ^ 2.0;
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	tmp = 0.0;
	if (t_2 <= 4e-279)
		tmp = sqrt(((2.0 * U) * (n * (t + (((l_m ^ 2.0) / Om) * -2.0)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) + (2.0 * (l_m * (l_m * (-1.0 / Om))))))));
	else
		tmp = l_m * (sqrt((U * U_42_)) * (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[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = 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] + N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-279], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t + N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0), $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[(N[(n * N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l$95$m * N[(l$95$m * N[(-1.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.00000000000000022e-279

   1. Initial program 18.7%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in U around 0 35.4%

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

      1. associate-/l* 33.2%

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

      2. unpow2 33.2%

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

      3. unpow2 33.2%

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

      4. times-frac 40.5%

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

      5. unpow2 40.5%

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

      6. neg-mul-1 40.5%

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

      7. distribute-lft-neg-out 40.5%

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

      8. *-commutative 40.5%

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

   7. Simplified 40.5%

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

   8. Taylor expanded in n around 0 47.2%

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

      1. associate-*r* 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. *-commutative 47.2%

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

   10. Simplified 47.2%

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

   if 4.00000000000000022e-279 < (*.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 63.1%

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

   3. Simplified 68.1%

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

      1. sub-neg 68.1%

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

      2. distribute-lft-in 57.3%

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

   6. Applied egg-rr 57.3%

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

      1. distribute-lft-out 68.1%

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

      2. sub-neg 68.1%

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

      3. associate-*r* 69.1%

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

      4. *-commutative 69.1%

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

   8. Simplified 69.1%

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

      1. div-inv 69.1%

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

   10. Applied egg-rr 69.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.3%

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

      1. sub-neg 0.3%

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

      2. distribute-lft-in 0.3%

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

   6. Applied egg-rr 0.3%

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

      1. distribute-lft-out 0.3%

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

      2. sub-neg 0.3%

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

      3. associate-*r* 3.0%

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

      4. *-commutative 3.0%

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

   8. Simplified 3.0%

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

      1. div-inv 3.0%

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

   10. Applied egg-rr 3.0%

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

   11. Taylor expanded in U* around inf 30.7%

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

       1. associate-/l* 33.3%

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

       2. associate-*r/ 33.3%

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

       3. associate-*r* 33.3%

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

       4. *-commutative 33.3%

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

       5. associate-*r/ 33.3%

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

   13. Simplified 33.3%

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

   14. Taylor expanded in U around 0 33.3%

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

       1. associate-*r/ 33.3%

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

       2. *-commutative 33.3%

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

   16. Simplified 33.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) + 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{-1}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{U \cdot U*} \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot t\_1\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{{l\_m}^{2}}{Om} \cdot -2\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left(t\_1 \cdot \left(U* - U\right)\right) - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \left(\sqrt{U \cdot U*} \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l_m Om) 2.0))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 (/ (* l_m l_m) Om))) (* (* n t_1) (- U* U))))))
   (if (<= t_2 4e-279)
     (sqrt (* (* 2.0 U) (* n (+ t (* (/ (pow l_m 2.0) Om) -2.0)))))
     (if (<= t_2 INFINITY)
       (sqrt
        (*
         (* 2.0 (* n U))
         (+ t (- (* n (* t_1 (- U* U))) (* 2.0 (* l_m (/ l_m Om)))))))
       (* l_m (* (sqrt (* U U*)) (* 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 = pow((l_m / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-279) {
		tmp = sqrt(((2.0 * U) * (n * (t + ((pow(l_m, 2.0) / Om) * -2.0)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = l_m * (sqrt((U * U_42_)) * (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 = Math.pow((l_m / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-279) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t + ((Math.pow(l_m, 2.0) / Om) * -2.0)))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = l_m * (Math.sqrt((U * U_42_)) * (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 = math.pow((l_m / Om), 2.0)
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)))
	tmp = 0
	if t_2 <= 4e-279:
		tmp = math.sqrt(((2.0 * U) * (n * (t + ((math.pow(l_m, 2.0) / Om) * -2.0)))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = l_m * (math.sqrt((U * U_42_)) * (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(l_m / Om) ^ 2.0
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * t_1) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 4e-279)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t + Float64(Float64((l_m ^ 2.0) / Om) * -2.0)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * Float64(t_1 * Float64(U_42_ - U))) - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(l_m * Float64(sqrt(Float64(U * U_42_)) * 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_)
	t_1 = (l_m / Om) ^ 2.0;
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * t_1) * (U_42_ - U)));
	tmp = 0.0;
	if (t_2 <= 4e-279)
		tmp = sqrt(((2.0 * U) * (n * (t + (((l_m ^ 2.0) / Om) * -2.0)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (t_1 * (U_42_ - U))) - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = l_m * (sqrt((U * U_42_)) * (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[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = 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] + N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-279], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t + N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0), $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[(N[(n * N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.00000000000000022e-279

   1. Initial program 18.7%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in U around 0 35.4%

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

      1. associate-/l* 33.2%

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

      2. unpow2 33.2%

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

      3. unpow2 33.2%

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

      4. times-frac 40.5%

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

      5. unpow2 40.5%

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

      6. neg-mul-1 40.5%

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

      7. distribute-lft-neg-out 40.5%

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

      8. *-commutative 40.5%

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

   7. Simplified 40.5%

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

   8. Taylor expanded in n around 0 47.2%

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

      1. associate-*r* 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. *-commutative 47.2%

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

   10. Simplified 47.2%

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

   if 4.00000000000000022e-279 < (*.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 63.1%

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

   3. Simplified 68.1%

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

      1. sub-neg 68.1%

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

      2. distribute-lft-in 57.3%

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

   6. Applied egg-rr 57.3%

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

      1. distribute-lft-out 68.1%

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

      2. sub-neg 68.1%

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

      3. associate-*r* 69.1%

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

      4. *-commutative 69.1%

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

   8. Simplified 69.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.3%

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

      1. sub-neg 0.3%

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

      2. distribute-lft-in 0.3%

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

   6. Applied egg-rr 0.3%

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

      1. distribute-lft-out 0.3%

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

      2. sub-neg 0.3%

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

      3. associate-*r* 3.0%

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

      4. *-commutative 3.0%

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

   8. Simplified 3.0%

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

      1. div-inv 3.0%

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

   10. Applied egg-rr 3.0%

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

   11. Taylor expanded in U* around inf 30.7%

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

       1. associate-/l* 33.3%

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

       2. associate-*r/ 33.3%

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

       3. associate-*r* 33.3%

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

       4. *-commutative 33.3%

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

       5. associate-*r/ 33.3%

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

   13. Simplified 33.3%

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

   14. Taylor expanded in U around 0 33.3%

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

       1. associate-*r/ 33.3%

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

       2. *-commutative 33.3%

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

   16. Simplified 33.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{U \cdot U*} \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;t \leq 1.25 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + \left(n \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot U*\right) - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= t 1.25e+233)
     (sqrt
      (*
       t_1
       (+ t (- (* n (* (pow (/ l_m Om) 2.0) U*)) (* 2.0 (* l_m (/ l_m Om)))))))
     (* (sqrt t_1) (sqrt t)))))

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 * (n * U);
	double tmp;
	if (t <= 1.25e+233) {
		tmp = sqrt((t_1 * (t + ((n * (pow((l_m / Om), 2.0) * U_42_)) - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = sqrt(t_1) * sqrt(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) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (n * u)
    if (t <= 1.25d+233) then
        tmp = sqrt((t_1 * (t + ((n * (((l_m / om) ** 2.0d0) * u_42)) - (2.0d0 * (l_m * (l_m / om)))))))
    else
        tmp = sqrt(t_1) * sqrt(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 t_1 = 2.0 * (n * U);
	double tmp;
	if (t <= 1.25e+233) {
		tmp = Math.sqrt((t_1 * (t + ((n * (Math.pow((l_m / Om), 2.0) * U_42_)) - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = Math.sqrt(t_1) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = 2.0 * (n * U)
	tmp = 0
	if t <= 1.25e+233:
		tmp = math.sqrt((t_1 * (t + ((n * (math.pow((l_m / Om), 2.0) * U_42_)) - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = math.sqrt(t_1) * math.sqrt(t)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (t <= 1.25e+233)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(n * Float64((Float64(l_m / Om) ^ 2.0) * U_42_)) - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(sqrt(t_1) * sqrt(t));
	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 * (n * U);
	tmp = 0.0;
	if (t <= 1.25e+233)
		tmp = sqrt((t_1 * (t + ((n * (((l_m / Om) ^ 2.0) * U_42_)) - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = sqrt(t_1) * sqrt(t);
	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[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.25e+233], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(n * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.25000000000000002e233

   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 52.6%

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

      1. sub-neg 52.6%

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

      2. distribute-lft-in 44.8%

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

   6. Applied egg-rr 44.8%

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

      1. distribute-lft-out 52.6%

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

      2. sub-neg 52.6%

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

      3. associate-*r* 53.3%

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

      4. *-commutative 53.3%

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

   8. Simplified 53.3%

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

   9. Taylor expanded in U around 0 53.3%

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

       1. neg-mul-1 53.3%

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

   11. Simplified 53.3%

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

   if 1.25000000000000002e233 < t

   1. Initial program 9.8%

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

   3. Simplified 21.3%

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

   5. Taylor expanded in t around inf 21.6%

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

      1. pow1/2 22.0%

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

      2. associate-*r* 21.8%

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

      3. associate-*r* 21.8%

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

      4. unpow-prod-down 65.2%

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

      5. pow1/2 65.2%

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

      6. pow1/2 65.2%

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

   7. Applied egg-rr 65.2%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot U*\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{-131} \lor \neg \left(n \leq 2.6 \cdot 10^{-42}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left(n \cdot {l\_m}^{2}\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= n -2.6e-131) (not (<= n 2.6e-42)))
   (sqrt (* (* (* 2.0 n) U) (- t (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
   (sqrt (+ (* 2.0 (* U (* n t))) (* -4.0 (/ (* 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 <= -2.6e-131) || !(n <= 2.6e-42)) {
		tmp = sqrt((((2.0 * n) * U) * (t - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	} else {
		tmp = sqrt(((2.0 * (U * (n * t))) + (-4.0 * ((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 <= (-2.6d-131)) .or. (.not. (n <= 2.6d-42))) then
        tmp = sqrt((((2.0d0 * n) * u) * (t - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))))
    else
        tmp = sqrt(((2.0d0 * (u * (n * t))) + ((-4.0d0) * ((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 <= -2.6e-131) || !(n <= 2.6e-42)) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)))));
	} else {
		tmp = Math.sqrt(((2.0 * (U * (n * t))) + (-4.0 * ((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 <= -2.6e-131) or not (n <= 2.6e-42):
		tmp = math.sqrt((((2.0 * n) * U) * (t - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_)))))
	else:
		tmp = math.sqrt(((2.0 * (U * (n * t))) + (-4.0 * ((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 <= -2.6e-131) || !(n <= 2.6e-42))
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * Float64(n * t))) + Float64(-4.0 * Float64(Float64(U * 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 <= -2.6e-131) || ~((n <= 2.6e-42)))
		tmp = sqrt((((2.0 * n) * U) * (t - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)))));
	else
		tmp = sqrt(((2.0 * (U * (n * t))) + (-4.0 * ((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, -2.6e-131], N[Not[LessEqual[n, 2.6e-42]], $MachinePrecision]], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(U * N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.59999999999999996e-131 or 2.6e-42 < n

   1. Initial program 50.0%

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

      1. add-sqr-sqrt 24.1%

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

      2. times-frac 25.4%

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

   4. Applied egg-rr 25.4%

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

      1. unpow2 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in t around inf 53.2%

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

   if -2.59999999999999996e-131 < n < 2.6e-42

   1. Initial program 41.2%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in Om around inf 49.6%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{-131} \lor \neg \left(n \leq 2.6 \cdot 10^{-42}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{{l\_m}^{2}}{Om} \cdot -2\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -1.2e-142)
   (sqrt (* (* (* 2.0 n) U) (- t (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
   (sqrt (* (* 2.0 U) (* n (+ t (* (/ (pow l_m 2.0) Om) -2.0)))))))

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 <= -1.2e-142) {
		tmp = sqrt((((2.0 * n) * U) * (t - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	} else {
		tmp = sqrt(((2.0 * U) * (n * (t + ((pow(l_m, 2.0) / Om) * -2.0)))));
	}
	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 <= (-1.2d-142)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))))
    else
        tmp = sqrt(((2.0d0 * u) * (n * (t + (((l_m ** 2.0d0) / om) * (-2.0d0))))))
    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 <= -1.2e-142) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)))));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * (t + ((Math.pow(l_m, 2.0) / Om) * -2.0)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= -1.2e-142:
		tmp = math.sqrt((((2.0 * n) * U) * (t - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_)))))
	else:
		tmp = math.sqrt(((2.0 * U) * (n * (t + ((math.pow(l_m, 2.0) / Om) * -2.0)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -1.2e-142)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t + Float64(Float64((l_m ^ 2.0) / Om) * -2.0)))));
	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 <= -1.2e-142)
		tmp = sqrt((((2.0 * n) * U) * (t - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)))));
	else
		tmp = sqrt(((2.0 * U) * (n * (t + (((l_m ^ 2.0) / Om) * -2.0)))));
	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, -1.2e-142], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t + N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.19999999999999994e-142

   1. Initial program 53.3%

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

      1. add-sqr-sqrt 27.7%

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

      2. times-frac 28.9%

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

   4. Applied egg-rr 28.9%

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

      1. unpow2 28.9%

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

   6. Simplified 28.9%

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

   7. Taylor expanded in t around inf 60.0%

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

   if -1.19999999999999994e-142 < n

   1. Initial program 43.2%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in U around 0 40.8%

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

      1. associate-/l* 40.8%

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

      2. unpow2 40.8%

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

      3. unpow2 40.8%

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

      4. times-frac 47.4%

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

      5. unpow2 47.4%

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

      6. neg-mul-1 47.4%

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

      7. distribute-lft-neg-out 47.4%

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

      8. *-commutative 47.4%

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

   7. Simplified 47.4%

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

   8. Taylor expanded in n around 0 44.9%

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

      1. associate-*r* 44.9%

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

      2. cancel-sign-sub-inv 44.9%

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

      3. metadata-eval 44.9%

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

      4. *-commutative 44.9%

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

   10. Simplified 44.9%

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

Alternative 9: 44.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5.3 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 5.3e-223)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l_m 2.0) Om)))))))
   (* (sqrt (* 2.0 (* n U))) (sqrt 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 <= 5.3e-223) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l_m, 2.0) / Om)))))));
	} else {
		tmp = sqrt((2.0 * (n * U))) * sqrt(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 <= 5.3d-223) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l_m ** 2.0d0) / om)))))))
    else
        tmp = sqrt((2.0d0 * (n * u))) * sqrt(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 <= 5.3e-223) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l_m, 2.0) / Om)))))));
	} else {
		tmp = Math.sqrt((2.0 * (n * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= 5.3e-223:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l_m, 2.0) / Om)))))))
	else:
		tmp = math.sqrt((2.0 * (n * U))) * math.sqrt(t)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 5.3e-223)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om)))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(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 <= 5.3e-223)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l_m ^ 2.0) / Om)))))));
	else
		tmp = sqrt((2.0 * (n * U))) * sqrt(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, 5.3e-223], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.30000000000000027e-223

   1. Initial program 50.0%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in n around 0 46.9%

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

   if 5.30000000000000027e-223 < t

   1. Initial program 40.6%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in t around inf 31.1%

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

      1. pow1/2 33.2%

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

      2. associate-*r* 36.0%

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

      3. associate-*r* 36.0%

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

      4. unpow-prod-down 45.0%

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

      5. pow1/2 45.0%

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

      6. pow1/2 45.0%

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

   7. Applied egg-rr 45.0%

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

Alternative 10: 40.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.9 \cdot 10^{+109}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \left(\sqrt{U \cdot U*} \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.9e+109)
   (pow (* (* n t) (* 2.0 U)) 0.5)
   (* l_m (* (sqrt (* U U*)) (* 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 <= 2.9e+109) {
		tmp = pow(((n * t) * (2.0 * U)), 0.5);
	} else {
		tmp = l_m * (sqrt((U * U_42_)) * (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 <= 2.9d+109) then
        tmp = ((n * t) * (2.0d0 * u)) ** 0.5d0
    else
        tmp = l_m * (sqrt((u * u_42)) * (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 <= 2.9e+109) {
		tmp = Math.pow(((n * t) * (2.0 * U)), 0.5);
	} else {
		tmp = l_m * (Math.sqrt((U * U_42_)) * (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 <= 2.9e+109:
		tmp = math.pow(((n * t) * (2.0 * U)), 0.5)
	else:
		tmp = l_m * (math.sqrt((U * U_42_)) * (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 <= 2.9e+109)
		tmp = Float64(Float64(n * t) * Float64(2.0 * U)) ^ 0.5;
	else
		tmp = Float64(l_m * Float64(sqrt(Float64(U * U_42_)) * 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 <= 2.9e+109)
		tmp = ((n * t) * (2.0 * U)) ^ 0.5;
	else
		tmp = l_m * (sqrt((U * U_42_)) * (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, 2.9e+109], N[Power[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(l$95$m * N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.9e109

   1. Initial program 50.7%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in t around inf 39.5%

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

      1. add-exp-log 17.3%

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

   7. Applied egg-rr 17.3%

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

      1. pow1/2 18.3%

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

      2. associate-*r* 18.3%

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

      3. rem-exp-log 41.4%

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

   9. Applied egg-rr 41.4%

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

   if 2.9e109 < l

   1. Initial program 23.6%

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

   3. Simplified 33.2%

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

      1. sub-neg 33.2%

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

      2. distribute-lft-in 20.7%

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

   6. Applied egg-rr 20.7%

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

      1. distribute-lft-out 33.2%

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

      2. sub-neg 33.2%

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

      3. associate-*r* 35.3%

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

      4. *-commutative 35.3%

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

   8. Simplified 35.3%

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

      1. div-inv 35.3%

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

   10. Applied egg-rr 35.3%

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

   11. Taylor expanded in U* around inf 18.6%

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

       1. associate-/l* 20.9%

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

       2. associate-*r/ 20.9%

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

       3. associate-*r* 20.9%

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

       4. *-commutative 20.9%

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

       5. associate-*r/ 20.9%

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

   13. Simplified 20.9%

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

   14. Taylor expanded in U around 0 20.9%

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

       1. associate-*r/ 20.9%

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

       2. *-commutative 20.9%

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

   16. Simplified 20.9%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{+109}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{U \cdot U*} \cdot \left(n \cdot \frac{\sqrt{2}}{Om}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-250}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 4.5e-250)
   (pow (* (* n t) (* 2.0 U)) 0.5)
   (* (sqrt (* 2.0 (* n U))) (sqrt 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 <= 4.5e-250) {
		tmp = pow(((n * t) * (2.0 * U)), 0.5);
	} else {
		tmp = sqrt((2.0 * (n * U))) * sqrt(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 <= 4.5d-250) then
        tmp = ((n * t) * (2.0d0 * u)) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (n * u))) * sqrt(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 <= 4.5e-250) {
		tmp = Math.pow(((n * t) * (2.0 * U)), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (n * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= 4.5e-250:
		tmp = math.pow(((n * t) * (2.0 * U)), 0.5)
	else:
		tmp = math.sqrt((2.0 * (n * U))) * math.sqrt(t)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 4.5e-250)
		tmp = Float64(Float64(n * t) * Float64(2.0 * U)) ^ 0.5;
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(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 <= 4.5e-250)
		tmp = ((n * t) * (2.0 * U)) ^ 0.5;
	else
		tmp = sqrt((2.0 * (n * U))) * sqrt(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, 4.5e-250], N[Power[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.49999999999999993e-250

   1. Initial program 50.6%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in t around inf 35.9%

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

      1. add-exp-log 15.7%

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

   7. Applied egg-rr 15.7%

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

      1. pow1/2 16.4%

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

      2. associate-*r* 16.4%

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

      3. rem-exp-log 37.9%

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

   9. Applied egg-rr 37.9%

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

   if 4.49999999999999993e-250 < t

   1. Initial program 40.1%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in t around inf 30.0%

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

      1. pow1/2 32.0%

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

      2. associate-*r* 34.7%

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

      3. associate-*r* 34.7%

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

      4. unpow-prod-down 44.3%

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

      5. pow1/2 44.3%

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

      6. pow1/2 44.3%

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

   7. Applied egg-rr 44.3%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-250}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq 10^{-247}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U 1e-247)
   (pow (* (* n t) (* 2.0 U)) 0.5)
   (* (sqrt (* 2.0 U)) (sqrt (* 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 (U <= 1e-247) {
		tmp = pow(((n * t) * (2.0 * U)), 0.5);
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((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 (u <= 1d-247) then
        tmp = ((n * t) * (2.0d0 * u)) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((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 (U <= 1e-247) {
		tmp = Math.pow(((n * t) * (2.0 * U)), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U <= 1e-247:
		tmp = math.pow(((n * t) * (2.0 * U)), 0.5)
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= 1e-247)
		tmp = Float64(Float64(n * t) * Float64(2.0 * U)) ^ 0.5;
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(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 (U <= 1e-247)
		tmp = ((n * t) * (2.0 * U)) ^ 0.5;
	else
		tmp = sqrt((2.0 * U)) * sqrt((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[U, 1e-247], N[Power[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1e-247

   1. Initial program 46.9%

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

   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 t around inf 36.9%

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

      1. add-exp-log 5.7%

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

   7. Applied egg-rr 5.7%

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

      1. pow1/2 7.1%

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

      2. associate-*r* 7.1%

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

      3. rem-exp-log 38.3%

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

   9. Applied egg-rr 38.3%

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

   if 1e-247 < U

   1. Initial program 45.6%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in t around inf 31.2%

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

      1. pow1/2 35.2%

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

      2. associate-*r* 35.2%

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

      3. unpow-prod-down 44.8%

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

      4. pow1/2 41.8%

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

   7. Applied egg-rr 41.8%

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

      1. unpow1/2 41.8%

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

   9. Simplified 41.8%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 10^{-247}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U* \leq 1.75 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U* 1.75e+79)
   (sqrt (* 2.0 (* U (* n t))))
   (pow (* 2.0 (* t (* n 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 (U_42_ <= 1.75e+79) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = pow((2.0 * (t * (n * 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 (u_42 <= 1.75d+79) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = (2.0d0 * (t * (n * 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 (U_42_ <= 1.75e+79) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U_42_ <= 1.75e+79:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 1.75e+79)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = Float64(2.0 * Float64(t * Float64(n * 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 (U_42_ <= 1.75e+79)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = (2.0 * (t * (n * 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[U$42$, 1.75e+79], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 1.7499999999999999e79

   1. Initial program 46.5%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in t around inf 37.0%

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

   if 1.7499999999999999e79 < U*

   1. Initial program 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in t around inf 25.4%

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

      1. pow1/2 31.1%

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

      2. associate-*r* 36.6%

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

      3. *-commutative 36.6%

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

   7. Applied egg-rr 36.6%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 1.75 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U* \leq 6200000000:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U* 6200000000.0)
   (sqrt (* 2.0 (* U (* n t))))
   (pow (* 2.0 (* n (* U t))) 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 (U_42_ <= 6200000000.0) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = pow((2.0 * (n * (U * t))), 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 (u_42 <= 6200000000.0d0) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = (2.0d0 * (n * (u * t))) ** 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 (U_42_ <= 6200000000.0) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if U_42_ <= 6200000000.0:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 6200000000.0)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 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 (U_42_ <= 6200000000.0)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = (2.0 * (n * (U * t))) ^ 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[U$42$, 6200000000.0], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 6.2e9

   1. Initial program 46.6%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in t around inf 36.8%

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

   if 6.2e9 < U*

   1. Initial program 45.7%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in t around inf 31.7%

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

      1. pow1/2 36.2%

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

      2. associate-*l* 36.2%

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

   7. Applied egg-rr 36.2%

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

Alternative 15: 38.6% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (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_) {
	return pow(((n * t) * (2.0 * U)), 0.5);
}

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 = ((n * t) * (2.0d0 * u)) ** 0.5d0
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.pow(((n * t) * (2.0 * U)), 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.4%

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

3. Simplified 51.0%

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

5. Taylor expanded in t around inf 34.6%

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

   1. add-exp-log 15.1%

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

7. Applied egg-rr 15.1%

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

   1. pow1/2 15.9%

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

   2. associate-*r* 15.9%

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

   3. rem-exp-log 37.1%

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

9. Applied egg-rr 37.1%

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

10. Final simplification 37.1%

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

Alternative 16: 36.6% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* 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_) {
	return sqrt((2.0 * (U * (n * t))));
}

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((2.0d0 * (u * (n * t))))
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((2.0 * (U * (n * t))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.4%

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

3. Simplified 51.0%

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

5. Taylor expanded in t around inf 34.6%

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

Reproduce

herbie shell --seed 2024143 
(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*))))))